To be published in Physics of Plasmas 2023

# Modification of Lie’s transform perturbation theory
for charged particle motion in a magnetic field

Linjin Zheng<EMAIL_ADDRESS>Institute for Fusion Studies,
University of Texas at Austin, Austin, TX 78712

###### Abstract

It is pointed out that the conventional Lie transform perturbation theory for
the guiding center motion of charged particles in a magnetic field needs to be
modified for ordering inconsistency. There are two reasons. First, the
ordering difference between the temporal variation of gyrophase and that of
the other phase space coordinates needs to be taken into account. Second, it
is also important to note that the parametric limit of the derivative of a
function is not equivalent to the derivative of the limit function. When these
facts are taken into account, the near identity transformation rule for one
form related to the Lagrangian is modified. With the modified near identity
transformation rule, the drift motion of charged particles can be described in
the first order, instead of in the second order and beyond through a tedious
expansion process as in the conventional formulation. This resolves the
discrepancy between the direct and Lie transform treatments in the Lagrangian
perturbation theory for charged particle motion in a magnetic field.

###### pacs:

52.53.Py, 52.55.Fa, 52.55.Hc

##  I Introduction

The guiding center motion of charged particles in a magnetic field is a
fundamental topic for magnetically confined fusion research. The standard
guiding center theory has been developed since the 1950s as reviewed in
reference [1]. The topic was later revisited using the Hamiltonian and
Lagrangian theories, for example in references [2, 3, 4, 5, 6]. This is mainly
related to the development of nonlinear gyrokinetic simulation as studied or
reviewed in references [8, 9, 10]. This is due to the concerns that the
standard guiding-center theory does not preserve Liouville’s theorem [11] and
the energy conservation law for time-independent systems. These properties are
especially important for long-time simulations. This has motived the further
development of the Hamiltonian and Lagrangian theories for guiding center
motion.

One of the important developments in the Hamiltonian and Lagrangian theories
for guiding center motion of charged particles is the introduction of the
phase space Lagrangian theories and the Lie transform perturbation technique.
The pioneer contributions in this procedure can be found in references [4, 5,
6, 7] and references therein. The detailed Lie transformation procedure for
guiding center motion of charged particles, which were omitted in the original
work in reference [4], were given in reference [12]. The Lie transform method
provides a systematic perturbation theory for guiding center motion and has
many advantages, such as the near identity transform process allowing the
expansion generator be determined in the order-by-order analyses. Also, the
backward transformation can be obtained easily from the forward
transformation.

However, there is a discrepancy between the direct and Lie transform
treatments in the Lagrangian perturbation theory in phase space for charged
particle motion in a magnetic field. In the direct method, the phase space
Lagrangian valid to the first order is given as follows [6]

$\displaystyle{{}^{d}}\Gamma=\left(\frac{e}{mc}{\bf A}+u{\bf
b}\right)~{}{\bf\cdot}~{}d{\bf X}+\frac{mc}{e}\mu
d\zeta-\left(\frac{u^{2}}{2}+\mu B+\frac{e}{m}\varphi\right)dt,$ (1)

while the standard Lie transform theory [4], which is detailed in the appendix
of reference [12] (Eq. (B18) with the zeroth order contribution added), yields
in the same order

$\displaystyle{{}^{d}}\Gamma$ $\displaystyle=$
$\displaystyle\left(\frac{e}{mc}{\bf A}+u{\bf b}\right)~{}{\bf\cdot}~{}d{\bf
X}-\left(\frac{u^{2}}{2}+\mu B+\frac{e}{m}\varphi\right)dt.$ (2)

Here, the general phase space coordinate system $\vec{Z}=\\{{\bf
X},\mu,u,\zeta;t\\}$ is used, ${\bf X}$ is related to the guiding center
coordinate, $\mu=v_{\bot}^{2}/2B$ is the magnetic moment, ${\bf v}$ denotes
the particle velocity with $v_{\bot}$ being the perpendicular component and
$u$ the parallel component, and $t$ represents time, ${\bf A}$ is the vector
potential, ${\bf B}$ is the magnetic field, ${\bf b}={\bf B}/B$, $\varphi$
denotes the electric scalar potential, $e$ is the charge, $c$ is the speed of
light, the boldface denotes the vector in configuration space, $\bar{(\cdot)}$
and $\vec{(\cdot)}$ are introduced to represent respectively the covariant and
contravariant vectors in the phase space with time included. The constant
factor $m$ for Lagrangian, for example in Eqs. (1) and (2), has been cast
aside. To distinguish one form (a covariant
vector)$\bar{\Gamma}=\\{\Gamma_{\mu}\\}$ and $\Gamma_{\mu}dZ^{\mu}$, the
notation for the scalar zero form ${}^{d}\Gamma=\Gamma_{\mu}dZ^{\mu}$ is
introduced. In these analyses, the gyrofrequency $\Omega=eB/mc$ is assumed to
be high, i.e., $(v/R)/\Omega\sim(\partial/\partial t)/\Omega\sim{\cal
O}(\epsilon)\ll 1$, where $R$ is the scale of electromagnetic field, which is
larger than the Larmor radius by an order of magnitude. It is also assumed
that $v_{\bf E}=|{\bf E}\times{\bf B}|/B^{2}\ll v$.

Comparing Eqs. (1) and (2) one can see that the term “$(mc/e)\mu d\zeta$” is
missing in the conventional Lie transformation treatment in the first order.
Note that

$\displaystyle\frac{(mc/e)\mu d\zeta}{u{\bf b}~{}{\bf\cdot}~{}d{\bf X}}$
$\displaystyle\sim$
$\displaystyle\frac{(mc/e)(v_{\bot}^{2}/B)(d\zeta/dt)}{u{\bf
b}~{}{\bf\cdot}~{}(d{\bf X}/dt)}$ (3) $\displaystyle\sim$
$\displaystyle\frac{(v_{\bot}^{2}/(eB/mc))(d\zeta/dt)}{u{\bf
b}~{}{\bf\cdot}~{}(d{\bf X}/dt)}\sim\frac{v_{\bot}^{2}}{u^{2}}\sim 1.$

Here, it has been used that $d\zeta/dt\sim eB/mc$ and $d{\bf X}/dt\sim{\bf
v}$. The ordering estimate in Eq. (3) shows that one cannot regard $(mc/e)\mu
d\zeta$ as $O(\epsilon)$ as compared to $u{\bf b}~{}{\bf\cdot}~{}d{\bf X}$.
Equation (2) is therefore ordering inconsistent because the term $u{\bf
b}~{}{\bf\cdot}~{}d{\bf X}$ is kept but the term $(mc/e)\mu d\zeta$ is
dropped, noting that they are of the same order.

