i_{1}({\gamma}_{x_{\tau}}^{2})+\nu_{1}({\gamma}_{x_{\tau}}^{2})+i_{-1}({\gamma}_{x_{\tau}}^{2})+\nu_{-1}({\gamma}_{x_{\tau}}^{2})$
(5.27) $\displaystyle\geq$ $\displaystyle n+0+1$ $\displaystyle=$
$\displaystyle n+1.$

If $r\geq 1$, then by Theorems 2.2 and 6.2 and (5.27) we have

$\displaystyle i_{-1}({\gamma}_{x_{\tau}}^{4r})$ $\displaystyle=$
$\displaystyle i_{-1}(({\gamma}_{x_{\tau}}^{2})^{2p})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{r}i_{{\omega}_{2r}^{2j-1}}({\gamma}_{x_{\tau}}^{4})$
$\displaystyle\geq$
$\displaystyle\sum_{j=1}^{r}(i_{1}({\gamma}_{x_{\tau}}^{4})+\nu_{1}({\gamma}_{x_{\tau}}^{4})-n)$
$\displaystyle=$ $\displaystyle
r(i_{1}({\gamma}_{x_{\tau}}^{4})+\nu_{1}({\gamma}_{x_{\tau}}^{4})-n)$
$\displaystyle\geq$ $\displaystyle r,$ (5.29)

where ${\omega}_{2r}=e^{\pi\sqrt{-1}/(2r)}$ as defined in Theorem 2.2.

By Theorem 3.2, we have

$i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{\tau}}^{4r+s})\geq
i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{\tau}}^{4r}).$ (5.30)

Then (5.22), (5.29) and (5.30) yield

$r\leq i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})+1.$ (5.31)

Thus for $i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})$ is odd,
by (5.31) we have

$4r+s\leq 4r+3\leq
4(i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0}))+7.$ (5.32)

Claim 3. For $i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0})$ is
even, the equality in (5.31) can not hold.

Otherwise, $r\geq 1$ and the equality in (5) holds i.e.,

$i_{{\omega}_{2r}^{2j-1}}({\gamma}_{x_{\tau}}^{4})=i_{1}({\gamma}_{x_{\tau}}^{4})+\nu_{1}({\gamma}_{x_{\tau}}^{4})-n=1,\quad
j=1,2,...,r.$ (5.33)

By the definition of ${\omega}_{2r}$, we have ${\omega}_{2r}^{2j-1}\neq-1$ for
$j=1,2,...,r$. So by (5.33) and 2 of Theorem 6.2, we have $I_{2p}\diamond
N_{1}(1,-1)^{\diamond q}\diamond
K\in{\Omega}^{0}({\gamma}_{x_{\tau}}^{4}(\tau))$ for some non-negative
integers $p$ and $q$ satisfying $0\leq p+q\leq n$ and $K\in{\rm Sp}(2(n-p-q))$
with ${\sigma}(K)\in{\bf U}\setminus\\{1\\}$ satisfying the condition that all
eigenvalues of $K$ located with the arc between $1$ and ${\omega}_{2r}$ in
${\bf U}^{+}\setminus\\{\pm 1\\}$ possess total multiplicity $n-p-q$. So there
are no eigenvalues of $K$ on the arc between ${\omega}_{2r}^{2j-1}$ and $-1$
except ${\omega}_{2r}^{2r-1}$ with $r=1$. However, whether
${\omega}_{2r}^{2r-1}\in{\sigma}({\gamma}_{x_{\tau}}^{4}(\tau))$ or not, we
always have

$\displaystyle S^{+}_{{\gamma}_{x_{\tau}}^{4}(\tau)}({\omega}_{2r}^{2r-1})=0,$
(5.34) $\displaystyle i_{{\omega}_{2r}^{2r-1}}({\gamma}_{x_{\tau}}^{4})=1.$
(5.35)

So (6.21) and Lemma 6.2, we have

$\displaystyle i_{-1}({\gamma}_{x_{\tau}}^{4})$ $\displaystyle=$
$\displaystyle
i_{{\omega}_{2r}^{2r-1}}({\gamma}_{x_{\tau}}^{4})+S^{+}_{{\gamma}_{x_{\tau}}^{4}(\tau)}({\omega}_{2r}^{2r-1})$
(5.36) $\displaystyle=$ $\displaystyle 1+0=1.$

