on the flow-box coordinates. Since $\tau$ is the time variable and
$(\mathcal{E},\mathcal{P})$ give the initial conditions on the cross-section
$S^{-}_{\delta,j}$, we can write (71) as

$q=q_{impact}+\delta^{1/\alpha}\tilde{q}(\frac{\tau-
t_{in}}{\delta^{1/\alpha}},\mathcal{E},\mathcal{P};\delta).$ (90)

Since the value of $t_{in}$ is independent of $(\mathcal{E},\mathcal{P})\in
S^{-}_{\delta,j}$, the required regularity of $q$ follows immediately from
Lemma 4.1. ∎

####  4.1.3 The periodic orbit $L^{*}_{\delta}$

Choose, near each regular impact point $M^{j}$, $j=1,\ldots,k^{*}$, of the
billiard orbit $L^{*}$, the cross-sections $S_{\delta,j}^{-}$ and
$S_{\delta,j}^{+}$ as in Lemma 4.2. The cross-sections depend continuously on
$\delta$ and stay bounded away from the billiard boundary. Therefore, in a
neighborhood of the piece of $L^{*}$ between the consecutive cross-sections
$S_{\delta,j}^{+}$ and $S^{-}_{\delta,j+1}$, the potential tends to zero as
$\delta\to 0$ and the motion tends to the constant speed motion, as in the
billiard. Hence, for the orbits of system (21) the flight time from
$S_{\delta,j}^{+}$ to $S^{-}_{\delta,j+1}$ tends to the billiard flight time
and the Poincaré map $S_{\delta,j}^{+}\rightarrow S^{-}_{\delta,j+1}$ tends to
the corresponding Poincaré map of the billiard flow, along with the
derivatives with respect to the initial conditions.

By Lemma 4.2, the same is true for the Poincaré map and the flight time
between $S_{\delta,j}^{-}$ and $S_{\delta,j}^{+}$ (the flight time does not
depend on initial conditions and equals to $2t_{in}$, so it tends to the
billiard flight time by Lemma 4.1). Thus, we obtain that the return map to
$S_{\delta,1}^{-}$ for the flow (21) is $C^{r}$ close, for all $r$, to the
return map of the billiard flow (the return map is the composition of the
Poincaré maps $S_{\delta,j}^{-}\to S_{\delta,j}^{+}$ and
$S_{\delta,j}^{+}\rightarrow S^{-}_{\delta,j+1\mod k^{*}}$ for
$j=1,\ldots,k^{*}$). Also, the return time to $S_{\delta,1}^{-}$ is $C^{r}$
close to the return time for the billiard flow.

By Assumption BD2, the intersection point of $L^{*}$ with $S_{0,1}^{-}$ is a
non-degenerate non-resonant (up to order 4) elliptic fixed point of the
billiard return map at the energy level $H=\frac{1}{2}$ (this corresponds to
the motion with the speed $1$). Such fixed points persist at small
perturbations, so the return map to $S_{\delta,1}^{-}$ also has a KAM-non-
degenerate elliptic fixed point at the energy level to $H=\frac{1}{2}$ for all
small $\delta$.

This gives us an elliptic periodic orbit $L^{*}_{\delta}$ of system (21) such
that $L^{*}_{\delta}\cap S_{\delta,1}^{-}$ tends to $L^{*}\cap S_{0,1}^{-}$ as
$\delta\to 0$. By the continuous dependence of the return time on $\delta$,
the period of $L^{*}_{\delta}$ tends to the period of the billiard orbit
$L^{*}$. Recall that we use the notation
$L^{*}=(q^{*}(\omega_{0}t),p^{*}(\omega_{0}t)$, where
$\omega_{0}=\frac{2\pi}{|L^{*}|}$ and $(q^{*},p^{*})$ are $2\pi$-periodic
functions, $q^{*}$ is continuous and piece-wise linear, and $p^{*}$ is
discontinuous and piece-wise constant. We denote
$L^{*}_{\delta}=(q^{*}_{\delta}(\omega_{0}(\delta)t),p_{\delta}^{*}(\omega_{0}(\delta)t)$
for some smooth $2\pi$-periodic functions $(q^{*}_{\delta},p^{*}_{\delta})$.
Here, $\omega_{0}(\delta)\to\omega_{0}$ as $\delta\to 0$, and
$(q^{*}_{\delta},p_{\delta}^{*})\to(q^{*},p^{*})$, with derivatives, if
$q^{*}$ stays away from the billiard boundary. Since $q^{*}_{\delta}$ is
uniformly Lipshitz and the time spent in the boundary layer is small, it
follows that $q^{*}_{\delta}\to q^{*}$ in $C^{0}$ for all $t$.

The same holds true in every energy level close to $H=\frac{1}{2}$. The
billiard has a periodic orbit $L^{*}(E)$ in the energy level $H=E$ which
follows the same path as $L^{*}$ in the $q$\- space, with the speed
$\|p\|=\sqrt{2E}$ and period $T(E)=\frac{|L^{*}|}{\sqrt{2E}}$. By the same
arguments as above, the system (21) for all sufficiently small $\delta$ has an
elliptic periodic orbit $L_{\delta}^{*}(E)$ in the energy level $H=E$, and the
family of the orbits $L_{\delta}^{*}(E)$ approaches $L^{*}(E)$ as $\delta\to
0$. In particular, the period $T_{\delta}(E)$ tends to $T(E)$ along with
derivatives with respect to $E$ (because the return time to $S_{\delta,1}^{-}$
tends to the billiard return time with derivatives with respect to the initial
conditions).

For every $\delta>0$, for each impact point $M^{j}$, $j=1,\dots,k^{*}$, let us
choose the region $\mathcal{U}_{\delta,j}$ as in Lemma 4.3 (i.e., this region
consists of the orbits of (21) that connect the cross-sections
$S_{\delta,j}^{-}$ and $S_{\delta,j}^{+}$). Since $L^{*}$ is regular periodic
orbit, its impacts are distinct, so the regions $\mathcal{U}_{\delta,j}$ do
not overlap for different $j$. Let $\mathcal{U}^{0}$ be a sufficiently small,
yet independent of $\delta$, open neighborhood of the part of $L^{*}_{\delta}$
which is not covered by the union of $\mathcal{U}_{\delta,j}$.

We use $(q,p)$ as coordinates in $\mathcal{U}^{0}$; note that for the points
$(q,p)\in\mathcal{U}^{0}$, the $q$-component is bounded away from the billiard
boundary. In $\mathcal{U}_{\delta,j}$, we use the flow-box coordinates
$(\tau,\mathcal{E},\mathcal{P})_{j}$ given by Lemma 4.3. We restrict the
freedom in the choice of the flow-box coordinates by the requirement that
$(\mathcal{E},\mathcal{P})_{j}=0$ on the periodic orbit $L_{\delta}^{*}$.

Thus, we have covered a neighborhood of $L_{\delta}^{*}$ by a system of
coordinate charts for all $\delta>0$. The overlap region between
$\mathcal{U}^{0}$ and $\mathcal{U}_{\delta,j}$ near the cross-sections
$S_{\delta,j}^{-}$ and $S_{\delta,j}^{+}$ stays bounded away from the billiard
boundary. Therefore, by Lemma 4.3 the transition map between the
$(q,p)$-coordinates in $\mathcal{U}^{0}$ and the flow-box coordinates in
$\mathcal{U}_{\delta,j}$ in the overlap region is symplectic, depends
continuously on $\delta$, and has a well-defined limit, in $C^{\infty}$, as
$\delta\to 0$.

In other words, we have introduced the structure of a smooth symplectic
manifold in the neighborhood of $L_{\delta}^{*}$ for $\delta>0$, and this
structure has a regular limit at $\delta=0$. The equations of motion also have
a regular limit in these coordinates: in $\mathcal{U}^{0}$ the equations
converge to

$\dot{q}=p,\qquad\dot{p}=0$

at $\delta=0$, and in $\mathcal{U}_{\delta,j}$ the equations are the same for
all $\delta$:

$\dot{\tau}=1,\qquad\dot{\mathcal{E}}=0,\qquad\dot{\mathcal{P}}=0.$

The elliptic periodic orbit $L^{*}_{\delta}$ tends, as $\delta\to 0$, to the
billiard periodic orbit $L^{*}=L^{*}_{0}$, which is a smooth curve in these
coordinates (for every $\delta\geqslant 0$, the curve $L^{*}_{\delta}$ has the
same equation $(\mathcal{E},\mathcal{P})=0$ in the flow-box coordinates). As
in Section 2.1, we can introduce action-angle coordinates near
$L^{*}_{\delta}$ for all $\delta\geqslant 0$, with the resulting Hamiltonian
as in (6), yet here the coefficients depend (continuously) on $\delta$:

$H_{0}(I_{0},\theta,z;\delta)=\frac{1}{2}+\omega(\delta)I+\frac{1}{2}I^{\top}A(\delta)I+g(I_{0},\theta,z;\delta),$
(91)

where
$g=g_{0}(I_{0};\delta)+g_{1}(I_{0};\delta)\hat{I}+O(\|z\|^{4}|I_{0}|+\|z\|^{5})$
with $g_{0}=O(I_{0}^{3})$, $g_{1}=O(I_{0}^{2})$ (see notations after (6)). We
stress that the symplectic transformation between the action-angle coordinates
$(I_{0},\theta,z)$ and the energy-time coordinates
$(\tau,\mathcal{E},\mathcal{P})$ defined in the near-impact regions,
$\mathcal{U}_{\delta,j}$, is smooth for all $\delta\geqslant 0$.

Similar to Section 2.1, we denote the relation between the action-angle
coordinates and the $(q,p)$-coordinates as
$(q,p)=(\hat{q}(I_{0},\theta,z),\hat{p}(I_{0},\theta,z))$. Away from the
billiard boundary, i.e., in $\mathcal{U}^{0}$, this coordinate transformation
is well-defined in the limit $\delta=0$. However, near the impacts (in the
regions $\mathcal{U}_{\delta,j}$) the relation between $(q,p)$ and the action-
angle coordinates acquires singularities at $\delta=0$.

The periodic orbit $L^{*}_{\delta}$ corresponds to $(I_{0},z)=0$. We have
$\dot{\theta}=\omega_{0}(\delta)$ on $L^{*}_{\delta}$, i.e.,
$\theta=\omega_{0}(\delta)t$. Thus, in our notations,
$L^{*}_{\delta}=(q^{*}_{\delta}(\theta),p_{\delta}^{*}(\theta))=(\hat{q}(0,\theta,0),\hat{p}(0,\theta,0))$.
On $L^{*}_{\delta}$, equation (90) near the $j$-th impact point, where for
$\delta=0$ the impact occurs at $\theta_{j}$, becomes

$q^{*}_{\delta}(\theta)=M^{j}+\delta^{1/\alpha}\tilde{q}_{j,\delta}\left(\frac{\theta-\theta_{j}}{\omega_{0}(\delta)\delta^{1/\alpha}}\right),$
(92)

where $\tilde{q}_{j,\delta}$ has all derivatives bounded, see Lemma 4.1. So,
near impacts, the $k$-th derivative of $q^{*}_{\delta}(\theta)$ is of order
$\delta^{-(k-1)/\alpha}$.

Like in Section 3, we will also perform the scaling
$z\rightarrow\delta^{1/4}z$, $I_{0}\rightarrow\delta^{1/2}I_{0}$. In the
scaled variables, the motion in a small neighborhood of $L^{*}_{\delta}$ is
described by the scaled single-particle Hamiltonian

$H_{0,scal}=\delta^{-1/2}\left(H_{0}(\delta^{1/2}I_{0},\theta,\delta^{1/4}z;\delta)-\frac{1}{2}\right)=\omega(\delta)I+\frac{1}{2}\delta^{1/2}I^{\top}A(\delta)I+O(\delta^{3/4}).$
(93)

Note that away from the billiard boundary, we have

$\hat{q}(\delta^{1/2}I_{0},\theta,\delta^{1/4}z)-q^{*}_{\delta}(\theta)=O(\delta^{1/4}),$
(94)

with all derivatives with respect to $\theta$ and the scaled variables $z$ and
$I_{0}$. Note also that the scaling of $I_{0}$ and $z$ induces the scaling
$(\mathcal{E},\mathcal{P})\rightarrow(\delta^{1/2}\mathcal{E},\delta^{1/4}\mathcal{P})$
for the flow-box coordinates near impacts; the scaled energy $\mathcal{E}$
equals to $H_{0,scal}$ in (93). Recall that the symplectic transformation
between the unscaled $(I_{0},\theta,z)$ and $(\tau,\mathcal{E},\mathcal{P})$
is smooth for all $\delta\geqslant 0$. By (93), scaling $I_{0}$ and
$\mathcal{E}$ by $\delta^{1/2}$ remains a smooth transformation in the limit
$\delta=0$. Similarly, since $\mathcal{P}$ depends smoothly on
$(\delta^{1/2}I_{0},\delta^{1/4}z)$ and vanishes at the origin, it follows
that its scaling by $\delta^{1/4}$ also remains smooth in this limit.
Therefore, we conclude that the symplectic transformation between the scaled
$(I_{0},\theta,z)$ and the scaled $(\tau,\mathcal{E},\mathcal{P})$ is smooth
for all $\delta\geqslant 0$.

####  4.1.4 Conditions SP1-SP3 for the orbit $L^{*}_{\delta}$.

######  Lemma 4.4.

Under Assumptions BD1 and BD2 on the billiard orbit, the periodic orbit
$L^{*}_{\delta}$ satisfies Assumptions SP1-SP3 for all sufficiently small
$\delta\geqslant 0$.

###### Proof.

For the family $L^{*}(E)$ of billiard periodic orbits that follow, with the
speed $\|p\|=\sqrt{2E}$, the same path as $L^{*}_{0}$ in the $q$\- space, the
period $T(E)$ equals to $\frac{|L^{*}|}{\sqrt{2E}}$. It decreases with the
energy $E$. Since for small $\delta$ the period of $L^{*}_{\delta}(E)$ is
close to the period of $L^{*}(E)$ with the derivatives with respect to $E$
(see Section 4.1.3), the period of $L^{*}_{\delta}(E)$ also decreases with
$E$, i.e., Assumption SP1 is verified.

By the closeness of the return maps near $L^{*}_{\delta}$ and $L^{*}_{0}$, the
multipliers of $L^{*}_{\delta}$ are close to the multipliers of $L^{*}_{0}$.
Hence, by assumption BD2, there are no low-order resonances for all small
$\delta$, i.e., assumption SP2 is satisfied.

The twist assumption SP3 for the orbit $L^{*}_{\delta}$ is an open condition
on coefficients of system (91). Because of the continuity in $\delta$, it is
enough to check this condition at $\delta=0$. We evaluate the necessary
coefficients of (91) at $\delta=0$ by analyzing the return map near
$L^{*}_{0}=(I_{0}=0,z=0)$.

The system of differential equations defined by the Hamiltonian (91) is, in
restriction to the energy level $H_{0}=E$ close to $E=\frac{1}{2}$, given by

$\begin{array}[]{l}\omega_{0}I_{0}=E-\frac{1}{2}-\hat{\omega}\hat{I}+O(\hat{I}^{2}),\\\
\dot{\theta}=\omega_{0}+(aI_{0}+b\hat{I})+O(I^{2}),\\\
\dot{x}_{j}=(\omega_{j}+b_{j}I_{0}+(\hat{A}\hat{I})_{j}+O(I_{0}^{2}))y_{j}+O(\|z\|^{3}|I_{0}|+\|z\|^{4}),\\\
\dot{y}_{j}=-(\omega_{j}+b_{j}I_{0}+(\hat{A}\hat{I})_{j}+O(I_{0}^{2}))x_{j}+O(\|z\|^{3}|I_{0}|+\|z\|^{4})\qquad(j=1,\ldots,d-1),\end{array}$
(95)

where all the coefficients are taken at $\delta=0$. Recall the notation:
$I=(I_{0},\hat{I})$, $z_{j}=(x_{j},y_{j})$, $I_{j}=\frac{1}{2}z_{j}^{2}$,
$\omega=(\omega_{0},\hat{\omega})$, $A=\begin{pmatrix}a&b\\\
b^{\top}&\hat{A}\\\ \end{pmatrix}$, see Section 2.1.

For each value of $E$, the line $(x,y)=0$ (equivalently $\hat{I}=0$)
corresponds to a periodic orbit. Its period equals to the travel time from
$\theta=0$ to $\theta=2\pi$, i.e.,

$T(E)=\frac{2\pi}{\omega_{0}+a\frac{E-\frac{1}{2}}{\omega_{0}}+O((E-\frac{1}{2})^{2})}=\frac{2\pi}{\omega_{0}}(1-a\frac{E-\frac{1}{2}}{\omega_{0}^{2}}+O((E-\frac{1}{2})^{2}))$

The linear part of the system for $(x_{j},y_{j})$ is rotation with the
frequency
$\omega_{j}(E)=\omega_{j}+b_{j}\frac{E-\frac{1}{2}}{\omega_{0}}+O((E-\frac{1}{2})^{2})$.
It follows that the linearization of the return map
$(x,y)\mapsto(\bar{x},\bar{y})$ at the fixed point $(x,y)=0$ is the rotation
of $(x_{j},y_{j})$, $j=1,\ldots,d-1$, by the angle

$\Delta_{j}=T(E)\omega_{j}(E)=\frac{2\pi}{\omega_{0}}(\omega_{j}+\frac{E-\frac{1}{2}}{\omega_{0}}(b_{j}-\frac{\omega_{j}a}{\omega_{0}})+O((E-\frac{1}{2})^{2})).$
(96)

System (91) at $\delta=0$ describes the billiard flow, so its return map is
the return map of a billiard flow. The billiard flow is invariant with respect
to the energy and time scaling, in particular the return map at the energy
level $E=\frac{1}{2}$ is conjugate to the return map at any $E$ by the scaling
$(q,p)\rightarrow(q,p/\sqrt{2E})$. The conjugacy implies that the rotation
angles $\Delta_{j}$ in (96) are independent of $E$. Therefore,

$b_{j}=\frac{\omega_{j}a}{\omega_{0}},\qquad j=1,\ldots,d-1.$ (97)

Now, at $E=\frac{1}{2}$ we rewrite system (95) as

$\begin{array}[]{l}\dot{\theta}=\omega_{0}+O(\|z\|^{4}),\\\
\dot{x}_{j}=(\omega_{j}-\frac{b_{j}}{\omega_{0}}\hat{\omega}\hat{I}+(\hat{A}\hat{I})_{j})y_{j}+O(\|z\|^{4}),\\\
\dot{y}_{j}=-(\omega_{j}-\frac{b_{j}}{\omega_{0}}\hat{\omega}\hat{I}+(\hat{A}\hat{I})_{j})x_{j}+O(\|z\|^{4})\qquad(j=1,\ldots,d-1),\end{array}$

The return map $(\theta=0)\to(\theta=2\pi)$ for this system coincides, up to
$O(\|z\|^{4})$-terms with the (nonlinear) rotation of $(x_{j},y_{j})$,
$j=1\ldots,d-1$, to the angles

$\frac{2\pi}{\omega_{0}}(\omega_{j}-\frac{b_{j}}{\omega_{0}}\hat{\omega}\hat{I}+(\hat{A}\hat{I})_{j}).$

Thus, the map coincides, up to $O(\hat{I}^{2})$-terms, with the normal form
(22) where the action-angle coordinates $(I_{j},\Phi_{j})$ are introduced such
that $(x_{j},y_{j})=\sqrt{2I_{j}}(\cos(\Phi_{j}),\sin(\Phi_{j}))$, and the
matrix $\Omega$ is given by

$\Omega_{jk}=\frac{2\pi}{\omega_{0}}(\hat{A}_{jk}-\frac{\omega_{k}b_{j}}{\omega_{0}})=\frac{2\pi}{\omega_{0}}(\hat{A}_{jk}-\frac{a\omega_{j}\omega_{k}}{\omega^{2}_{0}}).$

We have $\det\Omega\neq 0$ by Assumption BD2. This implies that conditions (8)
and (9) of Assumption SP3 are satisfied. Indeed, using (97), we find

$\det
A=a\det(\hat{A}-\frac{1}{a}b^{\top}b)=a\det(\frac{\omega_{0}}{2\pi}\Omega)\neq
0,$

and

$\det A_{\omega}=\det\left(\begin{array}[]{ccc}0&\omega_{0}&\hat{\omega}\\\
\omega_{0}&a&\frac{a}{\omega_{0}}\hat{\omega}\\\
\hat{\omega}^{\top}&\frac{a}{\omega_{0}}\hat{\omega}^{\top}&\hat{A}\end{array}\right)=\det\left(\begin{array}[]{ccc}-\frac{\omega_{0}^{2}}{a}&0&0\\\
\omega_{0}&a&\frac{a}{\omega_{0}}\hat{\omega}\\\
\hat{\omega}^{\top}&\frac{a}{\omega_{0}}\hat{\omega}^{\top}&\hat{A}\end{array}\right)=-\frac{\omega_{0}^{2}}{a}\det
A\neq 0.$

∎

###  4.2 Multi-particle system

Now we analyze the multi-particle system (20) near the minimal line (26) of
the averaged interaction potential $U$ of (25). We have established that for
all small $\delta\geqslant 0$ the single-particle Hamiltonian can be brought
to the form (91). In these variables the multi-particle system is of the form

$H=\sum_{n=1}^{N}\left[\omega(\delta)I^{(n)}+\frac{1}{2}{I^{(n)}}^{\top}A(\delta)I^{(n)}+g(I_{0}^{(n)},\theta^{(n)},z^{(n)};\delta)\right]+\delta\mathop{\sum_{n,m=1,\dots,N}}_{n\neq
m}W(q^{(n)}-q^{(m)}),$ (98)

where, as $\delta\rightarrow 0$, the dependence
$q^{(n)}=\hat{q}(I_{0}^{(n)},\theta^{(n)},z^{(n)})$ becomes singular near
impacts, as described in Section 4.1. This singularity is quite mild in the
flow-box coordinates $(\tau^{(n)},\mathcal{E}^{(n)},\mathcal{P}^{(n)})$: we
have, uniformly for all $\delta\geqslant 0$, continuous dependence of
$q^{(n)}$ on $(\tau^{(n)},\mathcal{E}^{(n)},\mathcal{P}^{(n)})$ and, for every
fixed value of $\tau^{(n)}$, smooth dependence on
$(\mathcal{E}^{(n)},\mathcal{P}^{(n)})$, see Lemma 4.3.