Note that the detailed derivation process was omitted in Ref. [4] . This leads
us to discuss directly Ref. [12]. Nevertheless, as pointed in Ref. [12], the
detailed derivation given in the Appendix B of the paper is similar to that in
Ref. [4] except the rotation effects being added. It is especially noted that
the ordering inconsistency in Eq. (B18) of Ref. [12] discussed above appears
also in Eq. (20) of Ref. [4]. Like a mathematical theorem, it persists unless
a counter-proof is given. The Lie transform theory has become a standard
perturbation theory after thorough reviews for example in the journal Reviews
of modern physics (see for instance Refs. [5, 10, 6, 7]). From the citation
list of relevant articles one can see that the classical works in Lie
transform are still actively used.

As will be seen, this is because in the conventional Lie transform
perturbation theory the following type of deduction has been employed

$\displaystyle\left.\frac{\partial F}{\partial
Z}\right|_{\epsilon=0}=\frac{\partial{\left.F\right|_{\epsilon=0}}}{\partial
Z}.$ (4)

Apparently, this is not generally applicable. The parametric limit of the
derivative of a function is not equivalent to the derivative of the limit
function. This becomes serious for the system with rapidly varying coordinate.
For example for the case with $F=\left.F\right|_{\epsilon=0}+\epsilon f(Z)$
and $\partial f/\partial Z\sim 1/\epsilon$, one has

$\displaystyle\left.\frac{\partial F}{\partial
Z}\right|_{\epsilon=0}=\frac{\partial{\left.F\right|_{\epsilon=0}}}{\partial
Z}+\frac{\partial f}{\partial
Z}\not=\frac{\partial{\left.F\right|_{\epsilon=0}}}{\partial Z}.$ (5)

The terms $\frac{\partial{\left.F\right|_{\epsilon=0}}}{\partial Z}$ and
$\frac{\partial f}{\partial Z}$ are actually of the same order in this case.
The Larmor radius expansion exactly has this feature. Although the Larmor
radius is small, but the gyrophase varies in time rapidly.

It is interesting to point out that a similar case in gyro-fluid models was
pointed out earlier in Ref. [13] and further clarified later in Refs. [14,
15]. It is also related to the use of the parametric expansion with respect to
the small parameter $(\partial/\partial t)/\Omega\ll 1$. When it is applied to
the second order fluid moment of Vlasov equation the inclusion of first order
Finite-Larmor-Radius (FLR) corrections to a double-adiabatic closure in a
fluid modelling causes the dispersion relation of magneto-acoustic waves
propagating perpendicularly to a background magnetic field to get a wrong sign
in their spatial dispersion. The unphysical result is fixed only if the
$\epsilon\ll 1$ expansion is directly performed on the fluid moment equations
which account for the full anisotropy of the second and third order velocity
moments.

In this paper, we modify the conventional Lie transformation formalism for the
guiding center motion of charged particles in a magnetic field by taking into
account the ordering difference between the temporal variations of gyrophase
and that of the other phase space coordinates and the fact that the parametric
limit of the derivative of a function is not equivalent to the derivative of
the limit function. This leads us to resolve the discrepancy between the
direct and Lie transform treatments in the Lagrangian perturbation theory in
the phase space. The modification presented in this paper is expected to
affect generally the Lie transform perturbation formulation for the systems
with a rapidly varying coordinate.

The manuscript is organized as follows. In Sec. II a brief review of the
conventional Lie transform theory is given; in Sec. III. the modification of
Lie’s transform perturbation theory for charged particle motion in a magnetic
field is described; In the last section, conclusions and discussion are
presented. Appendix A is introduced to double confirm the newly derived
transformation rule for one form.

##  II Review of the conventional Lie transform perturbation theory

In this section, we briefly review the conventional Lie transform perturbation
theory for the charged particle motion in a magnetic field [4, 6]. This will
pave the way for the modified theory to be described in the next section.

With a small parameter $\epsilon$, the near-identity coordinate transform can
be generally expressed as

$\displaystyle Z^{\mu}$ $\displaystyle=$ $\displaystyle z^{\mu}+\epsilon
Z_{1f}^{\mu}(\vec{z})+\epsilon^{2}Z_{2f}^{\mu}(\vec{z})+\cdots$

Here, the subscript “f” denotes the forward transformation from the current to
the new coordinates. For the charged particle motion in a magnetic field, the
forward transformation is simply the transformation to the guiding center. The
forward transformation can be generally denoted as

$\displaystyle Z^{\mu}=Z_{f}^{\mu}(\vec{z},\epsilon).$ (6)

We also introduce the backward transformation as

$\displaystyle z^{\mu}=Z_{b}^{\mu}(\vec{Z}_{f}(\vec{z},\epsilon),\epsilon).$
(7)

In the Lie transformation, the coordinate transformation is specified through
a generator $g^{\mu}$ such that

$\displaystyle\frac{\partial Z_{f}^{\mu}(\vec{z},\epsilon)}{\partial\epsilon}$
$\displaystyle=$ $\displaystyle
g^{\mu}(\vec{Z}_{f}(\vec{z},\epsilon))~{}~{}~{}\hbox{and}~{}~{}~{}Z_{f}^{\mu}(\vec{z},0)=z^{\mu}.$
(8)

Applying $\partial/\partial\epsilon$ on the backward transformation in Eq.
(7), one obtains

$\displaystyle\frac{\partial Z_{b}^{\mu}}{\partial Z_{f}^{\nu}}\frac{\partial
Z_{f}^{\nu}}{\partial\epsilon}+\frac{\partial
Z_{b}^{\nu}}{\partial\epsilon}=0.$

Here, the summation for repeated indices is implied as usual. Using Eq. (8),
one obtains

$\displaystyle\frac{\partial Z_{b}^{\mu}(\vec{z},\epsilon)}{\partial\epsilon}$
$\displaystyle=$ $\displaystyle-g^{\nu}(\vec{Z})\frac{\partial
Z_{b}^{\mu}}{\partial Z_{f}^{\nu}}.$ (9)