But by (5.26), Lemma 6.1, and Theorem 6.1, we have

$\displaystyle i_{-1}({\gamma}_{x_{\tau}}^{4r})$ $\displaystyle=$
$\displaystyle i_{-1}(({\gamma}_{x_{\tau}}^{2r})^{2})$ $\displaystyle=$
$\displaystyle
i_{\sqrt{-1}}({\gamma}_{x_{\tau}}^{2r})+i_{-\sqrt{-1}}({\gamma}_{x_{\tau}}^{2r})$
$\displaystyle=$ $\displaystyle 2i_{\sqrt{-1}}({\gamma}_{x_{\tau}}^{2r}).$

Then $i_{-1}({\gamma}_{x_{\tau}}^{4r})$ is an even integer, which yields a
contradiction to (5.36). So Claim 3 holds, and we have

$r\leq i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0}).$ (5.37)

Hence

$4r+s\leq 4r+3\leq
4(i_{\sqrt{-1}}^{L_{0}}(B_{0})+\nu_{\sqrt{-1}}^{L_{0}}(B_{0}))+3.$ (5.38)

Theorem 1.5 holds from (5.32) and (5.38).

Proof of Theorem 1.4. This is the case $B_{0}\equiv 0$ of Theorem 1.2. From
Theorem 3.1 it is easy to see that

$i_{\sqrt{-1}}^{L_{0}}(0)=0,\qquad\nu_{\sqrt{-1}}^{L_{0}}(0)=0.$ (5.39)

Then $i_{\sqrt{-1}}^{L_{0}}(0)+\nu_{\sqrt{-1}}^{L_{0}}(0)=0$ and is also even.
So Theorem 1.4 holds from Theorem 1.5.

Proof of Corollary 1.2. Since $0<T<\frac{\pi}{||B_{0}||}$, there is
$\varepsilon>0$ small enough such that

$0\leq B_{0}\leq||B_{0}||I_{2n}<(\frac{\pi}{T}-\varepsilon)I_{2n}.$ (5.40)

It is easy to see that

${\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}}(t)={\rm
exp}(({\frac{\pi}{T}-\varepsilon})tJ)\quad\forall t\in[0,\frac{T}{4}].$ (5.41)

Since

$\nu_{L_{0}}({\rm exp}(({\frac{\pi}{T}-\varepsilon})tJ))=0,\qquad\forall
t\in[0,\frac{T}{2}].$ (5.42)

We have

$i_{L_{0}}({\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}}^{2})=0,\;i_{L_{0}}({\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}})=0.$
(5.43)

So by Theorem 2.1 we have

$\displaystyle
i_{\sqrt{-1}}^{L_{0}}((\frac{\pi}{T}-\varepsilon)I_{2n})=i_{L_{0}}({\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}}^{2})-i_{L_{0}}({\gamma}_{(\frac{\pi}{T}-\varepsilon)I_{2n}})=0.$
(5.44)

Then by (5.40) and Lemma 3.1 and Corollary 3.1 we have

$0\leq i_{-1}(B_{0},\frac{T}{2})+\nu_{-1}(B_{0},\frac{T}{2})\leq
i_{-1}((\frac{\pi}{T}-\varepsilon)I_{2n},\frac{T}{2})=0.$ (5.45)

So we have

$i_{-1}(B_{0},\frac{T}{2})+\nu_{-1}(B_{0},\frac{T}{2})=0.$ (5.46)

Hence by Theorem 1.1 or Corollary 1.1, the conclusion of Corollary 1.2 holds.

Also a natural question is that can we prove the minimal period is $T$ in this
way? We have the following remark.

Remark 5.1. Only use the Maslov-type index theory to estimate the iteration
time of the symmetric $T$-periodic brake solution $x_{T}$ obtained in the
proof of Theorem 1.5 with $B_{0}=0$, we can not hope to prove $T$ is the
minimal period of $x_{T}$. Even $H^{\prime\prime}(z)>0$ for all $z\in{\bf
R}^{2n}\setminus\\{0\\}$. For $n=1$ and $T=6\pi$, we can not exclude the
following case:

$\displaystyle x_{T}(t)=\left(\begin{array}[]{c}\sin t\\\ \cos
t\end{array}\right),$ (5.49) $\displaystyle H^{\prime}(x_{T}(t))=x_{T}(t),$
$\displaystyle H^{\prime\prime}(x_{T}(t))\equiv I_{2n}.$ (5.50)

It is easy to check that ${\gamma}_{x_{T}}(t)=R(t)$ for $t\in[0,3\pi]$. Hence
by Theorem 2.1 and Lemma 5.1 of [30] or the proof of Lemma 3.1 of [42] we have

$i_{\sqrt{-1}}^{L_{0}}({\gamma}_{x_{T}})=\sum_{3\pi/4\leq
s<3\pi}\nu_{L_{0}}({\gamma}_{x_{T}})(s)=1.$ (5.51)

In this case the minimal period of $x_{T}$ is $\frac{T}{3}$. Similarly for
$n>1$ we can construct examples to support this remark.