Let us make the scaling $z^{(n)}\rightarrow\delta^{1/4}z^{(n)}$,
$I_{0}^{(n)}\rightarrow\delta^{1/2}I_{0}^{(n)}$. By (93), we obtain the scaled
version of (98):

$\begin{array}[]{ll}H_{scal}&=\delta^{-1/2}\;H\\!\left((\delta^{1/2}I_{0}^{(n)},\theta^{(n)},\delta^{1/4}z^{(n)})_{{}_{n=1,\dots,N}}\right)\\\
&\displaystyle=\sum_{n}\left[\omega(\delta)I^{(n)}+\delta^{1/2}\frac{1}{2}{I^{(n)}}^{\top}A(\delta)I^{(n)}+O(\delta^{3/4})\right]\\\
&\displaystyle\qquad\qquad+\delta^{1/2}\sum_{n\neq
m}W(\hat{q}(\delta^{1/2}I_{0}^{(n)},\theta^{(n)},\delta^{1/4}z^{(n)})-\hat{q}(\delta^{1/2}I_{0}^{(m)},\theta^{(m)},\delta^{1/4}z^{(m)})).\end{array}$
(99)

As we consider a neighborhood of the minimal line (26) which satisfies the
collision-free Assumption IP2, we can assume that $W$ is bounded with its
derivatives (as in Theorem 2). Therefore, away from the impacts,

$W(\hat{q}(I_{0}^{(n)},\theta^{(n)},z^{(n)})-\hat{q}(I_{0}^{(m)},\theta^{(m)},z^{(m)}))=W(q^{*}_{\delta}(\theta^{(n)};\delta)-q^{*}_{\delta}(\theta^{(m)};\delta))+O(\delta^{1/4}),$

see (94).

Thus, when all the particles stay away from the billiard boundary, the
Hamiltonian (99) is $O(\delta^{3/4})$-close, with all derivatives, to the
truncated Hamiltonian

$H_{trun}=\sum_{n}\omega(\delta)I^{(n)}+\delta^{1/2}\sum_{n}\frac{1}{2}{I^{(n)}}^{\top}A(\delta)I^{(n)}+\delta^{1/2}\sum_{n\neq
m}W(q_{\delta}^{*}(\theta^{(n)};\delta)-q^{*}_{\delta}(\theta^{(m)};\delta)).$
(100)

Note that we cannot claim the same near impacts, so there we need a more
accurate comparison of systems (99) and (100).

We consider the Hamiltonians (99) and (100) close to the line

$\mathcal{L}_{min}^{*}=\\{(I_{0}^{(n)}=0,\theta^{(n)}=\theta^{(n)}_{min}+c,z^{(n)}=0)_{n=1,..N},c\in
S^{1}\\},$

the phase space image of the minima line (26), i.e. the solution curve (3) in
action angle coordinates). Take the codimension-one hypersurface

$\Sigma_{0}:\quad\sum_{n=1}^{N}\theta^{(n)}=0\mod 2\pi.$ (101)

By (28), this surface intersects the line $\mathcal{L}_{min}^{*}$ at
$\theta^{(n)}=\theta^{(n)}_{min}$. By (29), in the neighborhood of this
intersection point, all $q^{(n)}$ are bounded away from the billiard boundary.
As
$\dot{\varphi}=\frac{1}{N}\sum_{n=1}^{N}\dot{\theta}^{(n)}=\omega_{0}(\delta)+O(\delta^{1/2})\neq
0\ $, the hyper-surface (101) is transverse to the flows of both Hamiltonians
(99) and (100). Now we show that despite the fact that we can guarantee the
closeness of these Hamiltonians to each other only away from the billiard
boundary, their Poincaré return maps to the cross-section (101) are
sufficiently close to each other.

######  Lemma 4.5.

When Assumption IP3 (non-simultaneous impacts) is satisfied, the Poincaré
return map of system (99) to the cross-section $\sum_{n=1}^{N}\theta^{(n)}=0$
is $o(\delta^{1/2})$-close, along with all derivatives, to the corresponding
return map of the truncated system (100).

###### Proof.

We take several cross-sections
$\Sigma_{c_{k}}:\frac{1}{N}\sum_{n=1}^{N}\theta^{(n)}=c_{k},\quad
0=c_{1}<c_{2}<\ldots<c_{K}=2\pi$, and consider the return map of the flow of
(99) to $\Sigma_{0}$ as a composition of the maps from each consecutive cross-
section to the next one (intermediate Poincaré maps). We show that each of
these maps is well defined in the neighborhood of $\mathcal{L}_{min}^{*}$ and
is sufficiently close to the corresponding map for (100).

In a small neighborhood of $\mathcal{L}_{min}^{*}$, by the non-simultaneous
impacts Assumption IP3, when $q^{(n)}$ is close to an impact with the billiard
boundary for particle $n$, all the other particles stay away from the billiard
boundary. Therefore, we can choose the cross-sections $\Sigma_{c_{k}}$ in such
a way that when the phase point travels between $\Sigma_{c_{k}}$ and
$\Sigma_{c_{k+1}}$, either all the particles are bounded away from the
billiard boundary, or one particle gets close to a regular impact while all
the other particles stay away from the boundary. In the first case, systems
(99) and (100) are uniformly $O(\delta^{3/4})$-close in $C^{r}$ for any $r$ in
the region of the phase space between these cross-sections. Since the flight
time between the cross-sections is bounded, this implies that the
corresponding intermediate Poincaré maps are also $O(\delta^{3/4})$-close, as
required.

It remains to consider the second case, when one of the particles gets close
to an impact. Let it be particle $n_{0}$. Let the cross-sections before and
after the impact be $\Sigma_{c_{k}}$ and $\Sigma_{c_{k+1}}$, and let the value
of $q^{(n_{0})}$ on these cross-sections stay at a distance of order $\eta>0$
from the impact for all small $\delta$. Let us show that the map
$\Sigma_{c_{k}}\to\Sigma_{c_{k+1}}$ by system (99) is $O(\eta\delta^{1/2})$
close, in $C^{r}$ for any $r$, to the corresponding map by the system

$\begin{array}[]{l}\displaystyle\hat{H}_{scal}=\sum_{n}\left[\omega(\delta)I^{(n)}+\delta^{1/2}\frac{1}{2}{I^{(n)}}^{\top}\\!A(\delta)I^{(n)}+O(\delta^{3/4})\right]\\\
\displaystyle\qquad\qquad+\;\delta^{1/2}\mathop{\sum_{n,m\neq n_{0}}}_{n\neq
m}W(\hat{q}(\delta^{1/2}I_{0}^{(n)},\theta^{(n)},\delta^{1/4}z^{(n)})-\hat{q}(\delta^{1/2}I_{0}^{(m)},\theta^{(m)},\delta^{1/4}z^{(m)}))\end{array}$
(102)

(i.e., we switched off the interaction with the particle $n_{0}$), and,
similarly, the map $\Sigma_{c_{k}}\to\Sigma_{c_{k+1}}$ by system (100 ) is
$O(\eta\delta^{1/2})$-close, in $C^{r}$ for any $r$, to the corresponding map
by the system

$\hat{H}_{trun}=\sum_{n}\left[\omega(\delta)I^{(n)}+\delta^{1/2}\frac{1}{2}{I^{(n)}}^{\top}A(\delta)I^{(n)})\right]+\delta^{1/2}\mathop{\sum_{n,m\neq
n_{0}}}_{n\neq m}W(q^{*}_{\delta}(\theta^{(n)})-q^{*}_{\delta}(\theta^{(m)}).$
(103)

Indeed, since the particle $n_{0}$ between $\Sigma_{c_{k}}$ and
$\Sigma_{c_{k+1}}$ is close to impact, we can use the scaled flow-box
coordinates $(\tau^{(n_{0})},\mathcal{E}^{(n_{0})},\mathcal{P}^{(n_{0})})$ for
the $n_{0}$-th particle. As explained at the end of section 4.1.3, the
transformation between the scaled
$(\tau^{(n_{0})},\mathcal{E}^{(n_{0})},\mathcal{P}^{(n_{0})})$ coordinates and
the scaled $(\theta^{(n_{0})},I_{0}^{(n_{0})},z^{(n_{0})})$ is non-singular in
the limit $\delta=0$. We use the notation
$(\tau=\tau^{(n_{0})},\mathcal{E}=\mathcal{E}^{(n_{0})},\mathcal{P}=\mathcal{P}^{(n_{0})},X=\\{(I_{0}^{(n)},\theta^{(n)},z^{(n)})\\}_{n\neq
n_{0}})$ in the layer between $\Sigma_{c_{k}}$ and $\Sigma_{c_{k+1}}$. The
Hamiltonian (99) in these scaled coordinates can be schematically written as

$H_{scal}=\mathcal{E}+H_{1}(X;\delta)+\delta^{1/2}H_{2}(\tau,\delta^{1/2}\mathcal{E},\delta^{1/4}\mathcal{P},X;\delta),$
(104)

and the Hamiltonian (102) is given by

$\hat{H}_{scal}=\mathcal{E}+H_{1}(X;\delta).$ (105)

Here $H_{1}$ is the Hamiltonian of the subsystem corresponding to all the
particles but $n_{0}$:

$\begin{array}[]{l}\displaystyle H_{1}=\sum_{n\neq
n_{0}}\left[\omega(\delta)I^{(n)}+\delta^{1/2}\frac{1}{2}{I^{(n)}}^{\top}\\!A(\delta)I^{(n)}+O(\delta^{3/4})\right]\\\
\displaystyle\qquad\qquad+\;\delta^{1/2}\mathop{\sum_{n,m\neq n_{0}}}_{n\neq
m}W(\hat{q}(\delta^{1/2}I_{0}^{(n)},\theta^{(n)},\delta^{1/4}z^{(n)})-\hat{q}(\delta^{1/2}I_{0}^{(m)},\theta^{(m)},\delta^{1/4}z^{(m)})).\end{array}$

Since these particles stay away of impacts, $H_{1}$ is $C^{r}$ for any fixed
$r$, uniformly for all small $\delta\geqslant 0$. The term $H_{2}$ describes
the interaction of the particle $n_{0}$ with the rest of the
particles666Recall that for the convenience of notation each term
$W(q^{(n)}-q^{(m)})$ appears twice in the double sum in (98).:

$H_{2}=2\sum_{m\neq
n_{0}}W(q(\tau,\delta^{1/2}\mathcal{E},\delta^{1/4}\mathcal{P})-\hat{q}(\delta^{1/2}I_{0}^{(m)},\theta^{(m)},\delta^{1/4}z^{(m)}))$

where we denote
$q(\tau,\delta^{1/2}\mathcal{E},\delta^{1/4}\mathcal{P})=\hat{q}(\delta^{1/2}I_{0}^{(n_{0})},\theta^{(n_{0})},\delta^{1/4}z^{(n_{0})})$.
By Lemma 4.3, $H_{2}$ is a $C^{\infty}$ function of
$(\delta^{1/2}\mathcal{E},\delta^{1/4}\mathcal{P},X)$ and depends
continuously, in $C^{\infty}$, on $\tau$ and $\delta\geqslant 0$. Moreover, as
long as $q^{(n_{0})}$ stays away from the billiard boundary, $H_{2}$ depends
smoothly on $\tau$ as well, for all $\delta\geqslant 0$.

Since $\Sigma_{c_{k}}$ and $\Sigma_{c_{k+1}}$ stay bounded away from the
impact, one obtains that near these cross-sections

$\frac{d}{dt}\tau=\partial_{\mathcal{E}}H_{scal}\ =1+O(\delta)$

uniformly for all $\delta\geqslant 0$, i.e., it stays bounded away from zero.
Hence, hypersurfaces of constant $\tau$ are cross-sections to the flow of
$H_{scal}$ for all $\delta\geqslant 0$; the same is true for the flow of
$\hat{H}_{scal}$. Thus, we choose constant-$\tau$ cross-sections,
$\Sigma^{\tau}_{k}$ and $\Sigma^{\tau}_{k+1}$ close to $\Sigma_{c_{k}}$ and,
respectively, $\Sigma_{c_{k+1}}$. Because all these cross-sections are bounded
away from impacts, the flows of $H_{scal}$ and $\hat{H}_{scal}$ are
$O(\delta^{1/2})$-close in $C^{r}$ in between $\Sigma_{c_{k}}$ and
$\Sigma^{\tau}_{k}$ and in between $\Sigma^{\tau}_{k+1}$ and
$\Sigma_{c_{k+1}}$, for any $r$. Since the flight time from $\Sigma_{c_{k}}$
and $\Sigma_{c_{k+1}}$ is $O(\eta)$, the flight times between the cross-
sections $\Sigma_{c_{k}}$ and $\Sigma^{\tau}_{k}$ and between
$\Sigma^{\tau}_{k+1}$ and $\Sigma_{c_{k+1}}$ are also bounded by $O(\eta)$, so
the corresponding maps for $H_{scal}$ and $\hat{H}_{scal}$ between these
cross-sections are $O(\eta\delta^{1/2})$-close.

We are left to show that the maps from $\Sigma^{\tau}_{k}$ to
$\Sigma^{\tau}_{k+1}$ for system (104) ($H_{scal}$) and system (105)
($\hat{H}_{scal}$) on any given energy level are $O(\eta\delta^{1/2})$-close
in $C^{r}$ for any $r$. The equations of motion defined by (104) on the energy
level $H_{scal}=E$ are given by

$\begin{array}[]{l}\displaystyle\dot{\tau}=1+\delta^{1/2}\partial_{\mathcal{E}}H_{2},\\\
\mathcal{\dot{P}}=\delta^{1/2}\\{\mathcal{P},H_{2}\\},\\\
\dot{X}=\\{X,H_{1}\\}+\delta^{1/2}\\{X,H_{2}\\},\end{array}$

where $\\{\cdot,\cdot\\}$ are the Poisson brackets in the $(\mathcal{P},X)$
space. To have the equations in a closed form, we substitute the energy
$\mathcal{E}$ of particle $(n_{0})$ by its expression which can be found from
(104) by the implicit function theorem:

$\mathcal{E}=E-H_{1}(X;\delta)-\delta^{1/2}H_{2}(\tau,\delta^{1/2}\mathcal{E},\delta^{1/4}\mathcal{P},X;\delta)=E-H_{1}(X;\delta)-\delta^{1/2}H_{3}(\tau,\delta^{1/4}\mathcal{P},X;\delta),$

where $H_{3}$ is continuous in $\tau$ and $C^{\infty}$-smooth in
$\delta^{1/4}\mathcal{P},X$ uniformly for all $\tau$ and $\delta\geqslant 0$.

Choosing $\tau$ as the time variable, we obtain a non-autonomous system of the
form

$\frac{d}{d\tau}\mathcal{\dot{P}}=O(\delta^{3/4}),\qquad\frac{d}{d\tau}X=\\{X,H_{1}\\}+O(\delta^{1/2}),$

where the right-hand sides are continuous in the new time $\tau$ and
$C^{\infty}$ in $\mathcal{P},X$ for all small $\delta\geqslant 0$. The right-
hand side of this system is $O(\delta^{1/2})$-close with all derivatives with
respect to $\mathcal{P},X$, to

$\frac{d}{d\tau}\mathcal{\dot{P}}=0,\qquad\frac{d}{d\tau}X=\\{X,H_{1}\\},$

which is the system (105) restricted to any constant energy level. The
constant-$\Delta\tau$ maps of these systems are therefore
$O(\Delta\tau\delta^{1/2})$-close in $C^{r}$ for any $r$. This is what we need
since $\Delta\tau$, the flight time from $\Sigma^{\tau}_{k}$ to
$\Sigma^{\tau}_{k+1}$, is $O(\eta)$.

This proves that the Poincaré map $\Sigma_{c_{k}}\to\Sigma_{c_{k+1}}$ for the
Hamiltonian $H_{scal}$ of (99) is $O(\eta\delta^{1/2})$-close, in $C^{r}$ for
any $r$, to the corresponding map for the Hamiltonian $\hat{H}_{scal}$ of
(102).

Next we show that the corresponding maps for systems (100) and (103) are also
$O(\eta\delta^{1/2})$-close in $C^{r}$ for any $r$. Here, we do not use the
flow-box coordinates and instead of showing the
$O(\eta\delta^{1/2})$-closeness of the Poincaré maps between constant-$\tau$
cross-sections we show the $O(\eta\delta^{1/2})$-closeness of the Poincaré
maps between constant-$\theta^{(n_{0})}$ cross-sections, $\Sigma^{\theta}_{k}$
before impact and $\Sigma^{\theta}_{k+1}$ after impact. Recall that only the
particle $n_{0}$ is near impact for the flow between
$\Sigma_{c_{k}}\to\Sigma_{c_{k+1}}$. Hence the flows of (100) and (103)
between $\Sigma_{c_{k}}$ and $\Sigma^{\theta}_{k}$ and between
$\Sigma^{\theta}_{k+1}$ and $\Sigma_{c_{k+1}}$ are $O(\delta^{1/2})$-close,
with derivatives, for all $\delta\geqslant 0$. Since the flight time between
these cross-sections is bounded by $\eta$, the difference between the
corresponding Poincaré maps for systems (100) and (103) is
$O(\eta\delta^{1/2})$ small. So we need to examine only the flow between
$\Sigma^{\theta}_{k}$ and $\Sigma^{\theta}_{k+1}$. As explained in Section
4.1.3, the terms
$W(q^{*}_{\delta}(\theta^{(n)})-q^{*}_{\delta}(\theta^{(m)}))$ in $H_{trun}$
of (100) and $\hat{H}_{trun}$ of (103) are $C^{\infty}$ functions of all the
variables if $n,m\neq n_{0}$. The terms
$W(q^{*}_{\delta}(\theta^{(n_{0})})-q^{*}_{\delta}(\theta^{(m)}))$ with $m\neq
n_{0}$ in (100) are $C^{\infty}$ functions of $\theta^{(m)}$ uniformly for all
$\theta^{(n_{0})}$ and $\delta\geqslant 0$. Restricting the flow to a constant
energy level and choosing $\theta^{(n_{0})}$ as a new time variable, the
right-hand sides of the corresponding non-autonomous systems obtained from
(100) and (103) are $O(\delta^{1/2})$-close with derivatives with all the
variables but $\theta^{(n_{0})}$. This implies the
$O(\eta\delta^{1/2})$-closeness of their constant-$\theta^{(n_{0})}$ maps,
which implies the required $O(\eta\delta^{1/2})$-closeness of the maps
$\Sigma_{c_{k}}\to\Sigma_{c_{k+1}}$ for the Hamiltonians $H_{trun}$ and
$\hat{H}_{trun}$.