We first consider the application of the Lie transformation on a scalar.
Suppose there is a forward transform from a scalar $s(\vec{z})$ to a new
scalar $S(\vec{Z},\epsilon)$ such that

$\displaystyle S(\vec{Z},\epsilon)=s(\vec{Z}_{b}(\vec{Z},\epsilon)).$ (10)

Applying $\partial/\partial\epsilon$ on it, one obtains

$\displaystyle\frac{\partial
S(\vec{Z},\epsilon)}{\partial\epsilon}=\frac{\partial
s(\vec{Z}_{b}(\vec{Z},\epsilon))}{\partial Z^{\mu}_{b}}\frac{\partial
Z^{\mu}_{b}}{\partial\epsilon}=\frac{\partial S(\vec{Z},\epsilon)}{\partial
Z^{\mu}_{b}}\frac{\partial Z^{\mu}_{b}}{\partial\epsilon}.$

Using Eq. (9) and the chain rule, one obtains

$\displaystyle\frac{\partial
S(\vec{Z},\epsilon)}{\partial\epsilon}=-g^{\mu}(\vec{Z})\frac{\partial
S(\vec{Z},\epsilon)}{\partial Z^{\mu}}.$ (11)

Defining the operator $L_{g}\equiv g^{\mu}(\partial/\partial Z^{\mu})$. Eq.
(11) becomes

$\displaystyle\frac{\partial S}{\partial\epsilon}=-L_{g}S.$ (12)

This further gives that

$\displaystyle\frac{\partial^{n}S}{\partial\epsilon^{n}}=\left(-L_{g}\right)^{n}S.$
(13)

We now expand $S(\vec{Z},\epsilon)$ in a Taylor series:

$\displaystyle S(\vec{Z},\epsilon)$ $\displaystyle=$
$\displaystyle\left.\sum_{n=0}^{+\infty}\frac{\epsilon^{n}}{n!}\frac{\partial^{n}S}{\partial\epsilon^{n}}\right|_{\epsilon=0}.$

If the limit and derivative are commutable, this equation becomes

$\displaystyle S(\vec{Z},\epsilon)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{+\infty}\frac{\epsilon^{n}}{n!}\frac{\partial^{n}S|_{\epsilon=0}}{\partial\epsilon^{n}}=\sum_{n=0}^{+\infty}\frac{\epsilon^{n}}{n!}\frac{\partial^{n}s}{\partial\epsilon^{n}}.$
(14)

Using Eqs. (13) and (14), one obtains the Lie transformation of a scalar:

$\displaystyle S=e^{-\epsilon L_{g}}s.$ (15)

As pointed out in the introduction section that the parametric limit of the
derivative of a function is not equivalent to the derivative of the limit
function, this transformation rule needs to be verified case by case. Because
of the operator expression, the inverse transform is simply

$\displaystyle s=e^{\epsilon L_{g}}S,$ (16)

evaluating at $\epsilon=0$.

The coordinate transformation defined in equations (6) and (7) indicates that
$z^{\alpha}=Z_{b}^{\alpha}(\vec{Z},\epsilon)$. Comparing with Eq. (10), one
can see that the coordinate transformation is just a special case of scalar
transformation. Let us introduce the coordinate function $I^{\alpha}$ defined
by $I^{\alpha}(\vec{z})=z^{\alpha}$ for a particular $\alpha$. One then has

$\displaystyle Z_{b}^{\alpha}=e^{-\epsilon L_{g}}I^{\alpha}(\vec{z}).$ (17)

Explicitly, one has [12]

$\displaystyle Z^{\alpha}(\vec{Z},\epsilon)$ $\displaystyle=$ $\displaystyle
z^{\alpha}-\epsilon
L_{g}z^{\alpha}+\frac{1}{2}\epsilon^{2}L_{g}\left(L_{g}z^{\alpha}\right)+\cdots$
(18) $\displaystyle=$ $\displaystyle z^{\alpha}-\epsilon
g_{1}^{\alpha}-\epsilon^{2}\left(g_{2}^{\alpha}-\frac{1}{2}g_{1}^{\beta}\frac{\partial
g_{1}^{\alpha}}{\partial z^{\beta}}\right)+\cdots,$

where $g_{\cdots}$ are the functions of $\vec{z}$. Since the transformation is
invertible, the inverse transformation is given by

$\displaystyle z^{\alpha}(\vec{z},\epsilon)$ $\displaystyle=$ $\displaystyle
Z^{\alpha}+\epsilon
g_{1}^{\alpha}+\epsilon^{2}\left(g_{2}^{\alpha}+\frac{1}{2}g_{1}^{\beta}\frac{\partial
g_{1}^{\alpha}}{\partial Z^{\beta}}\right)+\cdots,$ (19)

where $g_{\cdots}$ are the functions of $\vec{Z}$.

Next, we consider the Lie transform on a 1-form in the phase space:
$\gamma_{\mu}$. For a coordinate transformation $\vec{z}\to\vec{Z}$, the new
1-form $\Gamma_{\mu}$ follows the invariant relation:
$\Gamma_{\mu}dZ^{\mu}=\gamma_{\mu}dz^{\mu}$. This is completely equivalent to
the usual rule for transforming a covariant vector:

$\displaystyle\Gamma_{\mu}(\vec{Z},\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\partial{Z_{b}^{\nu}(\vec{Z},\epsilon)}}{\partial
Z^{\mu}}\gamma_{\nu}(\vec{Z}_{b}(\vec{Z},\epsilon)).$ (20)

Applying $\partial/\partial\epsilon$ on it and using Eq. (11), one obtains

$\displaystyle\frac{\partial\Gamma_{\mu}(\vec{Z},\epsilon)}{\partial\epsilon}$
$\displaystyle=$ $\displaystyle-\frac{\partial}{\partial
Z^{\mu}}\left[g^{\lambda}(\vec{Z})\frac{\partial
Z_{b}^{\nu}(\vec{Z},\epsilon)}{\partial
Z^{\lambda}}\right]\gamma_{\nu}(\vec{Z}_{b}(\vec{Z},\epsilon))$ (21)
$\displaystyle-g^{\lambda}(\vec{Z})\frac{\partial{Z_{b}^{\nu}(\vec{Z},\epsilon)}}{\partial
Z^{\mu}}\frac{\partial\gamma_{\nu}(\vec{Z}_{b}(\vec{Z},\epsilon))}{\partial
Z^{\lambda}}$ $\displaystyle=$
$\displaystyle-g^{\lambda}(\vec{Z})\left[\frac{\partial{\Gamma_{\mu}(\vec{Z},\epsilon)}}{\partial
Z^{\lambda}}-\frac{\partial{\Gamma_{\lambda}(\vec{Z},\epsilon)}}{\partial
Z^{\mu}}\right]-\frac{\partial}{\partial
Z^{\mu}}\left[g^{\lambda}(\vec{Z})\Gamma_{\gamma}(\vec{Z},\epsilon)\right].$