##  6 Appendix on Maslov-type indices $(i_{\omega},\nu_{\omega})$

We first recall briefly the Maslov-type index theory of
$(i_{\omega},\nu_{\omega})$. All the details can be found in [41].

For any ${\omega}\in{\bf U}$, the following codimension 1 hypersuface in ${\rm
Sp}(2n)$ is defined by:

${\rm Sp}(2n)_{\omega}^{0}=\\{M\in{\rm Sp}(2n)|{\rm
det}(M-{\omega}I_{2n})=0\\}.$

For any two continuous path $\xi$ and $\eta$: $[0,\tau]\to{\rm Sp}(2n)$ with
$\xi(\tau)=\eta(0)$, their joint path is defined by

$\eta*\xi(t)=\left\\{\begin{array}[]{lr}\xi(2t)&{\rm if}\,0\leq
t\leq\frac{\tau}{2},\\\ \eta(2t-\tau)&{\rm if}\,\frac{\tau}{2}\leq
t\leq\tau.\end{array}\right.$ (6.1)

Given any two $(2m_{k}\times 2m_{k})$\- matrices of square block form
$M_{k}=\left(\begin{array}[]{cc}A_{k}&B_{k}\\\ C_{k}&D_{k}\end{array}\right)$
for $k=1,2$, as in [41], the $\diamond$-product of $M_{1}$ and $M_{2}$ is
defined by the following $(2(m_{1}+m_{2})\times 2(m_{1}+m_{2}))$-matrix
$M_{1}\diamond M_{2}$:

$M_{1}\diamond M_{2}=\left(\begin{array}[]{cccc}A_{1}&0&B_{1}&0\\\
0&A_{2}&0&B_{2}\\\ C_{1}&0&D_{1}&0\\\ 0&C_{2}&0&D_{2}\end{array}\right).$

A special path $\xi_{n}$ is defined by

$\xi_{n}(t)=\left(\begin{array}[]{cc}2-\frac{t}{\tau}&0\\\
0&(2-\frac{t}{\tau})^{-1}\end{array}\right)^{\diamond n},\qquad\forall
t\in[0,\tau].$

Definition 6.1. For any ${\omega}\in{\bf U}$ and $M\in{\rm Sp}(2n)$, define

$\nu_{\omega}(M)=\dim_{\bf C}\ker(M-{\omega}I_{2n}).$ (6.2)

For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$, define

$\nu_{\omega}({\gamma})=\nu_{\omega}({\gamma}(\tau)).$ (6.3)

If ${\gamma}(\tau)\notin{\rm Sp}(2n)_{\omega}^{0}$, we define

$i_{\omega}({\gamma})=[{\rm Sp}(2n)_{\omega}^{0}\,:\,{\gamma}*\xi_{n}],$ (6.4)

where the right-hand side of (6.4) is the usual homotopy intersection number
and the orientation of ${\gamma}*\xi_{n}$ is its positive time direction under
homotopy with fixed endpoints.

If ${\gamma}(\tau)\in{\rm Sp}(2n)_{\omega}^{0}$, we let
$\mathcal{F}({\gamma})$ be the set of all open neighborhoods of ${\gamma}$ in
$\mathcal{P}_{\tau}(2n)$, and define

$i_{\omega}({\gamma})=\sup_{U\in\mathcal{F}({\gamma})}\inf\\{i_{\omega}(\beta)|\,\beta(\tau)\in
U\,{\rm and}\,\beta(\tau)\notin{\rm Sp}(2n)_{\omega}^{0}\\}.$ (6.5)

Then $(i_{\omega}({\gamma}),\nu_{\omega}({\gamma}))\in{\bf
Z}\times\\{0,1,...,2n\\}$, is called the index function of ${\gamma}$ at
${\omega}$.