Because there is no dependence on the $q$-coordinates of the particle $n_{0}$
in equations (102) and (103), and this is the only source of singularity at
$\delta=0$ (all the other particles are away from the impacts), the
Hamiltonians $\hat{H}_{scal}$ and $\hat{H}_{trun}$ are $O(\delta^{3/4})$-close
in $C^{r}$ for any $r$. Therefore, their Poincaré maps are also
$O(\delta^{3/4})$-close. This, finally, proves that the maps
$\Sigma_{c_{k}}\to\Sigma_{c_{k+1}}$ by system (99) and system (100) are
$O(\eta\delta^{1/2})$-close.

Taking the composition of the Poincaré maps from each consecutive cross-
section to the next one, we obtain that the return maps to $\Sigma_{0}$ for
systems (99) and (100) are $O(\eta\delta^{1/2})$-close, in $C^{r}$ for any
$r$. Since $\eta$ can be taken as small as we want, we can also allow it to go
sufficiently slowly to zero as $\delta\to 0$, thus completing the proof of the
lemma. ∎

We now evaluate the Poincaré map for the truncated system (100). We average
the truncated Hamiltonian $H_{trun}$ with respect to the motion along the
periodic orbit $L^{*}_{\delta}$ to establish an analogue of Lemma 3.1, but,
due to the loss of smoothness at impacts, the proof is different – we utilize
explicitly the pairwise structure of the interaction potential instead of
using the Fourier expansion of Lemma 3.1.

First, we introduce the following terminology:

######  Definition 4.6.

A function $G$ of
$\left((I_{0}^{(1)},\theta^{(1)},z^{(1)}),\ldots,(I_{0}^{(N)},\theta^{(N)},z^{(N)})\right)$
is called weakly regular if it is $C^{\infty}$ at $\delta>0$ and satisfies the
following conditions (i) and (ii) in the limit $\delta\to 0$:

(i) When all $q^{(n)}(I_{0}^{(n)},\theta^{(n)},z^{(n)})$ stay bounded away
from the billiard boundary, the function $G$ is of class $C^{\infty}$ for all
$\delta\geqslant\ 0$ and depends continuously, in $C^{\infty}$, on
$\delta\geqslant 0$;

(ii) When exactly one of the particles is in the billiard boundary layer,
i.e., for some $n_{0}$,
$q^{(n_{0})}(I_{0}^{(n_{0})},\theta^{(n_{0})},z^{(n_{0})})$ is near an impact
point, the function $G$ is $C^{\infty}$ with respect to
$(I^{(n_{0})},z^{(n_{0})})$ and
$(\\{I_{0}^{(n)},\theta^{(n)},z^{(n)}\\}_{n\neq n_{0}})$ for each fixed
$\theta^{(n_{0})}$ and all $\delta\geqslant 0$, and it depends continuously,
as a $C^{\infty}$ function of $(I^{(n_{0})},z^{(n_{0})})$ and
$(\\{I_{0}^{(n)},\theta^{(n)},z^{(n)}\\}_{n\neq n_{0}})$, on
$\theta^{(n_{0})}$ and $\delta\geqslant 0$.

In particular, since we showed that if the particle $n_{0}$ is close to impact
$q^{*}_{\delta}(\theta^{(n_{0})})$ depends continuously on $\theta^{(n_{0})},$
the functions $W(q^{*}_{\delta}(\theta^{(n)})-q^{*}_{\delta}(\theta^{(m)}))$
in (100) are weakly regular. Note that we do not consider the case when there
are more than one particle in the boundary layer because it is not possible by
the non-simultaneous impacts assumption IP3.

######  Lemma 4.7.

There exists a weakly regular, symplectic change of coordinates which brings
the truncated Hamiltonian (100) to the form

$\begin{array}[]{l}\displaystyle
H=\sum_{n=1}^{N}\omega(\delta)I^{(n)}+\delta^{1/2}\sum_{n=1}^{N}\frac{1}{2}I^{(n)}A(\delta)I^{(n)}+\delta^{1/2}U(\theta^{(1)},\dots,\theta^{(N)};\delta)+\delta^{3/4}G.\end{array}$
(106)

Here $G$ is a weakly regular function, and the averaged potential is given by:

$U(\theta^{(1)},\dots,\theta^{(N)};\delta)=\mathop{\sum_{n,m=1,\dots,N}}_{n\neq
m}W_{avg}(\theta^{(n)}-\theta^{(m)};\delta),$ (107)

where

$W_{avg}(\theta^{(n)}-\theta^{(m)};\delta)=\frac{1}{2\pi}\int_{0}^{2\pi}W(q^{*}_{\delta}(s+\theta^{(n)})-q^{*}_{\delta}(s+\theta^{(m)}))ds.$
(108)

All the derivatives of $W_{avg}$ with respect to $\theta$ depend continuously
on $\delta$ for small $\delta\geqslant 0.$

###### Proof.

Let $(u,v)\
=(\frac{1}{2}(\theta^{(n)}+\theta^{(m)}),\frac{1}{2}(\theta^{(n)}-\theta^{(m)}))$.
Define

$\Psi_{0}(u,v;u_{0},\delta)=\frac{1}{\omega_{0}(\delta)}\left[\int_{u_{0}}^{u}W(q^{*}_{\delta}(s+v)-q^{*}_{\delta}(s-v))ds-(u-u_{0})W_{avg}(2v;\delta)\right],$
(109)

so $\Psi_{0}(u,-v;u_{0},\delta)=\Psi_{0}(u,v;u_{0},\delta)$. It follows from
(108) and the $2\pi$-periodicity of $q^{*}_{\delta}$ that

$\Psi_{0}(u+2\pi,v;u_{0},\delta)=\Psi_{0}(u,v;u_{0},\delta).$ (110)

Let

$\Psi(\theta^{(1)},\ldots,\theta^{(N)};u_{0},\delta)=\sum_{m_{1}\neq
m_{2}}\Psi_{0}(\frac{1}{2}(\theta^{(m_{1})}+\theta^{(m_{2})}),\frac{1}{2}(\theta^{(m_{1})}-\theta^{(m_{2})});u_{0},\delta).$
(111)

For positive $\delta$, it is identical to the function (48) of Section 3.
Consider the symplectic coordinate change (its smoothness properties in the
limit $\delta\rightarrow 0$ are discussed below)

$I_{0}^{(n)}\rightarrow
I_{0}^{(n)}-\delta^{1/2}\partial_{\theta^{(n)}}\Psi,\qquad n=1,\dots,N.$ (112)

By (110), the right-hand side does not change when we add $2\pi$ to all phases
$\theta^{(1)},\dots,\theta^{(N)}$, i.e., this is indeed a well-defined
coordinate transformation in a neighborhood of $\mathcal{L}_{min}^{*}$.
Differentiating (109), we obtain

$\omega_{0}(\delta)\partial_{u}\Psi=\omega_{0}(\delta)\partial_{\theta^{(n)}}\Psi+\omega_{0}(\delta)\partial_{\theta^{(m)}}\Psi=W(q^{*}_{\delta}(\theta^{(n)})-q^{*}_{\delta}(\theta^{(m)}))-W_{avg}(\theta^{(n)}-\theta^{(m)};\delta).$

As $\partial_{\theta^{(n)}}\Psi=\sum_{m\neq
n}\partial_{u}\Psi_{0}(\frac{1}{2}(\theta^{(n)}+\theta^{(m)}),\frac{1}{2}(\theta^{(n)}-\theta^{(m)});u_{0},\delta)$,
the transformation (112) makes

$\sum_{n=1}^{N}\omega_{0}(\delta)I_{0}^{(n)}\rightarrow\sum_{n=1}^{N}\omega_{0}(\delta)(I_{0}^{(n)}-\delta^{1/2}\sum_{m\neq
n}\partial_{u}\Psi_{0}(\frac{1}{2}(\theta^{(n)}+\theta^{(m)}),\frac{1}{2}(\theta^{(n)}-\theta^{(m)});u_{0},\delta)$

so it brings the Hamiltonian (100) to the required form (106).

We need to show that one can choose $u_{0}$ such that the expression

$\partial_{\theta^{(n)}}\Psi_{0}(\frac{1}{2}(\theta^{(n)}+\theta^{(m)}),\frac{1}{2}(\theta^{(n)}-\theta^{(m)});u_{0},\delta)$

in the formula for the coordinate transformation (112) is weakly regular for
all $n$ and $m$. Let

$\Psi_{1}(u,v;u_{0},\delta)=\int_{u_{0}}^{u}W(q^{*}_{\delta}(s+v)-q^{*}_{\delta}(s-v))ds.$

######  Lemma 4.8.

For $u_{0}$ chosen such that
$({u_{0}}\pm\frac{1}{2}(\theta_{min}^{(n)}-\theta_{min}^{(m)}))/\omega_{0}$
are bounded away from the impact moments $t^{j}$ for all $m$, $n$, and $j$,
the function
$\Psi_{1}(\frac{\theta^{(n)}+\theta^{(m)}}{2},\frac{\theta^{(n)}-\theta^{(m)}}{2};u_{0},\delta)$
and its derivatives with respect to $\theta^{(n)},\theta^{(m)}$ are weakly
regular.

###### Proof.

Let us divide the interval of integration to subintervals
$[u_{j},u_{j+1}],j=0,\ldots,J$ such that the following holds: the last
interval is $[u_{J},u_{J+1}=\frac{\theta^{(n)}+\theta^{(m)}}{2}]$; on each
sub-interval at most one of the particles is in the boundary layer (i.e., for
all $s$ in each sub-interval, either
$q^{*}_{\delta}(s+\frac{1}{2}(\theta^{(n)}-\theta^{(m)}))$, or
$q^{*}_{\delta}(s-\frac{1}{2}(\theta^{(n)}-\theta^{(m)}))$, or both are
outside the boundary layer); and all end points but the last one are at a
finite distance from the impact points:
$({u_{j}}\pm\frac{1}{2}(\theta^{(n)}_{min}-\theta^{(m)}_{min}))/\omega_{0}$
are bounded away from the impact moments for $j=0,\ldots,J$.

Such choice of sufficiently small intervals is possible: because we consider
$\theta^{(n)}$ and $\theta^{(m)}$ close to $\theta^{(n)}_{min}$ and
$\theta^{(m)}_{min}$, the non-simultaneous impacts assumption IP3 implies that
$\theta^{(n)}-\theta^{(m)}$ can not get close to $\omega_{0}(t^{j}-t^{k})$ for
$n\neq m$, hence it is impossible that
$s_{1}+\frac{1}{2}(\theta^{(n)}-\theta^{(m)})$ and
$s_{2}-\frac{1}{2}(\theta^{(n)}-\theta^{(m)})$ get simultaneously close to the
impact phases $\omega_{0}t^{j}$ and $\omega_{0}t^{k}$ if $s_{1}$ and $s_{2}$
belong to the same small interval.

By construction,

$\Psi_{1}(\frac{\theta^{(n)}+\theta^{(m)}}{2},\frac{\theta^{(n)}-\theta^{(m)}}{2};u_{0},\delta)=\sum_{j=0}^{J}\Psi_{1}(u_{j+1},\frac{\theta^{(n)}-\theta^{(m)}}{2};u_{j},\delta).$
(113)

For all $j<J$, we show that
$\Psi_{1}(u_{j+1},\frac{\theta^{(n)}-\theta^{(m)}}{2};u_{j},\delta)$ is
$C^{\infty}$ for all small $\delta\geqslant 0$ (hence it is weakly regular,
and its derivatives are weakly regular as well). Indeed, if both
$q^{*}_{\delta}(s\pm\frac{\theta^{(n)}-\theta^{(m)}}{2})$ are not in the
boundary layer, for all $s$ from the integration interval $[u_{j},u_{j+1}]$,
then the integrand
$W(q^{*}_{\delta}(s+\frac{\theta^{(n)}-\theta^{(m)}}{2})-q^{*}_{\delta}(s-\frac{\theta^{(n)}-\theta^{(m)}}{2};\delta))$
is a $C^{\infty}$ function for all $\delta\geqslant 0$ and the claim follows.
If, say, the first term,
$q^{*}_{\delta}(s+\frac{\theta^{(n)}-\theta^{(m)}}{2})$ is in the boundary
layer for some value of $s\in(u_{j},u_{j+1})$, then we shift the integration
interval by $\frac{1}{2}(\theta^{(n)}-\theta^{(m)})$ to establish

$\Psi_{1}(u_{j+1},\frac{\theta^{(n)}-\theta^{(m)}}{2};u_{j},\delta)=\int_{u_{j}+\frac{1}{2}(\theta^{(n)}-\theta^{(m)})}^{u_{j+1}+\frac{1}{2}(\theta^{(n)}-\theta^{(m)})}W(q^{*}_{\delta}(s^{\prime})-q^{*}_{\delta}(s^{\prime}-(\theta^{(n)}-\theta^{(m)})))ds^{\prime}.$
(114)

The term $q^{*}_{\delta}(s^{\prime}-(\theta^{(n)}-\theta^{(m)}))$ is away from
the billiard boundary for the integration interval,
$q^{*}_{\delta}(s^{\prime})$ is away from the boundary layer at the limits of
integration. Therefore, the integrand
$W(q^{*}_{\delta}(s^{\prime})-q^{*}_{\delta}(s^{\prime}-(\theta^{(n)}-\theta^{(m)})))$
is a $C^{\infty}$-function of $\theta^{(n)}$ and $\theta^{(m)}$, uniformly
continuous in $C^{\infty}$ for all $s^{\prime}$ and all $\delta\geqslant 0$.
Moreover, it is $C^{\infty}$ in $s$ also near the limits of integration. It
immediately follows that $\Psi_{1}$ given by (114) is $C^{\infty}$ as
required. Similarly, if the second term,
$q^{*}(s-\frac{\theta^{(n)}-\theta^{(m)}}{2})$ is near an impact, we shift the
integration interval by $-\frac{1}{2}(\theta^{(n)}-\theta^{(m)})$ and
establish the same smoothness result.

For the last segment, since $u_{J+1}=\frac{\theta^{(n)}+\theta^{(m)}}{2}$, and
$v=\frac{\theta^{(n)}-\theta^{(m)}}{2}$, if both $\theta^{(n)}/\omega_{0}$ and
$\theta^{(m)}/\omega_{0}$ are bounded away from the impact moments, the same
arguments as above show that
$\Psi_{1}(\frac{\theta^{(n)}+\theta^{(m)}}{2},\frac{1}{2}(\theta^{(n)}-\theta^{(m)});u_{J},\delta)$
are smooth as required. On the other hand, if, say, the particle $n$ is in the
boundary layer (and hence the particle $m$ is not in the boundary layer) we
write the integral in the form of (114):

$\Psi_{1}(u_{J+1},\frac{\theta^{(n)}-\theta^{(m)}}{2};u_{J},\delta)=\int_{u_{J}+\frac{1}{2}(\theta^{(n)}-\theta^{(m)})}^{\theta^{(n)}}W(q^{*}_{\delta}(s^{\prime})-q^{*}_{\delta}(s^{\prime}-(\theta^{(n)}-\theta^{(m)})))ds^{\prime}.$
(115)

As above, we have that
$W(q^{*}_{\delta}(s^{\prime})-q^{*}_{\delta}(s^{\prime}-(\theta^{(n)}-\theta^{(m)})))$
is a $C^{\infty}$-function of $\theta^{(n)}$ and $\theta^{(m)}$, uniformly
continuous in $C^{\infty}$ for all $s^{\prime}$ and all $\delta\geqslant 0$.
Moreover, it is $C^{\infty}$ in $s$ also near the lower limit of integration.
The upper limit of integration does not depend on $\theta^{(m)}$, so we
conclude that the integral is $C^{\infty}$ with respect to $\theta^{(m)}$,
i.e., $\Psi_{1}$ and $\partial_{\theta^{(m)}}\Psi_{1}$ given by (115) are
weakly regular, and

$\begin{array}[]{l}\partial_{\theta^{(n)}}\Psi_{1}(u_{J+1},\frac{\theta^{(n)}-\theta^{(m)}}{2};u_{J},\delta)=W(q^{*}_{\delta}(\theta^{(n)})-q^{*}_{\delta}(\theta^{(m)}))\\\
\qquad\qquad\qquad-\frac{1}{2}W(q^{*}_{\delta}(u_{J}\\!+\\!\frac{1}{2}(\theta^{(n)}\\!-\\!\theta^{(m)}))\\!-\\!q^{*}_{\delta}(u_{J}\\!-\\!\frac{1}{2}(\theta^{(n)}\\!-\\!\theta^{(m)})))\\\
\qquad\qquad\qquad-\int_{u_{J}+\frac{1}{2}(\theta^{(n)}-\theta^{(m)})}^{\theta^{(n)}}W^{\prime}(q^{*}_{\delta}(s^{\prime})-q^{*}_{\delta}(s^{\prime}-(\theta^{(n)}-\theta^{(m)})))\partial_{\theta^{(n)}}q^{*}_{\delta}(s^{\prime}-(\theta^{(n)}-\theta^{(m)}))ds^{\prime}\end{array}$

is also weakly regular since $s^{\prime}-(\theta^{(n)}-\theta^{(m)})$ is away
from impact along this integration interval. The similar weakly regularity
results hold true if the particle $m$, and not the particle $n$, is in the
boundary layer.

Thus, all the terms in (113) are weakly regular, along with the derivatives
with respect to $\theta^{(n)}$ and $\theta^{(n)}$, which gives the lemma. ∎

Note that by the periodicity of $q^{*}_{\delta}$,

$W_{avg}(\theta^{(n)}-\theta^{(m)};\delta)=\frac{1}{2\pi}\Psi_{1}(u_{0}+2\pi,u_{0},\frac{\theta^{(n)}-\theta^{(m)}}{2};\delta),$
(116)

and so the right-hand side does not depend on $u_{0}$. In particular we can
choose $u_{0}$ as needed for Lemma 4.8, so the average potential $W_{avg}$ is
weakly regular. Since we can always shift both $\theta^{(n)}$ and
$\theta^{(m)}$ away from the impacts, the weak regularity of the average
potential $W_{avg}$ means that it is is $C^{\infty}$ for all small
$\delta\geqslant\ 0$.

Now, by Lemma 4.8, we obtain that since

$\Psi_{0}(u,v;u_{0},\delta)=\frac{1}{\omega_{0}}\Psi_{1}(u,u_{0},v;\delta)-\frac{u-u_{0}}{\omega_{0}}W_{avg}(2v;\delta),$

the transformation (112) is also weakly regular. ∎

Like in Lemma 4.5, omitting the weakly regular $O(\delta^{3/4})$-term in the
Hamiltonian (106) results only in $O(\delta^{3/4})$ corrections to the return
map to an interior cross-section near $\mathcal{L}_{min}^{*}$. Thus, the
return map for the truncated averaged Hamiltonian

$\begin{array}[]{l}\displaystyle
H(I,\theta)=\sum_{n=1}^{N}\omega(\delta)I^{(n)}+\delta^{1/2}\sum_{n=1}^{N}\frac{1}{2}I^{(n)}A(\delta)I^{(n)}+\delta^{1/2}U(\theta^{(1)},\dots,\theta^{(N)};\delta)\end{array}$
(117)

near $\mathcal{L}_{min}^{*}$ is $o(\delta^{1/2})$-close the return map for the
scaled Hamiltonian (99).

This Hamiltonian has the same form as the truncated Hamiltonian of (46). The
parameters $\omega(\delta)$, $A(\delta)$, and the averaged potential
$U(\theta^{(1)},\dots,\theta^{(N)};\delta)$ depend continuously on $\delta$
and satisfy, for all small $\delta\geqslant 0$, the non-degeneracy assumptions
as in Theorem 1. So we finish the proof of Theorem 3 in the same way as in
Theorem 1. Namely, applying Lemma 3.2, we find that the Poincaré return map
for system (117) is $O(\delta^{3/4})$-close, with all derivatives, to the
time-$\frac{2\pi}{\omega_{0}}$ map for system (55). Hence, the return map for
(99) is $o(\delta^{1/2})$-close to the time-$\frac{2\pi}{\omega_{0}}$ map for
system (55). Now, the same arguments as in Lemma 3.3 show that if we make
$O(\delta^{-1/2})$ iterations of the Poincaré map of (99), the result is
$o(1)_{\delta\to 0}$-close to the time-1 map for the Hamiltonian (58). The
latter map has, by Lemma 3.4, a positive measure set of invariant KAM tori,
hence so does the rescaled system (99), as well as the original system (98).