Let $\xi_{\mu}$ be an arbitrary 1-form and $L_{g}\bar{\xi}$ another 1-form
whose components given by

$\displaystyle(L_{g}\bar{\xi})_{\mu}$ $\displaystyle=$ $\displaystyle
g^{\nu}\left(\partial_{\nu}\xi_{\mu}-\partial_{\mu}\xi_{\nu}\right).$ (22)

To be specific, we point out that Eq. (22) is just Eq.(46) in Ref. [5], in
which the last term of Eq. (21) has been dropped in view of that the total
derivative does not contribute to the variational principle. With this
definition, Eq. (21) can be expressed as

$\displaystyle\frac{\partial\bar{\Gamma}}{\partial\epsilon}$ $\displaystyle=$
$\displaystyle-
L_{g}\bar{\Gamma}-\bar{\partial}(\vec{g}~{}{\bf\cdot}~{}\bar{\Gamma}),$ (23)

which is just Eq.(47) in Ref. [5]. Here, $\bar{\partial}=\\{\partial/\partial
Z^{\mu}\\}$.

Noting the symmetry property, one can prove that $L_{g}\bar{\partial}$ always
vanishes (similar to ${\bf\nabla}~{}{\bf\times}~{}\nabla=0$ in the three
dimensional case) and $g^{\mu}(L_{g}\bar{\xi})_{\mu}=0$. Therefore, Eq. (23)
can be applied inductively to yield

$\displaystyle\frac{\partial^{n}\bar{\Gamma}}{\partial\epsilon^{n}}$
$\displaystyle=$
$\displaystyle\left(-L_{g}\right)^{n}\bar{\Gamma}+\left(-\bar{\partial}\vec{g}~{}{\bf\cdot}~{}\right)^{n}\bar{\Gamma}.$
(24)

If the limit and and derivative are commutable, Eq. (24) leads to

$\displaystyle\bar{\Gamma}(\vec{Z},\epsilon)$ $\displaystyle=$
$\displaystyle\left.\sum_{n=0}^{+\infty}\frac{\epsilon^{n}}{n!}\frac{\partial^{n}\bar{\Gamma}}{\partial\epsilon^{n}}\right|_{\epsilon=0}=\sum_{n=0}^{+\infty}\frac{\epsilon^{n}}{n!}\frac{\partial^{n}\bar{\Gamma}|_{\epsilon=0}}{\partial\epsilon^{n}}=\sum_{n=0}^{+\infty}\frac{\epsilon^{n}}{n!}\frac{\partial^{n}\bar{\gamma}}{\partial\epsilon^{n}}.$

Using this equation and Eq. (13), one obtains

$\displaystyle\bar{\Gamma}$ $\displaystyle=$ $\displaystyle e^{-\epsilon
L_{g}}\bar{\gamma}+\bar{\partial}S$ (25)

Here, $\bar{\partial}S$ results from the second term on the right hand side of
Eq. (24) and gives rise to an exact differential in the variational principle.
The inverse of Eq. (25) is simply

$\displaystyle\bar{\gamma}$ $\displaystyle=$ $\displaystyle e^{\epsilon
L_{g}}\bar{\Gamma}+\bar{\partial}s,$ (26)

where $s$ is a scalar different from $S$. As pointed out in the introduction
section that the parametric limit of the derivative of a function is not
equivalent to the derivative of the limit function, this transformation rule
needs to be verified case by case.

Using the transformation formula for scalar and 1-form in equations (15) and
(25) (together with Eq. (22)) respectively, one can develop the high order
perturbation theory. We assume that the 1-form in the phase space which can be
expanded as follows

$\displaystyle\bar{\gamma}=\bar{\gamma}^{(0)}+\epsilon\bar{\gamma}^{(1)}+\epsilon^{2}\bar{\gamma}^{(2)}+\cdots.$
(27)

It is also assumed that the lowest-order dynamics with $\bar{\gamma}^{(0)}$ is
well understood, i.e., its solutions are known or can be easily obtained.
Therefore, the lowest-order trajectory can be used to find the solutions of
higher orders.

In order to simplify the 1-form to sufficient orders, the following overall
transformation operator, which is a composition of individual Lie transforms,
is introduced

$\displaystyle T$ $\displaystyle=$ $\displaystyle\cdots T_{3}T_{2}T_{1}$ (28)

with

$\displaystyle T_{n}$ $\displaystyle=$ $\displaystyle e^{-\epsilon^{n}L_{n}}.$
(29)

Here, $L_{n}$ denotes $L_{g_{n}}$ as defined for scalar and 1-form in
equations (15) and (25) (together with Eq. (22)) respectively. The generators
$g^{\mu}_{n}$ with $n=1,\cdots,n$ will be used to simply the fundamental
1-form in Eq. (27) to order $n$. The inverse of the transformation vector is

$\displaystyle T^{-1}$ $\displaystyle=$ $\displaystyle
T_{1}^{-1}T_{2}^{-1}T_{3}^{-1}\cdots$ (30)

with

$\displaystyle T_{n}^{-1}$ $\displaystyle=$ $\displaystyle
e^{\epsilon^{n}L_{n}}.$ (31)

When successive Lie transforms are applied in this manner, Eq. (25) becomes,
noting that $T_{i}\bar{\partial}$ always vanishes,

$\displaystyle\bar{\Gamma}$ $\displaystyle=$ $\displaystyle
T\bar{\gamma}+\bar{\partial}S,$ (32)

where $S$ collects all possible scalar contributions. Expanding $\bar{\Gamma}$
and $S$ in powers of $\epsilon$ as well as $\bar{\gamma}$ in Eq. (27) and
collecting terms in each order, one obtains