Lemma 6.1. (Lemma 5.3.1 of [41]) For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$
and ${\omega}\in{\bf U}$, there hold

$\displaystyle
i_{\omega}({\gamma})=i_{\bar{{\omega}}}({\gamma}),\qquad\nu_{\omega}({\gamma})=\nu_{\bar{{\omega}}}({\gamma}).$
(6.6)

As in [38], for any $M\in{\rm Sp}(2n)$ we define

$\displaystyle{\Omega}(M)=\\{P\in{\rm Sp}(2n)$ $\displaystyle|$
$\displaystyle{\sigma}(P)\cap{\bf U}={\sigma}(M)\cap{\bf U}$ (6.7)
$\displaystyle{\rm
and}\,\nu_{\lambda}(P)=\nu_{\lambda}(M),\;\;\forall{\lambda}\in{\sigma}(M)\cap{\bf
U}\\}.$

We denote by ${\Omega}^{0}(M)$ the path connected component of ${\Omega}(M)$
containing $M$, and call it the homotopy component of $M$ in ${\rm Sp}(2n)$.

The following symplectic matrices were introduced as basic normal forms in
[41]:

$\displaystyle D({\lambda})=\left(\begin{array}[]{cc}{\lambda}&0\\\
0&{\lambda}^{-1}\end{array}\right),\qquad$ $\displaystyle{\lambda}=\pm 2,$
(6.10) $\displaystyle
N_{1}({\lambda},b)=\left(\begin{array}[]{cc}{\lambda}&b\\\
0&{\lambda}\end{array}\right),\qquad$ $\displaystyle{\lambda}=\pm 1,\,b=\pm
1,\,0,$ (6.13) $\displaystyle
R(\theta)=\left(\begin{array}[]{cc}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\end{array}\right),\qquad$
$\displaystyle\theta\in(0,\pi)\cup(\pi,2\pi),$ (6.16) $\displaystyle
N_{2}({\omega},b)=\left(\begin{array}[]{cc}R(\theta)&b\\\
0&R(\theta)\end{array}\right),\qquad$
$\displaystyle\theta\in(0,\pi)\cup(\pi,2\pi),$ (6.19)

where $b=\left(\begin{array}[]{cc}b_{1}&b_{2}\\\
b_{3}&b_{4}\end{array}\right)$ with $b_{i}\in{\bf R}$ and $b_{2}\neq b_{3}$.

For any $M\in{\rm Sp}(2n)$ and ${\omega}\in{\bf U}$, splitting number of $M$
at ${\omega}$ is defined by

$S_{M}^{\pm}=\lim_{\epsilon\to 0^{+}}i_{{\omega}{\rm
exp}(\pm\sqrt{-1}\epsilon)}({\gamma})-i_{\omega}({\gamma})$ (6.20)

for any path ${\gamma}\in\mathcal{P}_{\tau}(2n)$ satisfying
${\gamma}(\tau)=M$.

Splitting numbers possesses the following properties.

Lemma 6.2. (cf. [40], Lemma 9.1.5 and List 9.1.12 of [41]) Splitting number
$S_{M}^{\pm}({\omega})$ are well defined; that is they are independent of the
choice of the path ${\gamma}\in\mathcal{P}_{\tau}(2n)$ satisfying
${\gamma}(\tau)=M$. For ${\omega}\in{\bf U}$ and $M\in{\rm Sp}(2n)$,
$S_{N}^{\pm}({\omega})$ are constant for all $N\in{\Omega}^{0}(M)$. Moreover
we have

(1) $(S_{M}^{+}(\pm 1),S_{M}^{-}(\pm 1))=(1,1)$ for $M=\pm N_{1}(1,b)$ with
$b=1$ or $0$;

(2) $(S_{M}^{+}(\pm 1),S_{M}^{-}(\pm 1))=(0,0)$ for $M=\pm N_{1}(1,b)$ with
$b=-1$;

(3) $(S_{M}^{+}(e^{\sqrt{-1}\theta}),S_{M}^{-}(e^{\sqrt{-1}\theta}))=(0,1)$
for $M=R(\theta)$ with $\theta\in(0,\pi)\cup(\pi,2\pi)$;

(4) $(S_{M}^{+}({\omega}),S_{M}^{-}({\omega}))=(0,0)$ for ${\omega}\in{\bf
U}\setminus{\bf R}$ and $M=N_{2}({\omega},b)$ is trivial i.e., for

sufficiently small $\alpha>0$, $MR((t-1)\alpha)^{\diamond n}$ possesses no
eigenvalues on ${\bf U}$ for $t\in[0,1)$.

(5) $(S_{M}^{+}({\omega}),S_{M}^{-}({\omega})=(1,1)$ for ${\omega}\in{\bf
U}\setminus{\bf R}$ and $M=N_{2}({\omega},b)$ is non-trivial.