##  5 Gas in a rectangular box

Here we study the motion near the family of fast vertical periodic orbits in a
box, proving Theorem 4 (the non-simultaneous impacts case, see Sections 5.1,
5.2 and 5.3) and Theorem 5 (the simultaneous impacts case, Sections 5.1, 5.3
and 5.4).

###  5.1 Action-angle coordinates near vertical periodic orbits.

Since the single-particle degrees of freedom decouple, and only the vertical
direction dynamics are fast and billiard-like, the single-particle theory
developed in Section 4.1 applies to the one-dimensional vertical motion, where
$q=q_{d}\in R^{1}$ and the billiard corresponds to a particle bouncing between
the end points of the interval $[0,\pi]$. At positive $\delta$, the vertical
motion of a single particle is described by the Hamiltonian

$H_{d}=\frac{p^{2}}{2}+\delta
V_{d}(q)=\frac{p^{2}}{2}+\frac{\delta}{Q_{d}(q)^{\alpha}},$ (118)

see (30), (31). One can introduce action-angle variables $(I,\theta)$ for this
system. The periodic orbit that lies in the energy level $H=\frac{1}{2}$ is
denoted as $q=q^{*}_{\delta}(\theta)$; it tends to the saw-tooth (36) as
$\delta\to 0$. The vertical action $I$ is a one-to-one, smooth function of the
vertical energy and is a constant of motion. Thus, the single-particle
vertical motion is governed by

$H_{d}(I;\delta)=\frac{(\hat{p}_{d}(I,\theta;\delta))^{2}}{2}+\delta
V_{d}(\hat{q}_{d}(I,\theta;\delta))-\frac{1}{2}=\omega_{0}(\delta)I+\frac{1}{2}a(\delta)I^{2}+O(I^{3}),\qquad
I\in\mathbb{R}^{1}.$ (119)

This is analogous to formula (91) of Section 4.1 but, contrary to the
multidimensional single-particle theory, there are no $z$-variables nor the
corresponding $\mathcal{P}$-variables.

Using the action-angle coordinates in the vertical direction, the multi-
particle Hamiltonian (33) takes the following form:

$\begin{split}H&=\sum^{N}_{n=1}H_{d}(I^{(n)};\delta)+\delta\sum_{n=1,\dots,N}\sum_{i=1}^{d-1}\left[\frac{(p_{\xi,i}^{(n)})^{2}}{2}+V_{i}(\xi_{i}^{(n)})\right]\\\
&\qquad+\delta\mathop{\sum_{n,m=1,\dots,N}}_{n\neq m}\
W(\hat{q}_{d}(I^{(n)},\theta^{(n)};\delta)-\hat{q}_{d}(I^{(m)},\theta^{(m)};\delta),\xi^{(n)}-\xi^{(m)}));\end{split}$
(120)

recall that$(\xi,p_{\xi})$ denote the non-vertical coordinates:
$\xi_{i}^{(n)}=q_{i}^{(n)},p_{\xi,i}^{(n)}=p_{i}^{(n)},i=1,\dots,d-1,\
n=1,\dots N$.

Scaling the vertical actions as $\delta^{1/2}I$ and dividing the Hamiltonian
by $\delta^{1/2}$, we obtain

$\begin{split}H&{}_{scal}=\sum^{N}_{n=1}\omega_{0}(\delta)I^{(n)}+\delta^{1/2}\sum^{N}_{n=1}\left(\frac{1}{2}a(\delta)(I^{(n)})^{2}+\sum_{i=1}^{d-1}\left[\frac{(p_{\xi,i}^{(n)})^{2}}{2}+V_{i}(\xi_{i}^{(n)})\right]\right)+O(\delta)\\\
&\qquad+\delta^{1/2}\mathop{\sum_{n,m=1,\dots,N}}_{n\neq m}\
W(\hat{q}_{d}(\delta^{1/2}I^{(n)},\theta^{(n)};\delta)-\hat{q}_{d}(\delta^{1/2}I^{(m)},\theta^{(m)};\delta),\xi^{(n)}-\xi^{(m)})).\end{split}$
(121)

Below, we establish the existence of an elliptic periodic orbit for system
(121) by comparing it with the averaged system defined by the Hamiltonian

$\begin{split}H_{avg}&=\sum^{N}_{n=1}\omega_{0}(\delta)I^{(n)}+\delta^{1/2}\sum^{N}_{n=1}\left(\frac{1}{2}a(\delta)(I^{(n)})^{2}+\sum_{i=1}^{d-1}\left[\frac{(p_{\xi,i}^{(n)})^{2}}{2}+V_{i}(\xi_{i}^{(n)})\right]\right)\\\
&\qquad+\delta^{1/2}\mathop{\sum_{n,m=1,\dots,N}}_{n\neq
m}W_{avg}(\theta^{(n)}-\theta^{(m)},\xi^{(n)}-\xi^{(m)};\delta)+\delta\tilde{G}(\theta,I,\xi,p;\delta).\end{split}$
(122)

This is an analogue of the averaged system (106) (just the higher order terms
are of order $\delta$ and not of order $\delta^{3/4}$ as in (106)). Here
$W_{avg}$ is the pairwise interaction potential averaged over the vertical
oscillations in the system where the interaction between particles is switched
off (see (120)):

$H=\sum^{N}_{n=1}H_{d}(I^{(n)};\delta)+\delta\sum_{n=1,\dots,N}\sum_{i=1}^{d-1}\left[\frac{(p_{\xi,i}^{(n)})^{2}}{2}+V_{i}(\xi_{i}^{(n)})\right].$

The vertical oscillations at $I^{(n)}=0$ correspond to the choreographic
solution (cf. (3)):

$\mathbf{L}^{*}(\theta,\xi)=\\{q_{1,\ldots
d-1}^{(n)}=\xi_{i}^{(n)},q^{(n)}_{d}=q_{\delta}^{*}(\omega_{0}t+\theta^{(n)}),\;\;p_{1,\ldots
d-1}^{(n)}=0,p_{d}^{(n)}=p_{\delta}^{*}(\omega_{0}t+\theta^{(n)})\\}^{N}_{n=1}.$
(123)

Thus,

$W_{avg}(\theta^{(n)}-\theta^{(m)},\xi^{(n)}-\xi^{(m)};\delta)=\frac{1}{2\pi}\int_{0}^{2\pi}W(q^{*}_{\delta}(s+\theta^{(n)})-q^{*}_{\delta}(s+\theta^{(m)}),\xi^{(n)}-\xi^{(m)})ds.$
(124)

We denote the potential of the averaged system (122) by

$U_{\delta}(\theta,\xi)=\mathop{\sum_{n,m=1,\dots,N}}_{n\neq
m}W_{avg}(\theta^{(n)}-\theta^{(m)},\xi^{(n)}-\xi^{(m)};\delta)+\sum^{N}_{n=1}\sum_{i=1}^{d-1}V_{i}(\xi_{i}^{(n)}),$
(125)

so $U_{0}(\theta,\xi)=U(\theta,\xi)$ of (25). The regularity properties of the
potential $U_{\delta}$ and of the correction term $\delta G$ in (122) in the
limit $\delta\to 0$, and their influence on the dynamics, are evaluated
differently in the case of non-simultaneous and simultaneous impacts.

###  5.2 The non-simultaneous impacts case

The proof of Theorem 4 is, essentially, the same as for Theorem 3 \- we just
do not have here the fast variables $z$ but, instead, have slow variables
$\xi$ (which, in fact, makes the situation simpler). As in Lemma 4.5 of
Section 4.2, in the case where the impacts are non-simultaneous, the Poincaré
map for the Hamiltonian (121) is $o(\delta^{1/2})$-close to the Poincaré map
for the truncated Hamiltonian $H_{trun}$:

$\begin{split}H_{trun}&=\sum^{N}_{n=1}\omega_{0}(\delta)I^{(n)}+\delta^{1/2}\sum^{N}_{n=1}\left(\frac{1}{2}a(\delta)(I^{(n)})^{2}+\sum_{i=1}^{d-1}\left[\frac{(p_{\xi,i}^{(n)})^{2}}{2}+V_{i}(\xi_{i}^{(n)})\right]\right)\\\
&\qquad+\delta^{1/2}\mathop{\sum_{n,m=1,\dots,N}}_{n\neq
m}W(q^{*}_{\delta}(\theta^{(n)})-q^{*}_{\delta}(\theta^{(m)}),\xi^{(n)}-\xi^{(m)}).\end{split}$
(126)

As in Lemma 4.7, we average the truncated Hamiltonian $H_{trun}$ by performing
the symplectic coordinate transformation defined by the $\delta^{1/2}$-time
map for the Hamiltonian $\Psi$ defined as in (111), where $\Psi_{0}$ is
defined as in (109), i.e.,

$\Psi(\theta^{(1)},\ldots,\theta^{(N)},\xi^{(1)},\ldots,\xi^{(N)};u_{0},\delta)=\sum_{m_{1}\neq
m_{2}}\Psi_{0}(\frac{1}{2}(\theta^{(m_{1})}+\theta^{(m_{2})}),\frac{1}{2}(\theta^{(m_{1})}-\theta^{(m_{2})}),\xi^{(m_{1})}-\xi^{(m_{2})};u_{0},\delta),$
(127)

where

$\Psi_{0}(u,v,w;u_{0},\delta)=\frac{1}{\omega_{0}(\delta)}\left[\int_{u_{0}}^{u}W(q^{*}_{\delta}(s+v)-q^{*}_{\delta}(s-v),w;\delta)ds-(u-u_{0})W_{avg}(2v,w;\delta)\right].$

The only difference with Section 4.2 is that now $\Psi$ also depends on the
$\xi$ variables. The same computations as in Lemma 4.7 show that the resulting
near-identity transformation
$(I^{(n)},p_{\xi}^{(n)})\rightarrow(I^{(n)},p_{\xi}^{(n)})-\delta^{1/2}(\partial_{\theta^{(n)}},\partial_{\xi^{(n)}})\Psi$
is weakly regular and brings the truncated Hamiltonian (126) to the averaged
form (122), where the error term $\delta G$ is weakly regular (see Definition
4.6); the potential $W_{avg}$ is a $C^{\infty}$ function and depends
continuously on $\delta$, along with all derivatives, for al $\delta\geq 0$.

Like in Theorem 3, since no more than one particle can be near the box
boundary at any given moment of time (this is the non-simultaneous impacts
assumption), the weak regularity of $G$ implies that omitting the $\delta G$
terms in the averaged system (122) results only in $O(\delta)$-corrections to
the Poincaré return map. Thus, we obtain that the Poincaré return map of the
original system (121) is $o(\delta^{1/2})$-close to the Poincaré return map
for the system

$\begin{split}H&=\sum^{N}_{n=1}\omega_{0}(\delta)I^{(n)}+\delta^{1/2}\sum^{N}_{n=1}\left(\frac{1}{2}a(\delta)(I^{(n)})^{2}+\sum_{i=1}^{d-1}\frac{(p_{\xi,i}^{(n)})^{2}}{2}\right)+\delta^{1/2}U_{\delta}(\theta,\xi)\end{split}$
(128)

where $U_{\delta}(\theta,\xi)$ is defined by (125). Since
$U_{\delta}(\theta,\xi)$ and all its derivatives depend on $\delta$
continuously, the non-degeneracy assumption Box2 implies that the minimal line
at $\delta=0$ is non-degenerate and persists for small $\delta$. So, we
introduce local normal coordinates near this line:

$(\theta^{(n)},I^{(n)})\rightarrow(\varphi,\psi,P,J),$

in the same way as in (51),(52). By the translation invariance of
$U_{\delta}$, it is independent of $\varphi$:

$U_{\delta}(\theta,\xi)=\hat{U}_{\delta}(\psi,\xi).$ (129)

The potential $\hat{U}_{\delta}$ has a non-degenerate minimum at
$(\psi=0,\xi=\xi_{min})$. The Hamiltonian $H$ of (128) becomes

$H(P,\varphi,J,\psi,p_{\xi},\xi;\delta)=\omega_{0}(\delta)P+\delta^{1/2}\left(\frac{1}{2N}a(\delta)P^{2}+\frac{1}{2N}a(\delta)J^{2}+\frac{p_{\xi}^{2}}{2}+\hat{U}_{\delta}(\psi,\xi)\right).$
(130)

This Hamiltonian is similar to that in (53) but it is simpler, as there is
only one fast degree of freedom $(\varphi,P)$. Moreover, $H$ is independent of
$\varphi$, so $P$ is an integral which controls the period of the fast motion.

With this simplification in mind, let us follow the same procedure as is
applied in Section 3 to Hamiltonian (53) and compute the Poincaré return map
from $\varphi=0$ to $\varphi=2\pi$. Since the flight time depends only on $P$,
the restriction of this map onto a fixed level of $P$ is a constant-time map
for the Hamiltonian (130) or, equivalently, an $O(\delta^{1/2})$-time map for
the Hamiltonian

$\begin{array}[]{ll}H_{P}(J,\psi,p_{\xi},\xi;\delta)&=\frac{1}{2N}a(\delta)J^{2}+\
\frac{p_{\xi}^{2}}{2}+\hat{U}_{\delta}(\psi,\xi).\end{array}$ (131)

Thus, as in Lemma 4.5, the Poincaré map of (121) is $o(\delta^{1/2})$-close to
the $O(\delta^{1/2})$-time map for the Hamiltonian (131). Since
$\hat{U}_{\delta}(\psi,\xi)$ depends continuously on $\delta$ with all
derivatives, it follows from the non-degeneracy assumption Box2 and the KAM
non-degeneracy assumption Box3 that the Hamiltonian (131) has a KAM-non-
degenerate elliptic fixed point near $(J,\psi,p_{\xi},\xi)=(0,0,0,\xi_{min})$
with a positive measure set of KAM tori around it. Hence the Poincaré map of
the Hamiltonian (121) also has such elliptic fixed point, proving Theorem 4.

###  5.3 Smoothness properties of the averaged potential

Before proceeding to the proof of Theorem 5 (the case of simultaneous impacts)
we investigate regularity properties of the averaged potential $W_{avg}$ in
(124). We start with the case $\delta=0$, i.e., let the function $q^{*}$ in
(124) be the billiard’s saw-tooth solution (34).

######  Lemma 5.1.

Provided $||\xi^{(n)}-\xi^{(m)}||>\rho$, the averaged potential along the saw-
tooth solution, $W_{avg}$ of (36), is $C^{\infty}$ for
$\theta^{(n)}-\theta^{(m)}\neq 0\mod\pi$. Under the parity assumption Box4, at
the singularities $\theta^{(n)}-\theta^{(m)}=0\mod\pi$, the potential
$W_{avg}$ is $C^{2}$-smooth, yet it is not $C^{3}$ smooth in general.

###### Proof.

Setting the shorthand notation $\vartheta=\theta^{(m)}-\theta^{(n)}$ and
$\zeta=\xi^{(n)}-\xi^{(m)}$, and using the periodicity of $q^{*}$, the
averaged potential of (36) becomes

$W_{avg}(\vartheta,\zeta)=\frac{1}{2\pi}\int_{0}^{2\pi}\
W(q^{*}(s)-q^{*}(s+\vartheta),\zeta)ds.$

By (34), for $\vartheta\in(0,\pi)$,

$\begin{array}[]{ll}2\pi W_{avg}(\vartheta,\zeta)&=\int_{0}^{\pi-\vartheta}\
W(-\vartheta,\zeta)ds+\int_{\pi-\vartheta}^{\pi}\
W(2s-2\pi+\vartheta,\zeta)ds\\\
&\qquad\qquad\qquad\qquad+\int_{\pi}^{2\pi-\vartheta}W(\vartheta,\zeta)ds+\int_{2\pi-\vartheta}^{2\pi}W(4\pi-2s-\vartheta,\zeta)ds\\\
&=(\pi-\vartheta)(W(-\vartheta,\zeta)+W(\vartheta,\zeta))\\\
&\qquad\quad+\int_{\pi-\vartheta}^{\pi}(W(2s-2\pi+\vartheta,\zeta)+W(-2s+2\pi-\vartheta,\zeta))ds\end{array}$
(132)

so for $\|\zeta\|>\rho$ it is $C^{\infty}$ for $\vartheta\in(0,\pi)$. Now,
recall that $W_{avg}(\vartheta,\zeta)$ is an even function, so its extension
to $\vartheta\in(-\pi,0)$ is simply
$W_{avg}(\vartheta,\zeta)=W_{avg}(-\vartheta,-\zeta)$, hence it is
$C^{\infty}$ for all $\vartheta$ away from the matching points
$\vartheta=0\mod\pi$.

It remains to prove the $C^{2}$-smoothness at the matching points. By the
periodicity and the parity assumption Box4,

$W_{avg}(\vartheta,\zeta)=W_{avg}(-\vartheta,\zeta)=W_{avg}(2\pi-\vartheta,\zeta),$

so it is enough to verify only that the first derivatives of $W_{avg}$ in
(132) vanish at $\vartheta=0,\pi$.

By the parity assumption the interaction potential $W$ is even in $\vartheta$,
so (132) becomes

$\begin{array}[]{ll}2\pi
W_{avg}(\vartheta,\zeta)&=2(\pi-\vartheta)W(\vartheta,\zeta)+2\int_{\pi-\vartheta}^{\pi}W(2s-2\pi+\vartheta,\zeta)ds\\\
&=2\pi
W(\vartheta,\zeta)+2\int_{0}^{\vartheta}(W(-2u+\vartheta,\zeta)-W(\vartheta,\zeta))du.\end{array}$
(133)

Therefore,

$\frac{\partial}{\partial\vartheta}W_{avg}(\vartheta,\zeta)=\frac{\partial}{\partial\vartheta}W(\vartheta,\zeta)+\frac{1}{\pi}\int_{0}^{\vartheta}\frac{\partial}{\partial\vartheta}(W(-2u+\vartheta,\zeta)-W(\vartheta,\zeta))du=(1-\frac{\vartheta}{\pi})\frac{\partial}{\partial\vartheta}W(\vartheta,\zeta),$

so
$\frac{\partial}{\partial\vartheta}W_{avg}(\vartheta,\zeta)|_{\vartheta=0,\pi}=0$,
as required.

Differentiating further, we find

$\frac{\partial^{2}}{\partial\vartheta^{2}}W_{avg}(\vartheta,\zeta)=(1-\frac{\vartheta}{\pi})\frac{\partial^{2}}{\partial\vartheta^{2}}W(\vartheta,\zeta)-\frac{1}{\pi}\frac{\partial}{\partial\vartheta}W(\vartheta,\zeta).$
(134)

Next, we obtain

$\frac{\partial^{3}}{\partial\vartheta^{3}}W_{avg}(\vartheta,\zeta)=(1-\frac{\vartheta}{\pi})\frac{\partial^{3}}{\partial\vartheta^{3}}W(\vartheta,\zeta)-\frac{2}{\pi}\frac{\partial^{2}}{\partial\vartheta^{2}}W(\vartheta,\zeta),$

so
$\frac{\partial^{3}}{\partial\vartheta^{3}}W_{avg}(\vartheta,\zeta)|_{\vartheta\rightarrow
0,\pi}=-\frac{2}{\pi}\frac{\partial^{2}}{\partial\vartheta^{2}}W(\vartheta,\zeta)|_{\vartheta\rightarrow
0,\pi}$. Generically, these values do not vanish, so the even extension of
$W_{avg}$ to negative $\vartheta$ cannot be $C^{3}$ (i.e., it is only
piecewise smooth). ∎

Let us now consider the case of $\delta>0$. As in the lemma above, we use the
notation $\vartheta=\theta^{(n)}-\theta^{(m)}$ and
$\zeta=\xi^{(n)}-\xi^{(m)}$.

######  Lemma 5.2.

Away from $\vartheta=0\mod\pi$, the averaged potential
$W_{avg}(\vartheta,\zeta;\delta)$ is $C^{\infty}$-close, for small $\delta$ to
the saw-tooth averaged potential $W_{avg}(\vartheta,\zeta;0)$.