$\displaystyle\bar{\Gamma}^{(0)}$ $\displaystyle=$
$\displaystyle\bar{\gamma}^{(0)},$ (33) $\displaystyle\bar{\Gamma}^{(1)}$
$\displaystyle=$
$\displaystyle\bar{\partial}S^{(1)}-L_{1}\bar{\gamma}^{(0)}+\bar{\gamma}^{(1)},$
(34) $\displaystyle\bar{\Gamma}^{(2)}$ $\displaystyle=$
$\displaystyle\bar{\partial}S^{(2)}-L_{2}\bar{\gamma}^{(0)}+\bar{\gamma}^{(2)}-L_{1}\bar{\gamma}^{(1)}+\frac{1}{2}L_{1}^{2}\bar{\gamma}^{(0)},$
(35) $\displaystyle\bar{\Gamma}^{(3)}$ $\displaystyle=$
$\displaystyle\bar{\partial}S^{(3)}-L_{3}\bar{\gamma}^{(0)}+\bar{\gamma}^{(3)}-L_{2}L_{1}\bar{\gamma}^{(0)}+\frac{1}{6}L_{1}^{3}\bar{\gamma}^{(0)}$
(36) $\displaystyle-
L_{2}\bar{\gamma}^{(1)}+\frac{1}{2}L_{1}^{2}\bar{\gamma}^{(1)}-L_{1}\bar{\gamma}^{(2)},$

and so on.

These complete the basic theoretical review of the conventional Lie transform
theory in references [4, 6]. The conventional Lie transform theory was applied
to study the charged particle motion in a magnetic field [4, 12]. In the
Appendix B of Ref. [12], the Lagrangian in the zeroth order was obtained as
follows

${}^{d}\Gamma^{(0)}$ $\displaystyle=$ $\displaystyle\frac{e}{mc}{\bf
A}~{}{\bf\cdot}~{}d{\bf X},$ (37)

while the first order Lagrangian is given as follows (Eq. (B18) in Ref. [12])

${}^{d}\Gamma^{(1)}$ $\displaystyle=$ $\displaystyle u{\bf
b}~{}{\bf\cdot}~{}d{\bf X}-\left(\frac{u^{2}}{2}+\mu
B+\frac{e}{m}\varphi\right)dt.$ (38)

Combining Eqs. (37) and (38) yields Eq. (2) discussed in the introduction.

In these reviews, the correction as pointed out in the introduction section
has not been included, especially as will be seen in the next section, the
expansions of the phase space Lagrangian in Eqs. (33) - (36) will be modified.
Consequently, the first order Lagrangian in Eq. (38) will be modified. The
modification is related to the difference between Eq. (38) and the result by
the direct approach in Eq. (1) . The term $(mc/e)\mu d\zeta$ is missing in Eq.
(38), although it is of the same order as the term $u{\bf
b}~{}{\bf\cdot}~{}d{\bf X}$ as shown in the ordering analysis in Eq. (3). The
term $(mc/e)\mu d\zeta$ is only picked up in the next order in the
conventional Lee transform approach (Eq. (B30) in Ref. [12] or Eq.(29) in Ref.
[4]). Note that Eq. (38) is obtained by strictly following the standard Lie
transform formulation. As explained in the next section, the problem lies in
that the commutation between the limit and derivative in deriving Eq. (25) is
illegitimate in the case with a fast varying coordinate. In the derivation of
Eq. (25), the commutation as shown in Eq. (4) was used to reduce
$\bar{\Gamma}$ to $\bar{\gamma}$ in the conventional Lie transform
formulation. As explained alternatively in Appendix A, the illegitimate
commutation is equivalent to assume that
$\Gamma_{\mu}dZ^{\mu}=\gamma_{\mu}dZ^{\mu}$, which is apparently invalid. The
correct one should be $\Gamma_{\mu}dZ^{\mu}=\gamma_{\mu}dz^{\mu}$.

##  III Modification of Lie transformation

In this section, we describe the modification of the conventional Lie
transformation formulation for the systems containing rapidly varying
coordinates. For the guiding center motion of a charged particle in a magnetic
field, the gyrophase is this type of coordinates. This helps solve the
inconsistency in the phase-space Lagrangian perturbation theories between the
direct derivation and the Lie transform formulation as pointed out in the
introduction section.

Strictly speaking, the transformation rule for 1-form in Eq. (22) is correct.
However, when applying this rule in equations (34)-(36), the ordering
inconsistency occurs. One needs to take into account the difference between
$\Gamma_{\nu}$ and $\gamma_{\nu}$. This is because the parametric limit of the
derivative of a function is not equivalent to the derivative of the limit
function. Equation (25) does not apply to the systems containing a rapidly
varying coordinate. To correct the transformation rule, we continue the
derivation of the transformation rule in Eq. (21) (or Eq. (22)) to include the
transformation from $\Gamma_{\nu}$ to $\gamma_{\nu}$ in Eq. (20). This is
carried out as follows

$\displaystyle-g^{\lambda}(\vec{Z})\left(\frac{\partial{\Gamma_{\nu}(\vec{Z},\epsilon)}}{\partial
Z^{\lambda}}-\frac{\partial{\Gamma_{\lambda}(\vec{Z},\epsilon)}}{\partial
Z^{\nu}}\right)dZ^{\nu}$ $\displaystyle=$
$\displaystyle-g^{\lambda}(\vec{Z})\left(\frac{\partial{\gamma_{\mu}(\vec{Z}_{b}(\vec{Z},\epsilon))}}{\partial
Z^{\lambda}}\frac{\partial Z_{b}^{\mu}}{\partial
Z^{\nu}}-\frac{\partial{\gamma_{\mu}(\vec{Z}_{b}(\vec{Z},\epsilon))}}{\partial
Z^{\nu}}\frac{\partial Z_{b}^{\mu}}{\partial Z^{\lambda}}\right)dZ^{\nu}.$

Using the coordinate transformation rule in Eq. (19), one obtains the modified
transform rule for 1 form

$\displaystyle(L_{g}\bar{\Gamma})_{\mu}$ $\displaystyle=$ $\displaystyle
g^{\nu}\left(\partial_{\nu}\gamma_{\mu}-\partial_{\mu}\gamma_{\nu}\right)$
(39)
$\displaystyle-g^{\nu}\left[\left(\partial_{\nu}\gamma_{\delta}\right)\left(\partial_{\mu}g_{1}^{\delta}\right)-\left(\partial_{\mu}\gamma_{\delta}\right)\left(\partial_{\nu}g_{1}^{\delta}\right)\right]+\cdots.$