(6) $(S_{M}^{+}({\omega}),S_{M}^{-}({\omega})=(0,0)$ for any ${\omega}\in{\bf
U}$ and $M\in{\rm Sp}(2n)$ with ${\sigma}(M)\cap{\bf U}=\emptyset$.

(7) $S_{M_{1}\diamond
M_{2}}^{\pm}({\omega})=S_{M_{1}}^{\pm}({\omega})+S_{M_{2}}^{\pm}({\omega})$,
for any $M_{j}\in{\rm Sp}(2n_{j})$ with $j=1,2$ and ${\omega}\in{\bf U}$.

By the definition of splitting numbers and Lemma 6.2, for
$0\leq\theta_{1}<\theta_{2}<2\pi$ and ${\gamma}\in\mathcal{P}_{\tau}(2n)$ with
$\tau>0$, we have

$\displaystyle i_{{\rm exp}(\sqrt{-1}\theta_{2})}({\gamma})$ $\displaystyle=$
$\displaystyle i_{{\rm
exp}(\sqrt{-1}\theta_{1})}+S^{+}_{{\gamma}(\tau)}(e^{\sqrt{-1}\theta_{1}})$
(6.21) $\displaystyle+$
$\displaystyle\sum_{\theta\in(\theta_{1},\theta_{2})}\left(S^{+}_{{\gamma}(\tau)}(e^{\sqrt{-1}\theta})-S^{-}_{{\gamma}(\tau)}(e^{\sqrt{-1}\theta})\right)-S^{-}_{{\gamma}(\tau)}(e^{\sqrt{-1}\theta_{2}}).$

For any symplectic path ${\gamma}\in\mathcal{P}_{\tau}(2n)$ and $m\in{\bf N}$,
we define its $m$th iteration in the periodic boundary sense
${\gamma}(m):[0,m\tau]\to{\rm Sp}(2n)$ by

${\gamma}(m)(t)={\gamma}(t-j\tau){\gamma}(\tau)^{j}\qquad{\rm for}\,j\tau\leq
t\leq(j+1)\tau,\;j=0,1,...,m-1.$ (6.22)

Definition 6.2.(cf.[40], [41]) For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$ and
${\omega}\in{\bf U}$, we define

$(i_{\omega}({\gamma},m),\nu_{\omega}({\gamma},m))=(i_{\omega}({\gamma}(m)),\nu_{\omega}({\gamma}(m))),\qquad\forall
m\in{\bf N}.$ (6.23)

We have the following Bott-type iteration formula.

Theorem 6.1. (cf. [40], Theorem 9.2.1 of [41]) For any $\tau>0$,
${\gamma}\in\mathcal{P}_{\tau}(2n)$, $z\in{\bf U}$, and $m\in{\bf N}$,

$\displaystyle
i_{z}({\gamma},m)=\sum_{{\omega}^{k}=z}i_{\omega}({\gamma}),\qquad\nu_{z}({\gamma},m)=\sum_{{\omega}^{m}=z}\nu_{\omega}({\gamma}).$
(6.24)

By Theorem 8.1.4 of [41], we have the following Lemma.

Lemma 6.3. For ${\gamma}\in\mathcal{P}_{\tau}(2)$ with $\tau>0$, the following
results hold.

1\. If $N_{1}(1,1)\in{\Omega}^{0}({\gamma}(\tau))$, then

$\displaystyle
i_{1}({\gamma},m)=m(i_{1}({\gamma})+1)-1,\qquad\nu_{1}({\gamma},m)=1,\quad\forall
m\in{\bf N},$ (6.25) $\displaystyle i_{1}({\gamma})\in 2{\bf Z}+1.$ (6.26)

2\. If $N_{1}(1,1)\in{\Omega}^{0}({\gamma}(\tau))$, then

$\displaystyle
i_{1}({\gamma},m)=m(i_{1}({\gamma})+1)-1,\qquad\nu_{1}({\gamma},m)=2,\quad\forall
m\in{\bf N},$ (6.27) $\displaystyle i_{1}({\gamma})\in 2{\bf Z}+1.$ (6.28)

3\. If $N_{1}(1,-1)\in{\Omega}^{0}({\gamma}(\tau))$, then

$\displaystyle
i_{1}({\gamma},m)=m(i_{1}({\gamma}),\qquad\nu_{1}({\gamma},m)=1,\quad\forall
m\in{\bf N},$ (6.29) $\displaystyle i_{1}({\gamma})\in 2{\bf Z}.$ (6.30)

Denote by ${\bf U}^{+}=\\{{\omega}\in{\bf U}|\,Im\,{\omega}\geq 0\\}$ and
${\bf U}^{-}=\\{{\omega}\in{\bf U}|\,Im\,{\omega}\leq 0\\}$. The following
theorem was proved by Liu and Long in [33, 34], which plays a important role
in the proof of our main results in Sections 4-5.