When the parity assumption Box4 is satisfied, the averaged potential near
$\vartheta=0\mod\pi$ is $C^{2}$-close to $W_{avg}(\vartheta,\zeta;0)$, along
with the derivatives with respect to $\zeta$. The higher order derivatives
with respect to $\vartheta$ do not, in general, have a continuous limit near
the singular values $\vartheta=0\mod\pi$ as $\delta\to 0$. The following
estimates hold true:

$\begin{array}[]{l}\frac{\partial^{3}}{\partial\vartheta^{3}}W_{avg}(\vartheta,\zeta;\delta)=O(1),\\\
\frac{\partial^{4}}{\partial\vartheta^{4}}W_{avg}(\vartheta,\zeta;\delta)=O(\delta^{-\frac{1}{\alpha}}),\\\
\frac{\partial^{5}}{\partial\vartheta^{5}}W_{avg}(\vartheta,\zeta;\delta)=O(\delta^{-\frac{2}{\alpha}});\end{array}$

differentiation with respect to $\zeta$ does not affect these estimates:

$\frac{\partial^{i+k}}{\partial\vartheta^{i}\partial\zeta^{k}}W_{avg}(\vartheta,\zeta;\delta)=O(\delta^{-\frac{i-3}{\alpha}}),\qquad
i=3,4,5;\ \ k\geqslant 0.$

###### Proof.

In order to establish the regularity for $\vartheta\neq 0\mod\pi$ (i.e., for
non-simultaneous impacts), we take $\vartheta\in(0,\pi)$ and let $\eta>0$ be a
sufficiently small number so that $\eta<\min(\vartheta,\pi-\vartheta)$. By the
periodicity of $q^{*}_{\delta}$, we write (124) as

$\begin{array}[]{ll}2\pi
W_{avg}(\vartheta,\zeta;\delta)&=\int_{-\eta}^{2\pi-\eta}W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds\\\
&=(\int_{-\eta}^{\pi-\vartheta-\eta}+\int_{\pi-\vartheta-\eta}^{\pi-\vartheta+\eta}+\int_{\pi-\vartheta+\eta}^{2\pi-\vartheta-\eta}+\int_{2\pi-\vartheta-\eta}^{2\pi-\eta})W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds\\\
&=\int_{-\eta}^{\pi-\vartheta-\eta}W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds+\int_{\pi-\eta}^{\pi+\eta}W(q_{\delta}^{*}(s-\vartheta)-q_{\delta}^{*}(s),\zeta)ds\\\
&+\int_{\pi-\vartheta+\eta}^{2\pi-\vartheta-\eta}W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds+\int_{2\pi-\eta}^{2\pi+\vartheta-\eta}W(q_{\delta}^{*}(s-\vartheta)-q_{\delta}^{*}(s),\zeta)ds.\end{array}$

The function $q^{*}_{\delta}$ is $C^{\infty}$ for all $\delta\geqslant 0$ when
its argument is bounded away from $0\mod\pi$, so every integral in this sum is
$C^{\infty}$ function of $\vartheta$ for all $\delta\geqslant 0$ (this is an
explicit version of a similar statement in Lemma 4.7). Similarly, one proves
the regularity of $W_{avg}$ for $\vartheta\in(\pi,2\pi)$.

Let us now examine the case where $\vartheta$ is close to $0$ or $\pi$. We
have

$2\pi W_{avg}(\vartheta,\zeta;\delta)\ =\int_{0}^{2\pi}\
W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds:=S(\vartheta,\zeta;\delta,\eta)+R(\vartheta,\zeta;\delta,\eta),$
(135)

where $S$, the “singular part”, corresponds to the integration intervals with
both particles $\eta$-close to impacts:

$S(\vartheta,\zeta;\delta,\eta)=\int_{-\eta}^{\eta}W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds+\int_{\pi-\eta}^{\pi+\eta}W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds,$
(136)

and $R$, the “regular part”, corresponds to the integration intervals for
which both particles are at a distance larger than $\eta$ from impacts:

$R(\vartheta,\zeta;\delta,\eta)=\int_{\eta}^{\pi-\eta}W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds+\int_{\pi+\eta}^{2\pi-\eta}W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds.$
(137)

For small $\eta$, and $\vartheta$ close to $0$ (both particles have the same
phase) or $\pi$ (the anti-phase state), the term
$R(\vartheta,\zeta;\delta,\eta)$ is $C^{\infty}$ for all $\delta\geqslant 0$.
Thus, we need to evaluate the singular term, $S$, and its derivatives. Recall
that $q_{\delta}^{*}(s)\rightarrow q_{0}^{*}(s)$ in $C^{0}$, so $S$ converges
in $C^{0}$ to the billiard limit $S(\vartheta,\zeta;0,\eta)$. We will show
that up to order 3 the derivatives of $S$ are uniformly bounded for all small
$\delta$. This proves, by compactness argument, the $C^{2}$-closeness of $S$
and, hence, of $W_{avg}$, to their $C^{0}$ limits at $\delta=0$, as claimed.

So, to prove the lemma, we only need to estimate the derivatives of $S$. In
order to do this, we use formula (92) for $q^{*}_{\delta}$, where the first
impact is at $M^{1}:q=0$ (this corresponds to $\theta_{1}=0$) and the second
impact is at $M^{2}:q=\pi$ (this corresponds to $\theta_{2}=\pi$). Recall that
$q^{*}_{\delta}$ is the periodic orbit in the energy level $H=\frac{1}{2}$ of
the system (118). Since this system has one degree of freedom and is
reversible, the periodic orbit $q^{*}_{\delta}(\theta)$ is even and
$2\pi$-periodic. Moreover, by the symmetry of the potential $V_{d}(q)$ (see
Assumption Box1), we have $q^{*}_{\delta}(\theta)=q^{*}_{\delta}(\pi-\theta)$.
Thus, we may write (92) as

$\begin{array}[]{l}\displaystyle
q^{*}_{\delta}(\theta)=\delta^{1/\alpha}\tilde{q}_{\delta}\left(\frac{\theta}{\omega_{0}(\delta)\delta^{1/\alpha}}\right)\;\mbox{
near }\;\theta=0,\\\ \displaystyle
q^{*}_{\delta}(\theta)=\pi-\delta^{1/\alpha}\tilde{q}_{\delta}\left(\frac{\theta-\pi}{\omega_{0}(\delta)\delta^{1/\alpha}}\right)\;\mbox{
near }\;\theta=\pi.\end{array}$ (138)

where $\tilde{q}_{\delta}(\cdot)$ is an even function with bounded derivatives
(uniformly for all $\delta\geqslant 0$). By (118)

$\frac{d}{dt}q^{*}_{\delta}=p^{*}_{\delta},\qquad\frac{d}{dt}p^{*}_{\delta}=-\delta
V_{d}^{\prime}(q^{*}_{\delta}).$

Therefore, if we denote

$\tilde{p}_{\delta}\left(\frac{\theta}{\omega_{0}(\delta)\delta^{1/\alpha}}\right)=\tilde{q}_{\delta}^{\prime}=\frac{d}{dt}q^{*}_{\delta}$
(139)

(see (71) and (92)), then

$\tilde{p}_{\delta}^{\prime}=-\frac{\alpha\delta^{1+1/\alpha}}{Q_{d}(\delta^{1/\alpha}\tilde{q})^{\alpha+1}}Q_{d}^{\prime}(\delta^{1/\alpha}\tilde{q})=-\;\frac{1}{\tilde{q}_{\delta}^{\alpha+1}}\left(\frac{\alpha}{(Q_{d}^{\prime}(0))^{\alpha}}+O(\delta^{1/\alpha})\right).$
(140)

Notice that by Lemma 4.1, the behavior of $\tilde{q}$ is asymptotically linear
at large $u$:

$\tilde{q}_{\delta}(u)=|u|+o(u),$ (141)

so

$\tilde{p}_{\delta}^{(k)}(u)=O(|u|^{-\alpha-k}),\qquad k\geqslant 1.$ (142)

Note also that the frequency $\omega_{0}$ of the vertical oscillations tends
to that of the billiard motion, i.e.,

$\omega_{0}(0)=1.$ (143)

Now we can return to analyze the behavior of the singular term $S$. For
$\vartheta$ close to $0$, using formulas (138) and that $W$ is even, we write
(136) as

$\begin{array}[]{l}\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S(\vartheta,\zeta;\delta,\eta)=(\int_{-\eta}^{\eta}+\int_{\pi-\eta}^{\pi+\eta})\
W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds=2\int_{-\eta}^{\eta}W\left(\delta^{1/\alpha}\tilde{q}_{\delta}(\frac{s}{\omega_{0}\delta^{1/\alpha}})-\delta^{1/\alpha}\tilde{q}_{\delta}(\frac{s+\vartheta}{\omega_{0}\delta^{1/\alpha}}),\
\zeta\right)ds\\\ \\\
\displaystyle\qquad\qquad=2\omega_{0}\delta^{1/\alpha}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}W\left(\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\vartheta}{\omega_{0}\delta^{1/\alpha}}),\
\zeta\right)du.\end{array}$

Similarly, near $\vartheta=\pi$, we have

$\begin{array}[]{l}\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!S(\vartheta,\zeta;\delta,\eta)=(\int_{-\eta}^{\eta}+\int_{\pi-\eta}^{\pi+\eta})W(q_{\delta}^{*}(s)-q_{\delta}^{*}(s+\vartheta),\zeta)ds\\\
\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\displaystyle=\int_{-\eta}^{\eta}\\!\\!W(\delta^{1/\alpha}\tilde{q}_{\delta}(\frac{s}{\omega_{0}\delta^{1/\alpha}})+\delta^{1/\alpha}\tilde{q}_{\delta}(\frac{s+\vartheta-\pi}{\omega_{0}\delta^{1/\alpha}})-\pi,\zeta)ds+\int_{\pi-\eta}^{\pi+\eta}\\!\\!W(\pi-\delta^{1/\alpha}\tilde{q}_{\delta}(\frac{s-\pi}{\omega_{0}\delta^{1/\alpha}})-\delta^{1/\alpha}\tilde{q}_{\delta}(\frac{s-2\pi+\vartheta}{\omega_{0}\delta^{1/\alpha}}),\zeta)ds\\\
\displaystyle=2\omega_{0}\delta^{1/\alpha}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}W\left(\pi-\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\vartheta-\pi}{\omega_{0}\delta^{1/\alpha}}),\
\zeta\right)\ ds.\end{array}$

These two formulas can be written in a unified way:

$S(\vartheta,\zeta;\delta,\eta)=2\omega_{0}\delta^{1/\alpha}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}W\left(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta\right)du,$

where one chooses $\sigma=0$, $\hat{\vartheta}=\vartheta$ and the plus sign in
front of $\delta^{1/\alpha}\tilde{q}_{\delta}(u)$ in the case of $\vartheta$
close to zero, and $\sigma=\pi$, $\hat{\vartheta}=\vartheta-\pi$ and the minus
sign in front of $\delta^{1/\alpha}\tilde{q}_{\delta}(u)$ in the case of
$\vartheta$ close to $\pi$.

The first derivative of $S$ with respect to $\vartheta$ is

$\frac{\partial}{\partial\vartheta}S(\vartheta,\zeta;\delta,\eta)=-2\delta^{1/\alpha}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial
W}{\partial
q}\left(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta\right)\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du,$

where $\tilde{p}_{\delta}$ is the derivative of $\tilde{q}_{\delta}$ (see
(139)), so it is bounded with all derivatives by Lemma 4.1. Since the
integrand is bounded (along with all derivatives with respect to $\zeta$), it
follows that $\frac{\partial S}{\partial\vartheta}=O(\eta)$, along with its
derivatives with respect to $\zeta$.

Next, we check the second derivative:

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\begin{array}[]{ll}\frac{\partial^{2}}{\partial\vartheta^{2}}S(\vartheta,\zeta;\delta,\eta)&=2\omega_{0}^{-1}\delta^{1/\alpha}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{2}W}{\partial
q^{2}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\
\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})^{2}du\\\
&-2\omega_{0}^{-1}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial
W}{\partial
q}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\
\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du.\end{array}$

As before, the first line of the right-hand side is $O(\eta)$. Since
$\tilde{p}^{\prime}$ decays as $|u|^{-\alpha-1}$ (see (142)), the integral in
the second line is uniformly convergent. Thus,
$\frac{\partial^{2}S}{\partial\vartheta^{2}}$ is uniformly bounded for all
small $\delta$. The same is true for its derivatives with respect to $\zeta$.

Differentiating further, we obtain that

$\begin{array}[]{l}\frac{\partial^{3}}{\partial\vartheta^{3}}S(\vartheta,\zeta;\delta,\eta)=-2\omega_{0}^{-2}\delta^{\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{3}W}{\partial
q^{3}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\
\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})^{3}du\\\
\qquad+6\omega_{0}^{-2}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{2}W}{\partial
q^{2}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\
\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du\\\
\qquad-2\omega_{0}^{-2}\delta^{-\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial
W}{\partial
q}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\
\tilde{p}_{\delta}^{\prime\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du.\end{array}$
(144)

As above, the first term in the right-hand side is $O(\eta)$. Integrating the
last term by parts, we obtain

$\begin{array}[]{l}\frac{\partial^{3}}{\partial\vartheta^{3}}S_{j}(\vartheta,\zeta;\delta,\eta)=O(\eta)+\\\
\quad\
+2\omega_{0}^{-2}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\\!\\!\frac{\partial^{2}W}{\partial
q^{2}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\
(2\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})\pm\tilde{p}_{\delta}(u))\
\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du\\\
\quad\ -2\delta^{-\frac{1}{\alpha}}[\frac{\partial W}{\partial
q}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\
\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})]_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\end{array}$

Since $\tilde{p}_{\delta}^{\prime}$ decays as $|u|^{-\alpha-1}$, all the terms
here are uniformly bounded, i.e.,
$\frac{\partial^{3}}{\partial\vartheta^{3}}S(\vartheta,\zeta;\delta,\eta)$ is
uniformly bounded, with all its $\zeta$-derivatives.

Similarly, differentiating (144) and integrating by parts, we obtain:

$\begin{array}[]{l}\frac{\partial^{4}}{\partial\vartheta^{4}}S(\vartheta,\zeta;\delta,\eta)=2\omega_{0}^{-3}\delta^{\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{4}W}{\partial
q^{4}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\
\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})^{4}du\\\
\quad\
-12\omega_{0}^{-3}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{3}W}{\partial
q^{3}}(\delta^{1/\alpha}\sigma\pm\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\
\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})^{2}\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du\\\
\quad\
+6\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\\!\\!\frac{\partial^{2}W}{\partial
q^{2}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)\
(\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}))^{2}du\\\
\quad\
+2\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\\!\\!\frac{\partial^{2}W}{\partial
q^{2}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)\
(3\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\delta^{1/\alpha}})\pm\tilde{p}_{\delta}(u))\
\tilde{p}_{\delta}^{\prime\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du\\\
\quad\ -2\omega_{0}^{-3}\delta^{-\frac{2}{\alpha}}[\frac{\partial W}{\partial
q}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)\
\tilde{p}_{\delta}^{\prime\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})]_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\end{array}$

As before, the first term is $O(\eta)$. By (142), we have
$\tilde{p}_{\delta}^{\prime}(u)\approx 1/|u|^{\alpha+1}$ and
$\tilde{p}_{\delta}^{\prime\prime}(u)\approx 1/|u|^{\alpha+2}$ at large $|u|$.
This implies that the integrals in the 2-4 lines are uniformly bounded, so the
second line is $O(1)$ and the third and fourth lines are
$O(\delta^{-\frac{1}{\alpha}})$. The fifth line is of order
O($\delta^{-\frac{2}{\alpha}}\delta^{\frac{\alpha+2}{\alpha}})=O(\delta)$. So
the fourth order derivative with respect to $\vartheta$ diverges, along with
its derivatives with respect to $\zeta$, at most as
$O(\delta^{-\frac{1}{\alpha}})$, in agreement with the claim of the lemma.
Namely

$\begin{array}[]{l}\frac{\partial^{4}}{\partial\vartheta^{4}}S(\vartheta,\zeta;\delta,\eta)=2\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{2}W}{\partial
q^{2}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\
\zeta)\\\
\qquad\qquad\times(3(\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}))^{2}+(3\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\delta^{1/\alpha}})\pm\tilde{p}_{\delta}(u))\tilde{p}_{\delta}^{\prime\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}))du+O(1).\end{array}$
(145)

Finally, we evaluate the fifth derivative, using the same procedure as above
(i.e., twice differentiating (144), integrating by parts, and using the
estimate (142) for the decay of the derivatives of $\tilde{p}$ at large
$|u|$):

$\begin{array}[]{l}\frac{\partial^{5}}{\partial\vartheta^{5}}S(\vartheta,\zeta;\delta,\eta)=-2\omega_{0}^{-4}\delta^{\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{5}W}{\partial
q^{5}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)\
\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})^{5}du\\\
\quad\
+20\omega_{0}^{-4}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{4}W}{\partial
q^{4}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)\
\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})^{3}\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du\\\
\quad\
-30\omega_{0}^{-4}\delta^{-\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{3}W}{\partial
q^{3}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)\
\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})(\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}))^{2}du\\\
\quad\
-20\omega_{0}^{-4}\delta^{-\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{3}W}{\partial
q^{3}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)\
\tilde{p}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})^{2}\tilde{p}_{\delta}^{\prime\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du\\\
\quad\
+20\omega_{0}^{-4}\delta^{-\frac{2}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{2}W}{\partial
q^{2}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)\
\tilde{p}_{\delta}^{\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})\tilde{p}_{\delta}^{\prime\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du\\\
\quad\
+2\omega_{0}^{-4}\delta^{-\frac{2}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\\!\\!\frac{\partial^{2}W}{\partial
q^{2}}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)(4\tilde{p}_{\delta}(u+\\!\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})\pm\tilde{p}_{\delta}(u))\tilde{p}_{\delta}^{\prime\prime\prime}(u+\\!\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})du\\\
\quad\ -2\omega_{0}^{-4}\delta^{-\frac{3}{\alpha}}[\frac{\partial W}{\partial
q}(\sigma\pm\delta^{1/\alpha}\tilde{q}_{\delta}(u)-\delta^{1/\alpha}\tilde{q}_{\delta}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}}),\zeta)\
\tilde{p}_{\delta}^{\prime\prime\prime}(u+\frac{\hat{\vartheta}}{\omega_{0}\delta^{1/\alpha}})]_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\\\
\quad\ =O(\delta^{-\frac{2}{\alpha}})\end{array}$

This completes the proof of the lemma. ∎

######  Lemma 5.3.

Under the parity assumption, the second derivative of the averaged potential
satisfies

$\frac{\partial^{2}}{\partial\vartheta^{2}}W_{avg}(0,\zeta;0)=\frac{\partial^{2}W}{\partial
q^{2}}(0,\zeta),\qquad\qquad\frac{\partial^{2}}{\partial\vartheta^{2}}W_{avg}(\pi,\zeta;0)=-\frac{1}{\pi}\frac{\partial
W}{\partial q}(\pi,\zeta).$ (146)

The fourth derivative satisfies

$\begin{array}[]{ll}\frac{\partial^{4}}{\partial\vartheta^{4}}W_{avg}(0,\zeta;\delta)&=-\;\delta^{-\frac{1}{\alpha}}\
Q_{d}^{\prime}(0)K(\alpha)\ \frac{\partial^{2}W}{\partial q^{2}}(0,\zeta)\
(1+o(1)_{\delta\to 0}),\\\ \\\
\frac{\partial^{4}}{\partial\vartheta^{4}}W_{avg}(\pi,\zeta;\delta)&=\;\;\;\delta^{-\frac{1}{\alpha}}\
Q_{d}^{\prime}(0)K(\alpha)\ \frac{\partial^{2}W}{\partial q^{2}}(\pi,\zeta)\
(1+o(1)_{\delta\to 0}),\end{array}$ (147)

where

$K(\alpha)=\int_{2^{1/\alpha}}^{{\infty}}\
\frac{4\alpha^{2}}{q^{2\alpha+2}\sqrt{1-\frac{2}{q^{\alpha}}}}\ dq\ >\ 0.$
(148)

###### Proof.

Since $W_{avg}(\vartheta,\zeta;\delta)$ tends to the billiard limit in $C^{2}$
as $\delta\to 0$, formulas (146) are just given by (134). So, in order to
prove (146), we only need to calculate the fourth order derivative. As shown
in Lemma 5.2, the derivatives of the regular part $(R$ in (135)) are uniformly
bounded for all $\delta\geqslant\ 0$, and the fourth order derivative of
$W_{avg}$ at $0$ and $\pi$ is dominated by the derivative of $S$. It follows
from (145) at $\vartheta=0$ (hence, $\hat{\vartheta}=0$ and $\sigma=0$) that