Here, the second term seems to be formally one order smaller than the first
term on the right. However, if the components of $dz^{\nu}$ are different in
order, the second term on the right hand side of Eq. (39) has to be kept for
rapidly varying components for ordering consistency. The gyrophase is an
example of rapidly varying coordinates in considering the charged particle
motion in a magnetic field. Since the transformation rule in Eq. (39) is
fundamentally important, an alternative derivation is provided in Appendix A.
In the Appendix A, it is also pointed out that the conventional transform rule
for 1 form reviewed in the previous section is actually obtained from the
formula $\Gamma_{\mu}dZ^{\mu}=\gamma_{\mu}dZ^{\mu}$, instead of
$\Gamma_{\mu}dZ^{\mu}=\gamma_{\mu}dz^{\mu}$. This is the consequence of the
non-consistent exchange between the limit ($\epsilon\to 0$) and derivative.

The phase space Lagrangian for a charged particle motion in the
electromagnetic field is given in references [4, 6]

${}^{d}\gamma$ $\displaystyle=$
$\displaystyle\left(\frac{e}{mc}A_{\mu}+v_{\mu}\right)dz^{\mu}+\left(\frac{v^{2}}{2}+\frac{e}{m}\varphi\right)dt.$

Here again, $``^{d}\gamma"$ has been used in order to show the individual
components of the one form $\bar{\gamma}$ explicitly. In the perturbation
analyses, ${}^{d}\gamma$ is expanded in $\epsilon$ as follows

${}^{d}\gamma_{0}$ $\displaystyle=$
$\displaystyle\frac{e}{mc}A_{\mu}(z)dz^{\mu},$ (40) ${}^{d}\gamma_{1}$
$\displaystyle=$ $\displaystyle
v_{\mu}dz^{\mu}+\left(\frac{v^{2}}{2}+\frac{e}{m}\varphi\right)dt.$ (41)

In the zeroth order, one has

${}^{d}\Gamma^{(0)}$ $\displaystyle=$ $\displaystyle
dS^{(0)}+\frac{e}{mc}A_{\mu}(Z)\frac{\partial z^{\mu}}{\partial
Z^{\nu}}dZ^{\nu}$ (42) $\displaystyle=$ $\displaystyle
dS^{(0)}+\frac{e}{mc}A_{\mu}(Z)dZ^{\mu}-\frac{e}{mc}A_{\mu}(Z)dg_{1}^{\mu}.$

Here, $dS=\partial_{\mu}SdZ^{\mu}$, the last term on the right is kept, since
$dg_{1}^{\mu}$ can be of order $1/\epsilon$ as can be proved a posteriori.
Letting $S_{0}=A_{\mu}g_{1}^{\mu}$, one has $dS^{(0)}-\epsilon
A_{\mu}dg_{1}^{\mu}=\epsilon g_{1}^{\mu}dA_{\mu}$. Noting that $dA_{\mu}$ is
of order unity, one obtains

${}^{d}\Gamma^{(0)}$ $\displaystyle=$ $\displaystyle\frac{e}{mc}{\bf A}({\bf
X})~{}{\bf\cdot}~{}d{\bf X}.$ (43)

This derivation is in fact similar to the direct derivation of Eq. (3.41) in
reference [6].

The first order Lagrangian in Eq. (34) becomes

${}^{d}\Gamma^{(1)}$ $\displaystyle=$ $\displaystyle
dS^{(1)}-L^{conv}_{1}~{}{{}^{d}}\gamma^{(0)}+{{}^{d}}\gamma^{(1)}-v_{\mu}dg_{1}^{\mu}-g_{1}^{\lambda}\frac{\partial\gamma^{(0)}_{\mu}}{\partial
Z^{\lambda}}dg_{1}^{\mu}.$ (44)

Here, $L^{conv}_{1}$ is the conventional operator given in Eq. (22). The last
two terms are the additional terms as compared to the conventional result as
reviewed in Eq. (34). The fourth term on the right derives from the correction
of the term $v_{\mu}dz^{\mu}$, similar to the last term on the right hand side
of Eq. (42). The last term comes from the correction to the 1-form
transformation rule as shown in Eq. (39).

As shown in the appendix of reference [12], the conventional contribution can
be reduced as fowllows

$\displaystyle L^{conv}_{1}~{}{{}^{d}}\gamma_{0}$ $\displaystyle=$
$\displaystyle-\frac{e}{mc}g_{1}^{i}\left(\frac{\partial{\bf A}_{j}}{\partial
X^{i}}-\frac{\partial{\bf A}_{i}}{\partial X^{j}}\right)dX^{j}$ (45)
$\displaystyle=$ $\displaystyle-\frac{e}{mc}{\bf g}_{1}^{\bf
X}~{}{\bf\times}~{}{\bf B}~{}{\bf\cdot}~{}d{\bf X},$

where ${\bf B}={\bf\nabla}~{}{\bf\times}~{}{\bf A}$ has been used. Thus, the
one form in Eq. (44) becomes

${}^{d}\Gamma^{(1)}$ $\displaystyle=$ $\displaystyle dS^{(1)}+\left[(u{\bf
b}+{\bf v}_{\bot})-\frac{e}{mc}{\bf B}~{}{\bf\times}~{}{\bf g}_{1}^{\bf
X}\right]~{}{\bf\cdot}~{}d{\bf X}-{\bf v}~{}{\bf\cdot}~{}d{\bf g}^{\bf
X}_{1}-\frac{e}{mc}{\bf g}_{1}^{\bf X}~{}{\bf\cdot}~{}{\bf\nabla}{\bf
A}~{}{\bf\cdot}~{}d{\bf g}^{\bf X}_{1}$ (46)
$\displaystyle-\left(\frac{u^{2}}{2}+\mu B+\frac{e}{m}\varphi\right)dt.$