Theorem 6.2. (Theorem 10.1.1 of [41])

1\. For any ${\gamma}\in\mathcal{P}_{\tau}(2n)$ and ${\omega}\in{\bf
U}\setminus\\{1\\}$, it always holds that

$i_{1}({\gamma})+\nu_{1}({\gamma})-n\leq i_{\omega}({\gamma})\leq
i_{1}({\gamma})+n-\nu_{\omega}({\gamma}).$ (6.31)

2\. The left equality in (6.31) holds for some ${\omega}\in{\bf
U}^{+}\setminus\\{1\\}$ (or ${\bf U}^{-}\setminus\\{1\\}$) if and only if
$I_{2p}\diamond N_{1}(1,-1)^{\diamond q}\diamond
K\in{\Omega}^{0}({\gamma}(\tau))$ for some non-negative integers $p$ and $q$
satisfying $0\leq p+q\leq n$ and $K\in{\rm Sp}(2(n-p-q))$ with
${\sigma}(K)\in{\bf U}\setminus\\{1\\}$ satisfying the condition that all
eigenvalues of $K$ located with the arc between $1$ and ${\omega}$ including
${\bf U}^{+}\setminus\\{1\\}$ (or ${\bf U}^{-}\setminus\\{1\\}$)possess total
multiplicity $n-p-q$. If ${\omega}\neq-1$, all eigenvalues of $K$ are in ${\bf
U}\setminus{\bf R}$ and those in ${\bf U}^{+}\setminus{\bf R}$ (or ${\bf
U}^{-}\setminus{\bf R}$) are all Krein-negative (or Krein-positive) definite.
If ${\omega}=-1$, it holds that $(-I_{2s})\diamond N_{1}(-1,1)^{\diamond
t}\diamond H\in{\Omega}^{0}({\gamma}(\tau))$ for some non-negative integers
$s$ and $t$ satisfying $0\leq s+t\leq n-p-q$, and some $H\in{\rm Sp}(2(n-p-q-
s-t))$ satisfying ${\sigma}(H)\subset{\bf U}\setminus{\bf R}$ and that all
elements in ${\sigma}(H)\cap{\bf U}^{+}$ (or ${\sigma}(H)\cap{\bf U}^{-}$) are
all Krein-negative (or Krein-positive) definite.

3\. The left equality of (6.31) holds for all ${\omega}\in{\bf
U}\setminus\\{1\\}$ if and only if $I_{2p}\diamond
N_{1}(1,-1)^{\diamond(n-p)}\in{\Omega}^{0}({\gamma}(\tau))$ for some integer
$p\in[0,n]$. Especially in this case, all the eigenvalues of ${\gamma}(\tau)$
are equal to $1$ and $\nu_{\gamma}=n+p\geq n$.

4\. The right equality in (6.31) holds for some ${\omega}\in{\bf
U}^{+}\setminus\\{1\\}$ (or ${\bf U}^{-}\setminus\\{1\\}$) if and only if
$I_{2p}\diamond N_{1}(1,1)^{\diamond r}\diamond
K\in{\Omega}^{0}({\gamma}(\tau))$ for some non-negative integers $p$ and $r$
satisfying $0\leq p+r\leq n$ and $K\in{\rm Sp}(2(n-p-r))$ with
${\sigma}(K)\in{\bf U}\setminus\\{1\\}$ satisfying the condition that all
eigenvalues of $K$ located with the arc between $1$ and ${\omega}$ including
${\bf U}^{+}\setminus\\{1\\}$ (or ${\bf U}^{-}\setminus\\{1\\}$)possess total
multiplicity $n-p-r$. If ${\omega}\neq-1$, all eigenvalues of $K$ are in ${\bf
U}\setminus{\bf R}$ and those in ${\bf U}^{+}\setminus{\bf R}$ (or ${\bf
U}^{-}\setminus{\bf R}$) are all Krein-positive (or Krein-negative) definite.
If ${\omega}=-1$, it holds that $(-I_{2s})\diamond N_{1}(-1,1)^{\diamond
t}\diamond H\in{\Omega}^{0}({\gamma}(\tau))$ for some non-negative integers
$s$ and $t$ satisfying $0\leq s+t\leq n-p-r$, and some $H\in{\rm Sp}(2(n-p-q-
r-t))$ satisfying ${\sigma}(H)\subset{\bf U}\setminus{\bf R}$ and that all
elements in ${\sigma}(H)\cap{\bf U}^{+}$ (or ${\sigma}(H)\cap{\bf U}^{-}$) are
all Krein-positive (or Krein-negative) definite.