$\begin{array}[]{l}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{\partial^{4}}{\partial\vartheta^{4}}S(\vartheta,\zeta;\delta,\eta)|_{\vartheta=0}=2\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\frac{\partial^{2}W}{\partial
q^{2}}(0,\zeta)\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}(3\tilde{p}_{\delta}^{\prime}(u)^{2}+4\tilde{p}_{\delta}(u)\tilde{p}_{\delta}^{\prime\prime}(u))du+O(1)\\\
\qquad\qquad=-2\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\
\frac{\partial^{2}W}{\partial
q^{2}}(0,\zeta)\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}(\tilde{p}_{\delta}^{\prime}(u))^{2}du+8\omega_{0}^{-3}\frac{\partial^{2}W}{\partial
q^{2}}(0,\zeta)\delta^{-\frac{1}{\alpha}}[\tilde{p}_{\delta}(u)\tilde{p}_{\delta}^{\prime}(u)]_{-\eta\omega_{0}\delta^{-\frac{1}{\alpha}}}^{\eta\omega_{0}\delta^{-\frac{1}{\alpha}}}+O(1).\end{array}$

Since $\tilde{p}_{\delta}^{\prime}$ decays sufficiently fast by (142),

$\begin{array}[]{l}\frac{\partial^{4}}{\partial\vartheta^{4}}S(\vartheta,\zeta;\delta,\eta)|_{\vartheta=0}=-2\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\
\frac{\partial^{2}W}{\partial
q^{2}}(0,\zeta)\int_{-\infty}^{\infty}(\tilde{p}_{\delta}^{\prime}(u))^{2}du+O(1).\end{array}$
(149)

Using (140)

$\int_{-\infty}^{\infty}(\tilde{p}_{\delta}^{\prime}(u))^{2}du=\frac{\alpha^{2}\
}{Q_{d}^{\prime}(0)^{2\alpha}}\int_{-\infty}^{+\infty}\tilde{q}_{0}(u)^{-2\alpha-2}\
du+o(1),$ (150)

where $(\tilde{q}_{0}(u),\tilde{p}_{0}(u))$ is the solution of the limit of
the Hamiltonian system (118), namely

$H=\frac{\tilde{p}^{2}}{2}+\frac{1}{(Q_{d}^{\prime}(0)\tilde{q})^{\alpha}}$

at $H=\frac{1}{2}$. Choosing symmetric parameterization of the time $u$ (so
$\tilde{p}_{0}(0)=0$, hence
$\tilde{q}_{0}(0)=2^{1/\alpha}/Q_{d}^{\prime}(0)$), we obtain

$\int_{-\infty}^{+\infty}\tilde{q}_{0}(u)^{-2\alpha-2}\
du=2\int_{2^{1/\alpha}/Q_{d}^{\prime}(0)}^{+\infty}\tilde{q}_{0}^{-2\alpha-2}\;\frac{d\tilde{q}_{0}}{\tilde{p}_{0}}=2Q_{d}^{\prime}(0)^{2\alpha+1}\int_{2^{1/\alpha}}^{+\infty}q^{-2\alpha-2}\
\sqrt{1-\frac{2}{q^{\alpha}}}dq,$

so

$\int_{-\infty}^{\infty}(\tilde{p}_{\delta}^{\prime}(u))^{2}du=2Q_{d}^{\prime}(0)\alpha^{2}\int_{2^{1/\alpha}}^{+\infty}q^{-2\alpha-2}\
\sqrt{1-\frac{2}{q^{\alpha}}}dq+o(1).$

Finally, by (149), and since $\omega_{0}(0)=1$,

$\frac{\partial^{4}}{\partial\vartheta^{4}}S(\vartheta,\zeta;\delta,\eta)|_{\vartheta=0}=-4\delta^{-\frac{1}{\alpha}}\
\frac{\partial^{2}W}{\partial q^{2}}(0,\zeta)\
Q_{d}^{\prime}(0)\alpha^{2}\int_{2^{1/\alpha}}^{+\infty}q^{-2\alpha-2}\
\sqrt{1-\frac{2}{q^{\alpha}}}dq+o(\delta^{-\frac{1}{\alpha}}),$

and (147) follows at $\vartheta=0$.

When $\vartheta=\pi$ (hence, $\hat{\vartheta}=0$ and $\sigma=\pi$ in (145)),
equation (145) gives

$\begin{array}[]{l}\frac{\partial^{4}}{\partial\vartheta^{4}}S(\pi,\zeta;\delta,\eta)=2\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{2}W}{\partial
q^{2}}(\pi-2\delta^{1/\alpha}\tilde{q}_{\delta}(u),\zeta)\
(3(\tilde{p}^{\prime}_{\delta}(u))^{2}+2\
\tilde{p}_{\delta}(u)\tilde{p}^{\prime\prime}_{\delta}(u))du+O(1)\\\
\qquad\qquad=2\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\frac{\partial^{2}W}{\partial
q^{2}}(\pi,\zeta)\ (3(\tilde{p}^{\prime}_{\delta}(u))^{2}+2\
\tilde{p}_{\delta}(u)\tilde{p}^{\prime\prime}_{\delta}(u))du+O(1)\\\
\qquad\qquad=2\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\frac{\partial^{2}W}{\partial
q^{2}}(\pi,\zeta)\int_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}\
(\tilde{p}^{\prime}_{\delta}(u))^{2}du+4\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\frac{\partial^{2}W}{\partial
q^{2}}(\pi,\zeta)\left[\
\tilde{p}_{\delta}(u)\tilde{p}^{\prime}_{\delta}(u)\right]_{-\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}^{\eta/(\omega_{0}\delta^{\frac{1}{\alpha}})}+O(1)\\\
\qquad\qquad=2\omega_{0}^{-3}\delta^{-\frac{1}{\alpha}}\frac{\partial^{2}W}{\partial
q^{2}}(\pi,\zeta)\int_{-\infty}^{\infty}\
(\tilde{p}^{\prime}_{0}(u))^{2}du+O(1).\end{array}$

The last line differs from (149) only by a factor, so (147) follows at
$\vartheta=\pi$. ∎

The above results allow us to characterize the behavior of the averaged
potential $U(\theta,\xi;\delta)$ defined by (125). It is given by Lemma 5.2
that $U(\theta,\xi;\delta)$ depends on $\delta\geqslant 0$ continuously with
the derivatives up to order 2. Therefore, by the non-degenerate minimum
assumption Box2, it has a non-degenerate minimum line near the line (37) for
all small $\delta$:

$\theta^{(n)}=\theta^{(n)}_{min}(\delta)+c,\qquad\xi^{(n)}=\xi^{(n)}_{min}(\delta).$

######  Lemma 5.4.

The potential $U(\theta,\xi;\delta)$ has the following expansion near the
minimum line:

$\begin{split}U(\theta,\xi_{min}(\delta);\delta)&=\sum^{N}_{n=2}\sum^{n-1}_{m=1}\
\gamma_{nm}(\delta)(\theta^{(n)}-\theta^{(m)}-\theta^{(n)}_{min}-\theta^{(m)}_{min})^{2}\\\
&\qquad+\delta^{-\frac{1}{\alpha}}{K(\alpha)}{Q_{d}^{\prime}(0)}\beta_{nm}(\delta)(\theta^{(n)}-\theta^{(m)}-\theta^{(n)}_{min}-\theta^{(m)}_{min})^{4}+\ldots,\end{split}$
(151)

where $\gamma_{nm}(\delta)$ and $\beta_{nm}(\delta)$ tend, as $\delta\to 0$,
to $\gamma_{nm}$ and, respectively, $\beta_{nm}$ defined by (41); the positive
coefficient $K(\alpha)$ is given by (148), and the dots stand for sixth and
higher order terms of the expansion in powers of
$(\theta^{(n)}-\theta^{(m)}-\theta^{(n)}_{min}-\theta^{(m)}_{min})$.

###### Proof.

By (125), the derivatives of $U(\theta,\xi;\delta)$ with respect to
$\theta^{(n)}$ are given by the sum over the corresponding derivatives of
$W_{avg}(\theta^{(n)}-\theta^{(m)},\xi^{(n)}-\xi^{(m)};\delta)$. The second
derivatives tend to those of
$W_{avg}(\theta^{(n)}-\theta^{(m)},\xi^{(n)}-\xi^{(m)};0)$ by Lemma 5.2, so
the values of $\gamma_{nm}$ follow from Lemma 5.3. Since
$W_{avg}(\vartheta,\zeta;\delta)$ is even and $2\pi$-periodic in $\vartheta$,
the third order derivatives with respect to $\vartheta$ vanish at
$\vartheta=0\mod\pi$, whereas the fourth order derivatives are given by Lemma
5.3, which determines the values of $\beta_{nm}$. ∎

###  5.4 The simultaneous impacts case

Since the billiard choreographic solution in the case of simultaneous impacts
satisfies, for any pair of particles,
$\theta_{min}^{(n)}-\theta_{min}^{(m)}=0\mod\pi$, the averaged potential in
the limit $\delta=0$ is not, in general $C^{3}$ smooth, as explained in
Section 5.3. Therefore, the limit $\delta=0$ is singular in this case. Indeed,
we estimated the derivatives of the averaged potential up to order 5 and
showed that starting with order 4 they tend to infinity as $\delta\to 0$.

To deal with this difficulty, we prove Theorem 5 by using different arguments
from those used to prove Theorems 3 and 4. In Sections 5.4.1 and 5.4.2 we
bring the system to the averaged form (122) and estimate the correction term
$\delta\tilde{G}$. In Section 5.4.3 we bring the truncated averaged system to
a Birkhoff normal form; the coefficients of the 4-th order terms in the normal
form diverge as $\delta\to 0$, nevertheless we show that it has a KAM non-
degenerate elliptic periodic orbit for all small $\delta>0$. In Section 5.4.4
we estimate the difference between the Birkhoff normal form for the full and
truncated systems and, under the requirement $\alpha>6$ show that the full
system has an elliptic periodic orbit surrounded by KAM tori.

####  5.4.1 Expansion near the orbit $\mathbf{L}^{*}(\theta,\xi)$ of the
uncoupled system.

######  Lemma 5.5.

In the case of simultaneous-impacts, provided $\alpha>2$ and
$\|\xi^{(n)}-\xi^{(m)}\|>\rho$ for all $n\neq m$, the expansion of the scaled
Hamiltonian (121) near $\mathbf{L}^{*}(\theta,\xi)$ of (123) is of the form

$\begin{split}H&=\sum^{N}_{n=1}\omega_{0}(\delta)I^{(n)}+\delta^{1/2}\sum^{N}_{n=1}\left(\frac{1}{2}a(\delta)(I^{(n)})^{2}+\sum_{i=1}^{d-1}\left[\frac{(p_{\xi,i}^{(n)})^{2}}{2}+V_{i}(\xi_{i}^{(n)})\right]\right)\\\
&\qquad+\delta^{1/2}\mathop{\sum_{n,m=1,\dots,N}}_{n\neq m}\
W(q^{*}_{\delta}(\theta^{(n)})-q^{*}_{\delta}(\theta^{(m)}),\xi^{(n)}-\xi^{(m)})+\delta\hat{G}(I,\theta,\xi;\delta),\end{split}$
(152)

where the derivatives of $\delta\hat{G}$ which include exactly $k$
differentiations with respect to $\theta$ are bounded by
$O(\delta^{1-k/\alpha})$.

###### Proof.

In the box case, for $\alpha>2$, the expansion (94) can be improved to

$\hat{q}_{d}(\delta^{1/2}I,\theta;\delta)=q^{*}_{\delta}(\theta)+\delta^{1/2}G_{d}(\theta,I;\delta)$
(153)

where derivatives of $G_{d}$ which include exactly $k$ differentiations with
respect to $\theta$ are bounded by $O(\delta^{-k/\alpha})$. Indeed, since
$I=0$ corresponds to the periodic orbit $L^{*}_{\delta}$, we have that
$\hat{q}(0,\theta,0):=q^{*}_{\delta}(\theta)$. Away from impacts
$\hat{q}_{d}(\delta^{1/2}I,\theta;\delta)$ is a uniformly smooth function of
$(\delta^{1/2}I,\theta)$, so (153) follows immediately. Near impacts, as the
vertical dynamics are one-dimensional, equation (90) becomes, in the scaled
coordinates (recall that here there are no $\mathcal{P}$ variables):

$\hat{q}_{d}(\delta^{1/2}I,\theta,\delta)=q_{impact}+\delta^{1/\alpha}\tilde{q}(\frac{\tau(\delta^{1/2}I,\theta;\delta)\
-t_{in}(\delta)}{\delta^{1/\alpha}},\mathcal{E}(\delta^{1/2}I);\delta),$

where $q_{impact}$ is either $0$ or $\pi$. Since $(\tau,\mathcal{E)}$ are
smooth functions of $(\theta,I)$, we can expand the above expression at $I=0:$

$\hat{q}_{d}(\delta^{1/2}I,\theta,\delta)=q_{\delta}^{*}(\theta)+\delta^{1/2}I\bar{G}_{d}(\theta,\delta^{1/2-1/\alpha}I;\delta).$

Each derivative with respect to $\theta$ generates a $\delta^{-1/\alpha}$
factor whereas each derivative with respect to $I$ generates a
$\delta^{1/2-1/\alpha}$ factor. Hence, for $\alpha>2$, the derivatives of
$G_{d}=I\bar{G}_{d}$ which include exactly $k$ derivatives with respect to
$\theta$ are bounded by $O(\delta^{-k/\alpha})$, as claimed.

Since $\|\xi^{(n)}-\xi^{(m)}\|>\rho$ for all $n\neq m$, the interaction
potential $W$ at any point along $\mathbf{L}^{*}$ is smooth, so, plugging
(153) in the interaction term of (121) gives

$W(\hat{q}_{d}(\delta^{1/2}I^{(n)},\theta^{(n)};\delta)-\hat{q}_{d}(\delta^{1/2}I^{(m)},\theta^{(m)};\delta),\xi^{(n)}-\xi^{(m)}))=W(q_{\delta}^{*}(\theta^{(n)})-q_{\delta}^{*}(\theta^{(m)}),\xi^{(n)}-\xi^{(m)}))+$
$+\delta^{1/2}\hat{G}_{d}(\theta^{(n)},\theta^{(m)},\delta^{1/2}I^{(n)},\delta^{1/2}I^{(m)},\xi^{(n)}-\xi^{(m)};\delta),$

where derivatives of $\hat{G}_{d}$ which include exactly $k$ derivatives with
respect to $\theta^{(n)}$ or $\theta^{(m)}$ are of order $\delta^{-k/\alpha}$.
Hence, (121), becomes of the required form (152), where $\hat{G}$ is the sum
of such functions $\hat{G}_{d}$ and regular terms coming from the
$O((\delta^{1/2}I^{(n)})^{3})$ terms of (119). ∎

####  5.4.2 Regularity of averaging

We average the Hamiltonian (152) by using the same transformation (111) as in
the non-simultaneous impacts case. To this aim, we establish that $\Psi$ of
(111) and its derivatives have the correct regularity as $\delta\rightarrow
0$. We first show that

$\Psi_{1}(u,u_{0},v;\xi^{(n)}-\xi^{(m)};\delta)=\int_{u_{0}}^{u}W(q^{*}_{\delta}(s+v)-q^{*}_{\delta}(s-v),\xi^{(n)}-\xi^{(m)})ds$
(154)

satisfies the following result (the weak regularity of Lemma 4.8 is replaced
by weaker estimates on the derivatives, thus allowing both particles to visit
simultaneously the impact regions):

######  Lemma 5.6.

Provided $\|\xi^{(n)}-\xi^{(m)}\|>\rho$, the derivatives of the function
$\Psi_{1}(\frac{\theta^{(n)}+\theta^{(m)}}{2},u_{0,}\frac{\theta^{(n)}-\theta^{(m)}}{2},\xi^{(n)}-\xi^{(m)};\delta)$
which include $k>0$ differentiations with respect to
$\theta^{(n)},\theta^{(m)}$ are of order $\delta^{-(k-1)/\alpha}$.

###### Proof.

Since $\|\xi^{(n)}-\xi^{(m)}\|>\rho$, and the $\xi$ variables are fixed along
the integration interval, $W$ is smooth and bounded function of its arguments.
So all the derivatives with respect to $\xi$ are bounded. When both particles
are away from impacts (i.e. when the argument of $q^{*}_{\delta}$ is away from
$\\{0,\pi\\}$), all the derivatives of $q^{*}_{\delta}$ are bounded.
Otherwise, by (92), the $k$-th derivative of $q^{*}_{\delta}(\theta)$ is of
order $\delta^{-(k-1)/\alpha}$, so, the same property is shared by
$W(q^{*}_{\delta}(s+\frac{\theta^{(n)}-\theta^{(m)}}{2})-q^{*}_{\delta}(s-\frac{\theta^{(n)}-\theta^{(m)}}{2}),\xi^{(n)}-\xi^{(m)})$
and its integral, $\Psi_{1}$. ∎

By (116), the function $W_{avg}$ has the same regularity properties as
$\Psi_{1}$. Now perform the transformation defined by $\Psi(;\delta)$ of
(111):
$(I^{(n)},p_{\xi}^{(n)})\rightarrow(I^{(n)},p_{\xi}^{(n)})-\delta^{1/2}(\partial_{\theta^{(n)}},\partial_{\xi^{(n)}})\Psi$.
Notice that here (as opposed to the non-simultaneous impacts case) we apply
the transormation to the non-truncated Hamiltonian (152) (i.e. we do not drop
the correction term $\delta\hat{G}$). Though we formally obtain the same
averaged system (122), now the $\delta\tilde{G}$ term includes the transformed
$\delta\hat{G}$ term. The derivatives with respect to $\xi$ and $I$ of the
$\delta\tilde{G}$ term remain of order $\delta$ whereas its derivatives which
include exactly $k$ differentiations with respect to $\theta^{(n)}$,
$n=1,\ldots,N$, are of order $\delta^{1-k/\alpha}$. In particular, for
$\alpha>6$ the correction term $\delta\tilde{G}$ in (122) and its derivatives
up to order 3 are $o(\delta^{1/2})$.

####  5.4.3 Normal form of the truncated averaged system.

Let us study now the truncation of system (122), namely the system (128) for
the case of simultaneous impacts. This system is translation-invariant (the
Hamiltonian does not change when the same constant is added to all
$\theta^{(n)}$). By Lemma 5.2, the potential $U_{\delta}(\theta,\xi)$ in (128)
has, even for the simultaneous impacts case, a $C^{2}$ limit at $\delta=0$. By
the non-degeneracy assumption Box2, the potential $U$ has a non-degenerate
line of minima for all small $\delta$. This line corresponds to an elliptic
periodic orbit777This orbit is translation invariant, i.e., it is a relative
equilibrium. of system (128). Our goal here is to establish the KAM-
nondegeneracy of it. This amounts to bringing the Poincaré map near the
periodic orbit to the 4-th order Birkhoff normal form (Lemma 5.8 below) and
verification of the twist condition (Lemma 5.9).

We introduce local normal coordinates
$(\theta^{(n)},I^{(n)})\rightarrow(\varphi,\psi,P,J)$ near the periodic orbit
in the same way as in (51),(52). Here
$\varphi=\frac{1}{N}(\theta^{(1)}+\ldots+\theta^{(N)})$, and $P$ is
symplectically conjugate to $\varphi$. The variables $\psi$ vary near zero and
are linear combinations of
$((\theta^{(n)}-\theta^{(n)}_{min})-(\theta^{(m)}-\theta^{(m)}_{min}))$,
$n,m=1,\ldots,N$; the variables $J$ are symplectically conjugate to $\psi$ and
also vary near zero. As in Section 5.2, the Poincaré return map of (128) at
the level of fixed $P$, equals to the $O(\delta^{1/2})$-map of the flow
defined by the Hamiltonian

$H_{P}(J,\psi,p_{\xi},\xi;\delta)=\frac{1}{2N}a(\delta)J^{2}+\
\frac{p_{\xi}^{2}}{2}+\hat{U}_{\delta}(\psi,\xi),$ (155)

where $\hat{U}_{\delta}$ is the reduced potential of (129). This Hamiltonian
is exactly of the same form as $H_{P}$ of (131), yet, the regularity
properties of the reduced potential $\hat{U}_{\delta}(\psi,\xi)$ are different
in the simultaneous impacts case.