Similar to the direct reduction procedure in reference [6], noting further
that

$\displaystyle\frac{1}{2}d\left({{\bf g}^{\bf
X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf A}~{}{\bf\cdot}~{}{{\bf g}^{\bf
X}_{1}}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(d{{\bf g}^{\bf
X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf A}~{}{\bf\cdot}~{}{{\bf g}^{\bf
X}_{1}}+{{\bf g}^{\bf X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf
A}~{}{\bf\cdot}~{}d{{\bf g}^{\bf X}_{1}}\right)$ $\displaystyle+{{\bf g}^{\bf
X}_{1}}~{}{\bf\cdot}~{}\left(d{\bf\nabla}{\bf A}\right)~{}{\bf\cdot}~{}{{\bf
g}^{\bf X}_{1}},$

one obtains

$\displaystyle{\bf g}^{\bf X}_{1}~{}{\bf\cdot}~{}{\bf\nabla}{\bf
A}~{}{\bf\cdot}~{}d{{\bf g}^{\bf X}_{1}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(d{{\bf g}^{\bf
X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf A}~{}{\bf\cdot}~{}{{\bf g}^{\bf
X}_{1}}-{{\bf g}^{\bf X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf
A}~{}{\bf\cdot}~{}d{{\bf g}^{\bf X}_{1}}\right)$
$\displaystyle+\frac{1}{2}\left(d{{\bf g}^{\bf
X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf A}~{}{\bf\cdot}~{}{{\bf g}^{\bf
X}_{1}}+{{\bf g}^{\bf X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf
A}~{}{\bf\cdot}~{}d{{\bf g}^{\bf X}_{1}}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(d{{\bf g}^{\bf
X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf A}~{}{\bf\cdot}~{}{{\bf g}^{\bf
X}_{1}}-{{\bf g}^{\bf X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf
A}~{}{\bf\cdot}~{}d{{\bf g}^{\bf X}_{1}}\right)+\frac{1}{2}d\left({{\bf
g}^{\bf X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf A}~{}{\bf\cdot}~{}{{\bf g}^{\bf
X}_{1}}\right)$ $\displaystyle-{{\bf g}^{\bf
X}_{1}}~{}{\bf\cdot}~{}\left(d{\bf\nabla}{\bf A}\right)~{}{\bf\cdot}~{}{{\bf
g}^{\bf X}_{1}}.$

Excluding the exact derivative and $O(\epsilon)$ terms, one has

$\displaystyle{{\bf g}^{\bf X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf
A}~{}{\bf\cdot}~{}d{{\bf g}^{\bf X}_{1}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{{\bf g}^{\bf X}_{1}}~{}{\bf\cdot}~{}\left(d{{\bf
g}^{\bf X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf A}-{{\bf g}^{\bf
X}_{1}}~{}{\bf\cdot}~{}{\bf\nabla}{\bf A}\right)$ (47) $\displaystyle=$
$\displaystyle\frac{1}{2}{{\bf g}^{\bf X}_{1}}~{}{\bf\times}~{}d{{\bf g}^{\bf
X}_{1}}~{}{\bf\cdot}~{}{\bf B}+O(\epsilon).$

Here, it has been noted that $d{\bf g}^{\bf
X}_{1}~{}{\bf\times}~{}{\bf\nabla}~{}{\bf\times}~{}{\bf A}={\bf\nabla}{\bf
A}~{}{\bf\cdot}~{}d{\bf g}^{\bf X}_{1}-d{\bf g}^{\bf
X}_{1}~{}{\bf\cdot}~{}{\bf\nabla}{\bf A}$. Therefore, one form in Eq. (46) is
further reduced to

${}^{d}\Gamma^{(1)}$ $\displaystyle=$ $\displaystyle dS^{(1)}+\left[(u{\bf
b}+{\bf v}_{\bot})-\frac{e}{mc}{\bf B}~{}{\bf\times}~{}{\bf g}_{1}^{\bf
X}\right]~{}{\bf\cdot}~{}d{\bf X}-{\bf v}~{}{\bf\cdot}~{}d{\bf g}^{\bf
X}_{1}-\frac{1}{2}\frac{e}{mc}{{\bf g}^{\bf X}_{1}}~{}{\bf\times}~{}d{{\bf
g}^{\bf X}_{1}}~{}{\bf\cdot}~{}{\bf B}$ (48)
$\displaystyle-\left(\frac{u^{2}}{2}+\mu B+\frac{e}{m}\varphi\right)dt.$

Again similar to the direct reduction procedure in reference [6], one can see
that, to reduce ${}^{d}\Gamma_{1}$ one can choose

$\displaystyle{\bf g}_{1}^{\bf X}$ $\displaystyle=$
$\displaystyle-{{\mbox{\boldmath$\rho$}}},$ (49)

where $-{{\mbox{\boldmath$\rho$}}}={\bf v}~{}{\bf\times}~{}{\bf b}/\Omega$. In
this case, the term ${\bf v}_{\bot}~{}{\bf\cdot}~{}d{\bf X}$ term is
cancelled, the term ${\bf v}~{}{\bf\cdot}~{}d{\bf g}^{\bf X}_{1}$ becomes of
order $\epsilon$, and the term $-\frac{1}{2}\frac{e}{mc}{{\bf g}^{\bf
X}_{1}}~{}{\bf\times}~{}d{{\bf g}^{\bf X}_{1}}~{}{\bf\cdot}~{}{\bf B}$ is
reduced to $(e/mc)\mu d\zeta$. Combing the contributions from
${}^{d}\Gamma^{(0)}$ and ${}^{d}\Gamma^{(1)}$, one finally obtains from Eq.
(48)

${}^{d}\Gamma=\left(\frac{e}{mc}{\bf A}+u{\bf b}\right)~{}{\bf\cdot}~{}d{\bf
X}+\frac{mc}{e}\mu d\zeta-\left(\frac{u^{2}}{2}+\mu
B+\frac{e}{m}\varphi\right)dt.$ (50)

This is the result obtained with the modified transform rule in Eq. (39).

Equation (50) agrees with the result using the direct approach in Eq. (1) [6],
This is different from the conventional results in references [4, 12] , as
cited in Eqs. (37) and (38), in which the same result (i.e., the term
$(mc/e)\mu d\zeta$) is only obtained in the second order, instead of the first
order. As indicated in the ordering analyses in Eq. (3), the conventional
result in Eq. (38) with $u{\bf b}~{}{\bf\cdot}~{}d{\bf X}$ being kept but the
term $(mc/e)\mu d\zeta$ dropped is ordering inconsistent.