5\. The right equality of (6.31) holds for all ${\omega}\in{\bf
U}\setminus\\{1\\}$ if and only if $I_{2p}\diamond
N_{1}(1,1)^{\diamond(n-p)}\in{\Omega}^{0}({\gamma}(\tau))$ for some integer
$p\in[0,n]$. Especially in this case, all the eigenvalues of ${\gamma}(\tau)$
are equal to $1$ and $\nu_{\gamma}=n+p\geq n$.

6\. Both equalities of (6.31) holds for all ${\omega}\in{\bf
U}\setminus\\{1\\}$ if and only if ${\gamma}(\tau)=I_{2n}$.

Acknowledgements Part of the work was finished during the author’s visit at
University of Michgan, he sincerely thanks Professor Yongbin Ruan for his
invitation and the Department of Mathematics of University of Michigan for its
hospitality.

## References

  * [1] A. Ambrosetti, V. Benci, Y. Long, A note on the existence of multiple brake orbits. Nonlinear Anal. T. M. A., 21 (1993) 643-649.
  * [2] A. Ambrosetti and V. Coti Zelati, Solutions with minimal period for Hamiltonian systems in a potential well. Ann. I. H. P. Anal. non lin$\acute{e}$aire 4 (1987), 242-275.
  * [3] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems. Math. Ann. 255 (1981), 405-421.
  * [4] T. An and Y. Long, Index theories of second order Hamiltonian systems. Nonlinear Anal.  27(1998), 585-592.
  * [5] V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems. Ann. I. H. P. Analyse Nonl.  1 (1984) 401-412.
  * [6] V. Benci, F. Giannoni, A new proof of the existence of a brake orbit. In “Advanced Topics in the Theory of Dynamical Systems”. Notes Rep. Math. Sci. Eng. 6 (1989) 37-49.
  * [7] S. Bolotin, Libration motions of natural dynamical systems. Vestnik Moskov Univ. Ser. I. Mat. Mekh. 6 (1978) 72-77 (in Russian).
  * [8] S. Bolotin, V.V. Kozlov, Librations with many degrees of freedom. J. Appl. Math. Mech. 42 (1978) 245-250 (in Russian).
  * [9] B. Booss and K. Furutani, The Maslov-type index - a functional analytical definition and the spectral flow formula. Tokyo J. Math. 21 (1998), 1-34.
  * [10] B. Booss and C. Zhu, General spectral flow formula for fixed maximal domain. Central Eur. J. Math. 3(3) (2005), 558-577.
  * [11] S. E. Cappell, R. Lee, and E. Y. Miller, On the Maslov-type index. Comm. Pure Appl. Math., 47 (1994) 121-186.
  * [12] C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations. Commu. Pure. Appl. Math. 37 (1984), 207-253.
  * [13] D. Dong and Y. Long, The Iteration Theory of the Maslov-type Index Theory with Applications to Nonlinear Hamiltonian Systems. Trans. Amer. Math. Soc. 349 (1997), 2619-2661.
  * [14] J. J. Duistermaat, Fourier Integral Operators, Birkhäuser, Basel, 1996.
  * [15] I. Ekeland, Convexity Methods in Hamiltonian Mechanics. Spring-Verlag. Berlin, 1990.
  * [16] I. Ekeland and E. Hofer, Periodic solutions with percribed period for convex autonomous Hamiltonian systems. Invent. Math. 81 (1985), 155-188.
  * [17] G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems. Nonlinear Anal.  27 (1996), 821-839.
  * [18] G. Fei, S.-K. Kim, and T. Wang, Minimal Period Estimates of Period Solutions for Superquadratic Hamiltonian Syetems. J. Math. Anal. Appl. 238 (1999), 216-233.
  * [19] G. Fei, S.-K. Kim, and T. Wang, Solutions of minimal period for even classical Hamiltonian systems. Nonlinear Anal. 43 (2001), 363-375.
  * [20] M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian equations, Nonlinear Anal.  7 (1983), 475-482.
  * [21] M. M. Girardi and M. Matzeu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem. Nonlinear Anal. TMA. 10 (1986), 371-382.
  * [22] M. Girardi and M. Matzeu, Periodic solutions of convex Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity. Ann. Math. Pura ed App. 147 (1987), 21-72.
  * [23] M. Girardi and M. Matzeu, Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic case. Nonlinear Anal. TMA. 17 (1991),481-497.
  * [24] N. Ghuussoub, Location, multiplicity and Morse indices of min-max critical points. J. reine angew. Math. 417 (1991), 27-76.