By the parity assumption Box4,
$W_{avg}(\theta^{(n)}-\theta^{(m)},\xi^{(n)}-\xi^{(m)};\delta)$ is even and
$2\pi$-periodic in $\theta$, so the phase differences along the minimum line
remain locked at $\theta^{(n)}=0\mod\pi$, $n=1,\ldots,N$, for all small
$\delta$. Therefore, as $\psi$ is a linear function of
$\theta^{(n)}-\theta^{(m)}-\theta^{(n)}_{min}-\theta^{(m)}_{min}$, the reduced
potential is even in $\psi$:

$\hat{U}_{\delta}(\psi,\xi)=\hat{\hat{U}}_{\delta}(-\psi,\xi).$ (156)

By this symmetry,

$\frac{\partial^{j+l}}{(\partial\psi)^{j}(\partial\xi)^{l}}\hat{U}_{\delta}(\psi,\xi)|_{\psi=0,\xi=\xi_{min}}=0\quad\mbox{for
odd}\;\;j.$ (157)

In particular,
$\frac{\partial^{2}}{\partial\psi\partial\xi}\hat{U}_{\delta}(\psi,\xi)|_{\psi=0,\xi=\xi_{min}}=0$,
so at $\psi=0,\xi=\xi_{min}$ the quadratic part of the Hamiltonian (155) is
block-diagonal. Therefore, the expansion of the Hamiltonian (155) up to order
four terms is of the form

$H_{4jet}=H^{z}_{2}(z,p;\delta)+H^{\theta}_{2}(\psi,J;\delta)+H_{3}^{\theta
z}(\psi,z;\delta)+H^{z}_{3}(z;\delta)+H^{z}_{4}(z;\delta)+H_{4}^{\theta
z}(\psi,z;\delta)+\delta^{-\frac{1}{\alpha}}H_{4}^{\theta}(\psi;\delta),$
(158)

where, hereafter, $z=\xi-\xi_{min},p=p_{\xi}$, and $H_{j}$ denotes a
homogeneous polynomial of order $j$. By Lemma 5.4, the coefficients of $H_{j}$
are uniformly bounded for all $\delta\geqslant 0$.

Define the $z$-Hamiltonian:

$H^{z}(z,p;\delta)=\frac{p^{2}}{2}+\hat{U}_{\delta}(0,\xi_{min}+z)=H^{z}_{2}(z,p;\delta)+H^{z}_{3}(z;\delta)+H^{z}_{4}(z;\delta)+O(z^{5}),\qquad$
(159)

and the $\theta$-Hamiltonian:

$H^{\theta}(\psi,J;\delta)=\frac{1}{2N}aJ^{2}+\hat{U}_{\delta}(\psi,\xi_{min})=H^{\theta}_{2}(\psi,J;\delta)+\delta^{-\frac{1}{\alpha}}H_{4}^{\theta}(\psi;\delta)+\ldots,\qquad$
(160)

where, hereafter, the dots stand for terms of order 5 and higher (with
coefficients that may diverge as $\delta\rightarrow 0$). Then the Hamiltonian
(155) is of the form

$H_{P}=H^{z}(z,p;\delta)+H^{\theta}(\psi,J;\delta)+H_{3}^{\theta
z}(\psi,z;\delta)+H_{4}^{\theta z}(\psi,z;\delta)+\ldots,$ (161)

where, by the symmetry $\psi\rightarrow-\psi$ (see (156)), the polynomials
$H_{3,4}^{\theta z}(\psi,z;\delta)$ have only monomials which are quadratic in
$\psi$.

Denote the fourth order Birkhoff normal form of the Hamiltonian (159) near the
fixed point at the origin by $H^{z}_{NF}(z,p;\delta)$ and the Birkhoff normal
form of the Hamiltonian (160) by $H^{\theta}_{NF}(\psi,J;\delta)$. Recall that
in such normal forms all the terms up to order 4 are resonant – all non-
resonant terms up to order 4 are eliminated by a sequence of symplectic
coordinate transformations. Each of these transformations is a time-1 map of a
certain polynomial Hamiltonian whose coefficients depend smoothly on the
coefficients of the 4-jet of the original system, see e.g. Chapter 7 of [3].
It is a well-known fact that the normalizing transformations can be chosen
such that the linear symmetries of the system are preserved. In particular,
the symmetry $(\psi,J)\to-(\psi,J)$ survives the transformations we describe
below.

######  Lemma 5.7.

There exists a symplectic transformation which depends continuously on
$\delta>0$ and brings the Hamiltonian $H^{\theta}$ of (160) to the normal form

$H^{\theta}_{NF}(\psi,J;\delta)=\tilde{H}^{\theta}_{2}(\psi,J;\delta)+\delta^{-\frac{1}{\alpha}}K(\alpha)\tilde{H}^{\theta}_{4}(\psi,J;\delta)+\dots\
,$ (162)

where $\tilde{H}^{\theta}_{k}$ are homogeneous polynomials of $(\psi,J)$ of
degree $k$ ($k=2,4$) with bounded coefficients depending continuously on
$\delta$. The polynomials $\tilde{H}^{\theta}_{2,4}$ have a limit at
$\delta=0$ such that the normal form of the Hamiltonian $H^{\theta,0}$ of
Assumption Box5 is

$H^{\theta,0}_{NF}(\psi,J)=\tilde{H}^{\theta}_{2}(\psi,J;0)+\tilde{H}^{\theta}_{4}(\psi,J;0)+\ldots~{}.$
(163)

###### Proof.

The normal form transformation we are going to describe is a composition of
several symplectic transformations. First, we do a linear transformation which
diagonalizes the quadratic part of the $\theta$-Hamiltonian (160). By Lemma
5.4, the quadratic part of the Hamiltonian (160) is, in the limit $\delta=0$,
identical to the quadratic part of $H^{\theta,0}$. Therefore, by Assumption
Box2, all the frequencies are distinct for small $\delta$; hence the quadratic
part of (160) is indeed symplectically diagonalizable and the limit of the
diagonalizing transformation at $\delta=0$ also diagonalizes the quadratic
part of $H^{\theta,0}$. Since the transformation is linear, it does not
introduce third order terms.

Thus, the Hamiltonian (160) becomes
$H^{\theta}(\psi,J;\delta)=\tilde{H}^{\theta}_{2}(\psi,J;\delta)+\delta^{-\frac{1}{\alpha}}K(\alpha)\hat{H}^{\theta}_{4}(\psi,J;\delta)+\ldots$,
where $\hat{H}^{\theta}_{4}(\psi,J;0)$ coincides with the 4-th order terms of
$H^{\theta,0}$ after the diagonalization. Next, we do a symplectic
transformation in order to eliminate all non-resonant fourth order terms,
i.e., to bring the $\theta$-Hamiltonian to its normal form $H_{NF}^{\theta}$.
It is well-known (see [3], Chapter 7) that such normalizing transformation
does not alter the resonant terms of order 4 or lower, hence,
$\tilde{H}^{\theta}_{4}(\psi,J;\delta)$ is obtained from
$\hat{H}^{\theta}_{4}(\psi,J;\delta)$ by throwing away the non-resonant terms.
It follows that even though the normalizing transformation does not have a
limit888The transformation is the time-1 map,
$(\psi,J)\to(\psi,J)+\delta^{-\frac{1}{\alpha}}K(\alpha)(\partial_{J}S,-\partial_{\psi}S)+O((\psi,J)^{4})$,
of a Hamiltonian flow defined by the Hamilton function
$\delta^{-\frac{1}{\alpha}}K(\alpha)S(\psi,J;\delta)$, where $S$ is a fourth-
degree homogeneous polynomial with bounded coefficients which solves the
equation
$\\{\tilde{H}_{2}^{\theta},S\\}=\tilde{H}^{\theta}_{4}-\hat{H}^{\theta}_{4}$.
The normalizing transformation for $H^{\theta,0}$ is the time-1 map for the
Hamiltonian $S(\psi,J;0)$. at $\delta=0$, the term
$\tilde{H}^{\theta}_{4}(\psi,J;\delta)$ has a limit, equal to the resonant
part of $\hat{H}^{\theta}_{4}(\psi,J;0)$, so the lemma follows. ∎

######  Lemma 5.8.

The Birkhoff normal form of the reduced Hamiltonian (155) is the sum of the
normal forms of the Hamiltonians (159) and (160) with bounded corrections of
order 4 which vanish at $(J,\psi)=0$:

$H_{NF}=H^{z}_{NF}(z,p;\delta)+H^{\theta}_{NF}(\psi,J;\delta)+H_{2}^{\psi
J}(\psi,J;\delta)H_{2}^{zp}(z,p;\delta)+H_{4}^{\psi J}(\psi,J;\delta)+\ldots\
.$ (164)

###### Proof.

Consider the symplectic transformation which brings the Hamiltonian
$H^{z}=H^{z}_{2}(z,p;\delta)+H^{z}_{3}(z;\delta)+H^{z}_{4}(z;\delta)+\dots$ to
its Birkhoff normal form up to order 4:

$H^{z}_{NF}(z,p;\delta)=\tilde{H}^{z}_{2}(z,p;\delta)+\tilde{H}^{z}_{4}(z,p;\delta)+\ldots\
.$ (165)

Since, by Lemma 5.2, the coefficients of all monomials in $H^{z}$ up to order
4 have well-defined finite limits as $\delta\rightarrow 0$, this
transformation also has a well-defined finite limit. Applying this
transformation to $H_{P}$ of (155) (equivalently, to (161)), we obtain, in the
new coordinates,

$\begin{array}[]{l}H_{P}=\tilde{H}^{z}_{2}(z,p;\delta)+\tilde{H}^{z}_{4}(z,p;\delta)+H^{\theta}_{2}(\psi,J;\delta)+\hat{H}_{3}^{\theta
z}(\psi,z,p;\delta)\\\ \qquad\qquad\qquad+\hat{H}_{4}^{\theta
z}(\psi,z,p;\delta)+\delta^{-1/\alpha}H_{4}^{\theta}(\psi;\delta)+\ldots,\end{array}$
(166)

where all monomials in $\hat{H}_{3,4}^{\theta z}(\psi,z,p;\delta)$ are
quadratic in $\psi$ (because of the symmetry $\psi\to-\psi$). Since all the
terms in $\hat{H}_{3}^{\theta z}(\psi,z,p;0)$ are non-resonant by Assumption
Box5 (as at the minimum of $\hat{U}$, the frequencies have no resonances of
third order), they remain non-resonant for small $\delta$. Therefore, there
exists a symplectic transformation
$(z,p)\rightarrow(z,p)+O((\psi,J)^{2}),(\psi,J)\rightarrow(\psi,J)+O((z,p))\cdot(\psi,J)$
which eliminates the cubic terms and brings the Hamiltonian $H_{P}$ to the
form

$)H_{P}=\tilde{H}^{z}_{2}(z,p;\delta)+\tilde{H}^{z}_{4}(z,p;\delta)+H^{\theta}_{2}(\psi,J;\delta)+{\bar{H}}_{4}^{\theta
z}(\psi,J,z,p;\delta)+\bar{H}_{4}^{\theta}(\psi,J;\delta)+\delta^{-1/\alpha}H_{4}^{\theta}(\psi;\delta)+\ldots,$
(167)

where ${\bar{H}}_{4}^{\theta z}(\psi,J,z,p;\delta)$ is a sum of monomials
quadratic in $(\psi,J)^{2}$ and $(z,p)^{2}$.

Now, we apply to $H_{P}$ the transformation of Lemma 5.7, which brings the
Hamiltonian
$H^{\theta}_{2}(\psi,J;\delta)+\delta^{-1/\alpha}H_{4}^{\theta}(\psi;\delta)+\dots$
to its Birkhoff normal form. This brings $H_{P}$ to the form (164), with some
of the 4th order terms in $H_{2}^{\psi J}$, $H_{2}^{zp}$, and $H_{4}^{\psi J}$
possibly non-resonant. All such non-resonant terms are eliminated by a
symplectic transformation without changing all other terms up of order four or
less. ∎

Because there are no resonances up to order 4 by Assumption Box5, the 4-jets
of the Birkhoff normal forms depend only on actions. Namely, we introduce
actions

$\begin{array}[]{l}I_{z_{j}}=\frac{1}{2}(z_{j}^{2}+p_{j}^{2}),\qquad
j=1,\ldots N(d-1),\\\ I_{\psi_{k}}=\frac{1}{2}(\psi_{k}^{2}+J_{k}^{2}),\qquad
k=1,\dots,N-1.\end{array}$ (168)

Then

$\begin{array}[]{ll}H^{z}_{NF}&=\omega_{z}(\delta)I_{z}+I_{z}^{T}B_{z}(\delta)I_{z}+\ldots,\\\
H^{\theta}_{NF}&=\omega_{\psi}(\delta)I_{\psi}+\delta^{-1/\alpha}K(\alpha)I_{\psi}^{T}B_{\psi}(\delta)I_{\psi}+\ldots,\end{array}$

where, by Lemmas 5.2 and 5.7 $\omega_{z}(\delta)$, $\omega_{\psi}(\delta)$,
$B_{z}(\delta)$, $B_{\psi}(\delta)$ all are continuous functions with bounded
limits at $\delta=0$. By Assumption Box5, the determinants of $B_{z}(0)$ and
$B_{\psi}(0)$ are non-zero.

Moreover, by Lemma 5.7, the Hamiltonian $H^{\theta,0}_{NF}$ is of the form

$H^{\theta,0}_{NF}=\omega_{\psi}(0)I_{\psi}+I_{\psi}^{T}B_{\psi}(0)I_{\psi}+\ldots\
.$

Also, by Lemma 5.8, the Birkhoff normal form $H_{NF}$ of (155) is given by

$H_{NF}=H^{z}_{NF}+H^{\psi}_{NF}+O(\|I_{z}\|+\|I_{\psi}\|)\cdot\
I_{\psi}+\ldots\ .$

######  Lemma 5.9.

The Birkhoff normal form of the Hamiltonian (155) is non-degenerate.

###### Proof.

We need to check that the twist condition for $H_{NF}$ is satisfied. Indeed,

$\begin{array}[]{ll}\det(\frac{\partial^{2}H_{NF}}{\partial
I^{2}})&=\begin{vmatrix}B_{z}(\delta)&O(1)\\\
O(1)&\delta^{-1/\alpha}K(\alpha)B_{\psi}(\delta)+O(1)\\\
\end{vmatrix}=\delta^{-2N/\alpha}K(\alpha)^{2N}\begin{vmatrix}B_{z}(\delta)&O(1)\\\
O(\delta^{1/\alpha})&B_{\psi}(\delta)+O(\delta^{1/\alpha})\end{vmatrix}\\\ \\\
&=\delta^{-2N/\alpha}K(\alpha)^{2N}(|B_{z}(\delta)|\cdot|B_{\psi}(\delta)|+O(\delta^{1/\alpha})).\end{array}$
(169)

Since the determinants $|B_{z}(\delta)|$ and $|B_{\psi}(\delta)|$ are bounded
away from zero for all sufficiently small $\delta$, we have
$\det(\frac{\partial^{2}H_{NF}}{\partial I^{2}})\neq~{}0$, i.e. the twist
condition is satisfied, so $H_{NF}$ is non-degenerate. ∎

####  5.4.4 Normal form of the full system.

We have shown that the truncation of system (122), namely system (128) for the
case of simultaneous impacts, has a stable KAM non-degenerate elliptic
periodic orbit for all $\delta>0$. To complete the proof of Theorem 5, we need
to show that the periodic orbit persists in the full system (122), which
includes the error term $\delta\tilde{G}$, and also remains KAM-non-
degenerate. This is non-trivial, as both the normal form and the error term
$\delta\tilde{G}$ diverge in $C^{4}$ as $\delta\rightarrow 0$. While the
persistence of the periodic orbit follows from the $C^{2}$ regularity of the
normal form and the error term, the KAM non-degeneracy is established by
revisiting the normal form computation of Section 5.4.3 applied to the full
system.

Rewriting (122) with the use of the coordinates
$(P,\varphi,J,\psi,p_{\xi},\xi)$ of (130), we obtain the perturbed
Hamiltonian:

$\begin{array}[]{ll}H(P,\varphi,J,\psi,p_{\xi},\xi;\delta)&=\omega_{0}P+\delta^{1/2}(\frac{1}{2N}aP^{2}+\frac{1}{2N}aJ^{2}+\frac{p_{\xi}^{2}}{2}+\hat{U}_{\delta}(\psi,\xi))\\\
&\qquad\qquad+\delta\tilde{G}(P,\varphi,J,\psi,p_{\xi},\xi;\delta).\end{array}$
(170)

The corresponding system is:

$\begin{array}[]{ll}\dot{\varphi}&=\omega_{0}+\frac{1}{N}a\delta^{1/2}P+\delta\frac{\partial}{\partial
P}\tilde{G}(P,\varphi,J,\psi,p_{\xi},\xi;\delta),\\\
\dot{P}&=-\delta\frac{\partial}{\partial\varphi}\tilde{G}(P,\varphi,J,\psi,p_{\xi},\xi;\delta),\\\
\dot{\psi}&=\frac{1}{N}a\delta^{1/2}J+\delta\frac{\partial}{\partial
J}\tilde{G}(P,\varphi,J,\psi,p_{\xi},\xi;\delta),\\\
\dot{J}&=-\delta^{1/2}\frac{\partial}{\partial\psi\
}\hat{U}_{\delta}(\psi,\xi)-\delta\frac{\partial}{\partial\psi\
}\tilde{G}(P,\varphi,J,\psi,p_{\xi},\xi;\delta),\\\ {\dot{\xi}}&=\delta^{1/2}\
p_{\xi}+\delta\frac{\partial}{\partial
p_{\xi}}\tilde{G}(P,\varphi,J,\psi,p_{\xi},\xi;\delta),\\\
{\dot{p}_{\xi}}&=-\delta^{1/2}\frac{\partial}{\partial\xi\
}\hat{U}_{\delta}(\psi,\xi)-\delta\frac{\partial}{\partial\xi\
}\tilde{G}(P,\varphi,J,\psi,p_{\xi},\xi;\delta).\\\ \end{array}$ (171)

Restricting to a given energy level (so $P$ is determined by all other
variables) and using $\bar{\varphi}=\delta^{1/2}\varphi$ as a new time:

$\begin{array}[]{ll}\frac{d\psi}{d\bar{\varphi}}&=\frac{1}{N\omega_{0}}aJ+\delta^{1/2}G_{1}(\bar{\varphi},J,\psi,p_{\xi},\xi;\delta),\\\
\frac{dJ}{d\bar{\varphi}}&=-\frac{1}{\omega_{0}}\frac{\partial}{\partial\psi\
}\hat{U}_{\delta}(\psi,\xi)-\delta^{\frac{1}{2}-\frac{1}{\alpha}}G_{2}(\bar{\varphi},J,\psi,p_{\xi},\xi;\delta),\\\
\frac{d\xi}{d\bar{\varphi}}&=\
\frac{1}{\omega_{0}}p_{\xi}+\delta^{1/2}G_{3}(\bar{\varphi},J,\psi,p_{\xi},\xi;\delta),\\\
\frac{dp_{\xi}}{d\bar{\varphi}}&=-\frac{1}{\omega_{0}}\frac{\partial}{\partial\xi\
}\hat{U}_{\delta}(\psi,\xi)-\delta^{1/2}G_{4}(\bar{\varphi},J,\psi,p_{\xi},\xi;\delta),\end{array}$
(172)

where
$\frac{\partial}{\partial\psi^{k}}G_{i}(\bar{\varphi},J,\psi,p_{\xi},\xi;\delta)=O(\delta^{-k/\alpha})$,
see the end of Section 5.4.2. The reduced system (172) is a fast-oscillating
non-autonomous Hamiltonian system with the Hamilton function

$\begin{array}[]{ll}H_{reduced}(\bar{\varphi},J,\psi,p_{\xi},\xi;\delta)&=\frac{1}{2N\omega_{0}}aJ^{2}+\
\frac{p^{2}}{2\omega_{0}}+\frac{1}{\omega_{0}}\hat{U}_{\delta}(\psi,\xi)+\delta^{1/2}G_{5}(\bar{\varphi},J,\psi,p_{\xi},\xi;\delta)\\\
&=\frac{1}{\omega_{0}}H_{P}(J,\psi,p_{\xi},\xi;\delta)+\delta^{1/2}G_{5}(\bar{\varphi},J,\psi,p_{\xi},\xi;\delta),\end{array}$
(173)

where $H_{P}$ is the Hamiltonian (155). As above, $G_{5}$ is a function with
bounded derivatives with respect to $J,p_{\xi},\xi$, whereas making $k$
differentiations of $G_{5}$ with respect to $\psi$ introduces a factor of
$O(\delta^{-k/\alpha})$. Now, as apposed to (155), $H_{reduced}$ is a periodic
function of the time variable $\bar{\varphi}$, with the period
$2\pi\delta^{1/2}$.