  * [25] H. Gluck, W. Ziller, Existence of periodic solutions of conservtive systems. Seminar on Minimal Submanifolds, Princeton University Press(1983), 65-98.
  * [26] E. W. C. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy. J. Math. Anal. Appl. 132 (1988), 1-12.
  * [27] K. Hayashi, Periodic solution of classical Hamiltonian systems. Tokyo J. Math. 6(1983), 473-486.
  * [28] A. Lazer and S. Solomini, Nontrivial solution of operator equations and Morse indices of critical points of min-max type. Nonlinear anal.  12 (1988), 761-775.
  * [29] C. Liu, A note on the monotonicity of Maslov-type index of Linear Hamiltonian systems with applications. Proceedings of the royal Society of Edinburg 135A (2005), 1263-1277.
  * [30] . C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions. Adv. Nonlinear Stud. 7 no.1 (2007), 131-161.
  * [31] C. Liu, Asymptotically Linear Hamiltonian Systems With Largrangian Boundary Conditions. Pacific J. Math. vol. 232 no.1 (2007), 233-255.
  * [32] C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete Contin. Dyn. Syst. Vol.27 no.1 (2010), 337-355.
  * [33] C. Liu and Y. Long, An optimal increasing estimate for iterated Maslov-type indices. Chinese Sci. Bull. 42 (1997), 2275-2277.
  * [34] C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications. J. Diff. Equa. 165 (2000), 355-376.
  * [35] C. Liu and D. Zhang, An iteration theory of Maslov-type index for symplectic paths associated with a Lagranfian subspace and Multiplicity of brake orbits in bounded convex symmetric domains, arXiv: 0908.0021vl [math. SG].
  * [36] Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems. Science in China, Series A. 7 (1990), 673-682 (Chinese edition), 33 (1990), 1409-1419.(English edition)
  * [37] Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials. Ann. I. H. P. Anal. non lin$\acute{e}$aire, 10-6 (1993), 605-626.
  * [38] Y. Long, The minimal period problem of period solutions for autonomous superquadratic second Hamiltonian systems. J. Diff. Equa. 111 (1994), 147-174.
  * [39] Y. Long, On the minimal period for periodic solution problem of nonlinear Hamiltonian systems. Chinese Ann. of math. 18B (1997) 481-484.
  * [40] Y. Long, Bott formula of the Maslov-type index theory. Pacific J. Math. 187 (1999), 113-149.
  * [41] Y. long, Index Theory for symplectic Paths with applications. Birkh$\ddot{a}$user, Basel, 2002.
  * [42] Y. Long, D. Zhang, C. Zhu, Multiple brake orbits in bounded convex symmetric domains. Advances in Math. 203 (2006) 568-635.
  * [43] P.H. Rabinowitz, Periodic solution of Hamiltonian systems. Commu. Pure Appl. Math. 31 (1978), 157-184.
  * [44] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conf. Ser. in Math., No. 65, A.M.S., Providence (1986)
  * [45] P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems. Nonlinear Anal. T. M. A. 11 (1987) 599-611.
  * [46] J. Robbin and D. Salamon, The Maslov indices for paths. Topology 32 (1993), 827-844.
  * [47] H. Seifert, Periodische Bewegungen mechanischer Systeme. Math. Z. 51 (1948), 197-216.
  * [48] S. Solimini, Morse indes estimates in min-max theorems. manuscription math. 63 (1989), 421-453.
  * [49] A. Szulkin, Cohomology and Morse theory for strongly indefinite functions. Math. Z. 209 (1992), 375-418.
  * [50] Y. Xiao, Periodic Solutions with Prescribed Minimal Period for Second Order Hamiltonian Systems with Even Potentials.Acta Math. Sinica, English Series No.5 Vol. 26 (2010), 825-830.
  * [51] D. Zhang, Symmetric Period Solutions With Prescribed Period For Even Autonomous Semipositive Hamiltonian Systems. preprint
  * [52] D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems. Science in China No. 6 vol. 50 (2007), 761-772.
  * [53] C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I), Chinese Ann. of Math. 208 (1999) 413-424.