######  Lemma 5.10.

Provided that $\alpha>6$, the system (172) has a KAM-nondegenerate elliptic
periodic orbit.

###### Proof.

As we mentioned in Section 5.4.3, due to the non-resonance assumption of Box2
and the continuous dependence of the quadratic part of the Hamiltonian
$H_{P}(J,\psi,p_{\xi},\xi;\delta)$ on $\delta$, the truncated system (155) has
a non-resonant elliptic fixed point at the origin. Since for $\alpha>4$ the
perturbation to (155), i.e. the term $\delta^{1/2}G_{5}$ in the Hamiltonian
(173), is $C^{2}$-small, the system (172) has, for all small $\delta,$ an
elliptic periodic orbit
$(J,\psi,p_{\xi},\xi)=\delta^{\frac{1}{2}-\frac{1}{\alpha}}X_{p}(\bar{\varphi};\delta)$
where $X_{p}(\bar{\varphi};\delta)$ is a continuous bounded function for all
$\delta\geqslant 0$. Let
$(J,\psi,p_{\xi},\xi)=\delta^{\frac{1}{2}-\frac{1}{\alpha}}X_{p}(\bar{\varphi};\delta)+X$.
Then

$H(\bar{\varphi},X;\delta):=H_{reduced}(\delta^{\frac{1}{2}-\frac{1}{\alpha}}X_{p}(\bar{\varphi};\delta)+X;\delta)=H_{P}(X;\delta)+G_{6}(\bar{\varphi},X;\delta),$
(174)

where the non-autonomous term $G_{6}$ is given by

$G_{6}(\bar{\varphi},X;\delta)=H_{P}(\delta^{\frac{1}{2}-\frac{1}{\alpha}}X_{p}(\bar{\varphi};\delta)+X;\delta)-H_{P}(X;\delta)+\delta^{\frac{1}{2}}G_{5}(\bar{\varphi},\delta^{\frac{1}{2}-\frac{1}{\alpha}}X_{p}(\bar{\varphi};\delta)+X;\delta).$

Next we show that if $\alpha>6$, the term $G_{6}(\bar{\varphi},X;\delta)$ and
its derivatives with respect to $X$ including up to 3 derivatives with respect
to $\psi$ are small, while its 4th derivatives with respect to $\psi$ are
$o(\delta^{-1/\alpha})$:

$\frac{\partial^{k+l}}{\partial\psi^{k}\partial(J,p_{\xi},\xi)^{l}}G_{6}=O(\delta^{\frac{1}{2}-\frac{k}{\alpha}}),\quad\
k=0,1,2,3,4.$ (175)

Indeed, first recall that $\frac{\partial}{\partial\psi^{k}}G_{5}$ is
$O(\delta^{-\frac{k}{\alpha}}),$ so for $\alpha>6$ the term
$\delta^{\frac{1}{2}}G_{5}$ gives a correct contribution to the estimates
(175). Next, by Lagrange formula,

$H_{P}(\delta^{\frac{1}{2}-\frac{1}{\alpha}}X_{p}(\bar{\varphi};\delta)+X;\delta)-H_{P}(X;\delta)=\delta^{\frac{1}{2}-\frac{1}{\alpha}}X_{p}(\bar{\varphi};\delta)\cdot\int_{0}^{1}\nabla
H_{P}(X+s\delta^{\frac{1}{2}-\frac{1}{\alpha}}X_{p}(\bar{\varphi};\delta);\delta)ds.$

Recall that by Lemma 5.2 the potential $\hat{U}_{\delta}(\psi,\xi)$, hence the
Hamiltonian $H_{P}$, has bounded derivatives up to order 3, yet its fourth
derivatives with respect to $\psi$ are of order $\delta^{-\frac{1}{\alpha}}$
and its fifth derivatives are of order $\delta^{-\frac{2}{\alpha}}$. Hence, by
(5.4.4),
$\frac{\partial}{\partial\psi^{k}}(H_{P}(\delta^{\frac{1}{2}-\frac{1}{\alpha}}X_{p}(\bar{\varphi};\delta)+X;\delta)-H_{P}(X;\delta)))$
for $k=0,1,2$ is of order $O(\delta^{\frac{1}{2}-\frac{1}{\alpha}})$, whereas
for $k=3$ the derivatives are of order
$\delta^{\frac{1}{2}-\frac{2}{\alpha}}$, and for $k=4$ they are of order
$\delta^{\frac{1}{2}-\frac{3}{\alpha}}$, all in agreement with (175).

In order to align with the notations of Section 5.4.3, we denote hereafter the
coordinates of $X$ by $(J,\psi,p,z)$. Then, by (161) and (174)

$\begin{array}[]{l}H(\bar{\varphi},X;\delta)=H^{z}_{2}(z,p;\delta)+H^{z}_{3}(z;\delta)+H^{z}_{4}(z;\delta)+H^{\theta}_{2}(\psi,J;\delta)+H_{3}^{\theta
z}(\psi,z;\delta)\\\ \qquad\qquad\qquad+H_{4}^{\theta
z}(\psi,z;\delta)+\delta^{-1/\alpha}H_{4}^{\theta}(\psi;\delta)+G_{6}(\bar{\varphi},X;\delta),\end{array}$
(176)

where the functions $H_{j}$ of (161) are homogeneous polynomials of degree $j$
with bounded coefficients (continuous in $\delta$) and $H_{3,4}^{\theta
z}(\psi,z;\delta)$ have only monomials quadratic in $\psi$.

Recall that $H_{P}$ was brought to its normal form by a sequence of coordinate
transformations described in Lemma 5.8. Next, we apply similar transformation
to the full Hamiltonian $H(\bar{\varphi},X;\delta)$ of (174). The difference
is that $H$ is periodic in time $\bar{\varphi}$, so the normalizing
transformations are also periodic in $\bar{\varphi}$. It is well-known that
one can make the coordinate transformations such that the resulting
Hamiltonian will become autonomous up to any given order in $X$, i.e. such a
transformation corresponds to averaging to any given order [3]. Our goal is to
bring $H(\bar{\varphi},X;\delta)$ to an autonomous normal form up to order $4$
and compare it with the 4-th order normal form $H_{NF}$ (see (164)) of the
autonomous Hamiltonian $H_{P}$. Since the period is small, of order
$\delta^{1/2}$, there are no additional resonant terms due to the dependence
on $\bar{\varphi}$. Still, one needs to check that the singular terms in (176)
do not destroy the twist condition.

Let us make the linear symplectic transformations which bring the Hamiltonians
$H^{\theta}_{2}(\psi,J;\delta)$ and $H^{z}_{2}(z,p;\delta)$ to the diagonal
form $\tilde{H}^{\theta}_{2}(\psi,J;\delta)$ and
$\tilde{H}^{z}_{2}(z,p;\delta)$ as in (162) and (165), respectively. Applying
these transformations to (176), we obtain a new Hamiltonian

$\begin{array}[]{ll}H(\bar{\varphi},X;\delta)=\tilde{H}^{z}_{2}(z,p;\delta)+H^{z}_{3}(z,p;\delta)+H^{z}_{4}(z,p;\delta)+\tilde{H}^{\theta}_{2}(\psi,J;\delta)+H_{3}^{\theta
z}(\psi,J,z,p;\delta)\\\ \qquad\qquad\qquad+H_{4}^{\theta
z}(\psi,J,z,p;\delta)+\delta^{-1/\alpha}H_{4}^{\theta}(\psi,J;\delta)+G_{7}(\bar{\varphi},X;\delta),\end{array}$
(177)

where $H_{j}$ are different functions from those in (176), yet they keep the
same structure - they are homogeneous polynomials of degree $j$ with bounded
coefficients (continuous in $\delta$), and $H_{3,4}^{\theta z}$ have only
monomials quadratic in $(\psi,J)$. Note also that since the linear normalizing
transformations have bounded coefficients for all $\delta\geqslant 0$, the
derivatives of the function $G_{7}$ are of the same order as the derivatives
of $G_{6}$, as given by (175).

Since $X=0$ is a periodic orbit of the Hamiltonian
$H(\bar{\varphi},X;\delta)$, it follows that the expansion of
$G_{7}(\bar{\varphi},X;\delta)$ in powers of $X$ starts with quadratic terms.
Moreover, the terms of the expansion which are independent of $(\psi,J)$ are
of order $\delta^{1/2}$, the terms linear in $(\psi,J)$ are of order
$\delta^{1/2-1/\alpha}$, and so on, e.g. the 4-th order terms in $(\psi,J)$
are $O(\delta^{1/2-4/\alpha})$, i.e., $o(\delta^{-1/\alpha})$ since
$\alpha>6$. Therefore, expanding $G_{7}$ in powers of $X$ we can rewrite (177)
as

$\begin{array}[]{l}H(\bar{\varphi},X;\delta)=\tilde{H}^{z}_{2}(z,p;\delta)+\tilde{H}^{\theta}_{2}(\psi,J;\delta)+\hat{H}_{2}^{\theta
z}(\bar{\varphi},\psi,J,z,p;\delta)\\\
\qquad\qquad+\hat{H}^{z}_{3}(\bar{\varphi},z,p;\delta)+\hat{H}_{3}^{\theta
z}(\bar{\varphi},\psi,J,z,p;\delta)+\hat{H}^{z}_{4}(\bar{\varphi},z,p;\delta)+\hat{H}_{4}^{\theta
z}(\bar{\varphi},\psi,J,z,p;\delta)\\\
\qquad\qquad\qquad+\delta^{-1/\alpha}\hat{H}_{4}^{\theta}(\bar{\varphi},\psi,J;\delta)+\ldots,\end{array}$
(178)

where, hereafter, the dots stand for the terms of order higher than $4$ (i.e.,
they are irrelevant for our purposes). The terms $\hat{H}_{j}$ are homogeneous
polynomials of $X$ of degree $j$ with periodic in $\bar{\varphi}$
coefficients, continuous and bounded for all $\delta\geqslant 0$. Moreover,
$\hat{H}_{2}^{\theta z}$ is $O(\delta^{\frac{1}{2}-\frac{2}{\alpha}})$-close
to zero. The polynomials $\hat{H}_{3}^{z}$ and $\hat{H}_{4}^{z}$ are the cubic
and, respectively quartic part of $H(\bar{\varphi},X;\delta)$ at $(\psi,J)=0$,
so they are $O(\delta^{1/2})$-close to $H^{z}_{3,4}(z,p;\delta)$ of (177). The
polynomial $\hat{H}_{3}^{\theta z}$ is
$O(\delta^{\frac{1}{2}-\frac{3}{\alpha}})$-close to $H_{3}^{\theta z}$ of
(177) (indeed, it has a linear in $(\psi,J)$ part which is
$O(\delta^{\frac{1}{2}-\frac{1}{\alpha}})$-close to zero, since $H_{3}^{\theta
z}$ has no linear part in $\psi$, a quadratic in $(\psi,J)$ part which is
$O(\delta^{\frac{1}{2}-\frac{2}{\alpha}})$-close to $H_{3}^{\theta z}$, and
the cubic in $(\psi,J)$ part is, again,
$O(\delta^{\frac{1}{2}-\frac{3}{\alpha}})$-close to zero). Similarly, the
polynomials $\hat{H}_{4}^{\theta z}$ and $\hat{H}_{4}^{\theta}$ are
$O(\delta^{\frac{1}{2}-\frac{3}{\alpha}})$-close to $H^{\theta z}_{4}$ and
$H_{4}^{\theta}$, respectively. Since $\alpha>6$, all these corrections are
small, i.e., they vanish at $\delta=0$.

Since $\hat{H}_{2}^{\theta z}=O(\delta^{1/2-2/\alpha})$, the
$\bar{\varphi}$-periodic, linear symplectic transformation of the variables
$X$ which diagonailzes the quadratic part
($\tilde{H}^{z}_{2}+\tilde{H}^{\theta}_{2}(\psi,J;\delta)+\hat{H}_{2}^{\theta
z}$) of $H$ and makes the quadratic part independent of $\bar{\varphi}$ is
$O(\delta^{\frac{1}{2}-\frac{2}{\alpha}})$-close to identity:
$X\rightarrow(Id+\delta^{\frac{1}{2}-\frac{2}{\alpha}}M(\bar{\varphi};\delta))X$,
for some periodic in $\bar{\varphi}$ matrix $M$, continuous and bounded for
all $\delta\geqslant 0$.

After this transformation, the Hamiltonian (178) becomes

$\begin{array}[]{l}H(\bar{\varphi},X;\delta)=\hat{H}^{z}_{2}(z,p;\delta)+\hat{H}^{\theta}_{2}(\psi,J;\delta)+\hat{H}^{z}_{3}(\bar{\varphi},z,p;\delta)+\hat{H}_{3}^{\theta
z}(\bar{\varphi},\psi,J,z,p;\delta)+\hat{H}^{z}_{4}(\bar{\varphi},z,p;\delta)\\\
\qquad\qquad\qquad+\hat{H}_{4}^{\theta
z}(\bar{\varphi},\psi,J,z,p;\delta)+\delta^{-1/\alpha}\hat{H}_{4}^{\theta}(\bar{\varphi},\psi,J;\delta)+\ldots,\end{array}$
(179)

where $\hat{H}_{2}^{z}$ and $\hat{H}_{2}^{\theta}$ are
$O(\delta^{\frac{1}{2}-\frac{2}{\alpha}})$-close to $\tilde{H}_{2}^{z}$ and,
respectively, $\tilde{H}_{2}^{\theta}$; the new homogeneous third degree
polynomials $\hat{H}^{z}_{3}$ and $\hat{H}_{3}^{\theta z}$ acquire
$O(\delta^{\frac{1}{2}-\frac{2}{\alpha}})$-corrections in comparison with
$\hat{H}^{z}_{3}$ and $\hat{H}_{3}^{\theta z}$ of (178), and the new
homogeneous fourth degree polynomials $\hat{H}^{z}_{4}$, $\hat{H}_{4}^{\theta
z}$, and $\hat{H}_{4}^{\theta}$ acquire
$O(\delta^{\frac{1}{2}-\frac{3}{\alpha}})$-corrections. It is only important
for us that these corrections vanish at $\delta=0$.

Next, we follow the same steps as in Lemma 5.8. We make a periodic in
$\bar{\varphi}$, symplectic transformation of the $(z,p)$-coordinates which
brings the Hamiltonian
$\hat{H}^{z}_{2}(z,p;\delta)+\hat{H}^{z}_{3}(\bar{\varphi},z,p;\delta)+\hat{H}^{z}_{4}(\bar{\varphi},z,p;\delta)$
to its Birkhoff normal form, independent of $\bar{\varphi}$ up to order 4
(recall that no new resonances can be created here by the
$\bar{\varphi}$-dependence because the frequency is large). This Hamiltonian
is close to the $z$-Hamiltonian of (159), therefore the resulting normal form
$\hat{H}_{NF}^{z}(z,p;\delta)$ is close to the normal form $H_{NF}^{z}$, see
(165); in particular,
$\hat{H}_{NF}^{z}=\hat{H}_{2}(z,p;\delta)+\hat{H}_{4}(z,p;\delta)$ does not
contain cubic terms, its quadratic and quartic terms $\hat{H}_{2,4}^{z}$
depend only on the actions $I_{z}$ (see (168)), and their limit as $\delta\to
0$ coincides with the limit of $\tilde{H}_{2,4}^{z}$.

After this transformation Hamiltonian (177) becomes

$\begin{array}[]{l}H(\bar{\varphi},X;\delta)=\hat{H}^{z}_{2}(z,p;\delta)+\hat{H}^{z}_{4}(z,p;\delta)+\hat{H}^{\theta}_{2}(\psi,J;\delta)+\hat{H}_{3}^{\theta
z}(\bar{\varphi},\psi,J,z,p;\delta)\\\ \qquad\qquad\qquad+\hat{H}_{4}^{\theta
z}(\bar{\varphi},\psi,J,z,p;\delta)+\delta^{-1/\alpha}\hat{H}_{4}^{\theta}(\bar{\varphi},\psi,J;\delta)+\ldots,\end{array}$
(180)

where the modified functions $\hat{H}_{3,4}$ have the same structure as before
and are close to their counterparts in (166). Since the third order terms in
the Hamiltonian are all non-resonant (by Assumption Box 5 and by the fact that
the period in $\bar{\varphi}$ is small), they are eliminated by a normalizing
symplectic transformation of $X$, which equals to the identity plus higher
order $\bar{\varphi}$-dependent terms and is close to that employed in Lemma
5.8 (the transformation from (166) to (167)). The Hamiltonian (180) becomes

$\begin{array}[]{l}H(\bar{\varphi},X;\delta)=\hat{H}^{z}_{2}(z,p;\delta)+\hat{H}^{z}_{4}(\varphi,z,p;\delta)+\hat{H}^{\theta}_{2}(\psi,J;\delta)\\\
\qquad\qquad\qquad+\hat{H}_{4}^{\theta
z}(\bar{\varphi},\psi,J,z,p;\delta)+\delta^{-1/\alpha}\hat{H}_{4}^{\theta}(\bar{\varphi},\psi,J;\delta)+\ldots,\end{array}$
(181)

where $\hat{H}_{4}$, the fourth-degree polynoimials in $X$, are close to their
counterparts in (167) (note that applying the transformation to the singular
term, $\delta^{-1/\alpha}\hat{H}_{4}^{\theta}$ of (179), introduces additional
singularities but only to the terms of order 5 or higher in (181)).

The last step is to bring the 4-th order terms to the autonomous normal form.
This is done by a symplectic transformation and the result is equivalent to
throwing away the non-resonant 4-th order terms and taking the average of the
resonant ones over $\bar{\varphi}$. Since all terms $\hat{H}$ in (181)
coincide with their counterparts in (167) at $\delta=0$, we immediately obtain
that the resulting 4-th order Birkhoff normal form of the Hamiltonian (173) is

$H_{NF4}=H^{z}_{2}(z,p;\delta)+H^{z}_{4}(z,p;\delta)+H^{\theta}_{2}(\psi,J;\delta)+\delta^{-1/\alpha}K(\alpha)H_{4}^{\theta}(\psi,J;\delta)+H_{4}^{\theta
z}(\psi,J,z,p;\delta),$ (182)

where $H^{x}_{j}(;\delta)$ have bounded coefficients, and, as
$\delta\rightarrow 0$ they approach the corresponding terms of (164). Hence,
the KAM-nondegeneracy of the elliptic orbit of the system (172) at $X=0$
follows from Lemma 5.9.∎

This completes the proof of Theorem 5.

The divergence as $\delta\rightarrow 0$ of the terms or order 5 and higher in
$X$ does not alter the KAM nondegeneracy result, yet, it implies that
quantitative estimates regarding the size of the stability island require
analysis of the asymptotic $\delta$-dependence of such terms.

##  6 Discussion

While the lack of ergodicity in Hamiltonian systems is expected, here we found
a specific mechanism for breaking the ergodicity, which persists for
arbitrarily high energy for any finite number of particles that interact by
repelling forces. We constructed coherent states of the multi-particle gas
that correspond to collision-free choreographic solutions which are stabilized
at high energies. We have also built similar solutions for systems of weakly
interacting particles and systems of attracting particles.

Let us list several future research directions.

Non-ergodicity of the gas of repelling particles in containers of dispersive
geometry. We have established the existence of KAM-stable choreographic
solutions for generic multi-particle systems in containers that support a
stable periodic billiard motion of one particle. Conjecturally, this includes
any generic container with a sufficiently smooth convex boundary. Yet, there
are open classes of billiards with piece-wise smooth boundaries which do not
allow for stable single-particle motions – the main example is given by
dispersive billiards whose boundary is built of strictly concave smooth
pieces. We propose that it should be possible to apply our method for finding
KAM-stable choreographies for such containers as well. Indeed, smoothing the