Equation Eq. (50) is obtained through the modified transformation rule for one
form in Eq. (39). Two key factors are taken into account in deriving Eq. (39).
First, the ordering difference between the temporal variation of gyrophase and
that of the other phase space coordinates needs to be taken into account.
Second, it is also important to note that the limit and derivative cannot be
commuted in general. One can expect that not only the transformation rule for
one form is changed, but also any higher forms involving $dz^{\mu}$ (for
example 2 or 3 forms, etc..) are affected as well. Nevertheless, it is
noteworthy to note that since the main change for guiding center motion theory
lies in the transformation rule for one form, i.e., the Lagrangian, and one
usually does not need a backward transformation for Lagrangian. The usual
transformation rule for zero form remains basically valid. This leads the
forward and backward guiding center coordinate transformation in the
conventional Lie transform formulation in Eq. (17) remain applicable. This
indicates that the backward coordinate transformation can still be easily
obtained by the inversion of the near identity exponential operator
$e^{-\epsilon L_{g}}$ from the forward transformation. Therefore, the modified
framework for the case with a fast varying coordinate remains convenient for
practical applications.

##  IV Conclusions and discussion

In this paper, we show that the conventional Lie transform perturbation theory
for the guiding center motion of charged particles in a magnetic field needs
to be modified for ordering inconsistency. The reasons are two folds. First,
the components of $dz^{\mu}$ can be different in order. In the case of the
guiding center motion of charged particles in a magnetic field, the temporal
variation of gyrophase is much faster than other phase space coordinates. This
is actually noted in the non-Lie-transform formulation in reference [6]. The
other is related to the basic calculus rule. The parametric limit of the
derivative of a function is not equivalent to the derivative of the limit
function. This leads to the change of the near identity transformation rule.

With this ordering correction made, it is shown that the Lie transform
approach can achieve the same phase space Lagrangian of guiding center motion
in the first order as obtained by the direct approach in reference [6].
Without the correction, the same Lagrangian can only be obtained by the
expansion in the second order and beyond [4, 12]. This is a mathematical
problem. Its results were confirmed in multiple ways. The recovery of the
results with direct approach in Eq. (1) is a direct justification. The
ordering analyses in Eq. (3) justify the current result in Eq. (50), instead
of the conventional one in Eq. (38). Also, the key result in the paper, i.e.,
the transformation rule for one form in Eq.(39), is double confirmed by an
alternative derivation in Appendix A.

Let’s discuss this further. The conventional Lie transform theory [4, 12]
actually belongs to the regular perturbation theory. Because $d{\bf X}$ and
$d\zeta$ are different in order, there needs to be a singular perturbation
theory. This is similar to the treatment of boundary layer problem in the
fluid theory, the boundary layer theory applied to tearing modes in plasma
physics [16], and the renormalization process in dealing with the divergent
issue in the quantum field theory. In many fields of physics, people have
experienced such type of perturbation theory evolution. When the regular
perturbation theory was found to be incorrect, the singular perturbation
theory is developed with some kind of “renormalization”. In this regard, J. R.
Cary and A. J. Brizard made an important contribution in Ref. [6].

The importance of current work lies in the modification of the transformation
rule for one form or the Lagrangian for the system with a fast-varying
coordinate. In mathematics, differential forms give a unified description for
defining integrands over curves, surfaces, solids, and higher-dimensional
manifolds [7, 17]. It has many applications, especially in physics, geometry,
and topology. This certainly includes plasma physics due to pioneer
contributions by R. G. Littlejohn, J. R. Cary, A. J. Brizard, et al. [5]. For
example, Hamilton’s principle of least action is directly related to one form.
The physics system often needs perturbation analyses. The Lie transform
formalism for near identity transformation in the phase space provides a
unique and powerful tool for analyzing the Lagrangian system. In plasma
physics, it has been used to study the charged particle motion in a magnetic
field, the nonlinear gyrokinetics, the magnetic field flow, et al. (see for
instance the review articles in Refs. [5, 10, 6]). The Lie transform
perturbation theory deals with the variations of various forms, i.e., the
integrands, with the change of the variable differentials taken into account.
The change of the one-form transform rule pointed out in this paper is
critically important to the Lie transform framework. To justify the need for
this modification on a solid basis, the charged particle motion in a magnetic
field is used as an example for demonstration since it can be compared with
the results derived directly and verified by the obvious ordering analyses.
The correction to the near identity transformation rule affects not only the
theory of the guiding center motion of charged particles in a magnetic field,
but also generally to the systems with a rapidly varying coordinate. In plasma
physics, for example, one can see that the earlier derivation of the nonlinear
gyrokinetic equation needs to be repaired since the last term in the newly
derived transform rule in Eq. (39) has not been taken into consideration. This
also affects the applications in the classical mechanics and even the
fundamental mathematical formulation of Lie transformation as well. The
principle for the modification of one form pointed out in this paper affects
the transform rules for other forms, for example the two form related to the
exterior derivative. Again, two factors are needed to consider for other forms
in the perturbation analyses. First, one cannot just treat the integrands,
i.e., various forms, order by order, but ignore the ordering difference
between the variable differentials in the integration. Second, one cannot
simply commute the limit and derivative. This requires a systematical
reformulation of Lie transform for higher forms and will be addressed in the
future work. Nevertheless, for studying the Lagrangian system, the treatment
of one form given in this work is usually sufficient. These discussion
indicate that the current work has a severe impact on the Lie perturbation
theory. It basically limits the applicability of the conventional Lie
transform theory only to the system without a fast-varying coordinate. Note
that one of the most important applications of Lie transform is to treat the
system with a fast-varying coordinate to obtain the averaged effects over the
fast-varying coordinate. These indicate that the results in this paper are
important.

This research is supported by Department of Energy Grants DE-FG02-04ER54742.

##  Appendix A Alternative derivation

In this Appendix, we provide an alternative derivation of the transformation
rule in Eq. (39). Note that

$\displaystyle\Gamma_{\mu}dZ^{\mu}=\gamma_{\mu}dz^{\mu}.$ (51)

Using the transformation rule for scale function in Eq. (12), one obtains

$\displaystyle\frac{\partial}{\partial\epsilon}\left(\Gamma_{\mu}dZ^{\mu}\right)$
$\displaystyle=$
$\displaystyle\frac{\partial}{\partial\epsilon}\left(\gamma_{\mu}dz^{\mu}\right)$
(52) $\displaystyle=$ $\displaystyle\gamma_{\mu}d\frac{\partial
z^{\mu}}{\partial\epsilon}+\frac{\partial\gamma_{\mu}}{\partial\epsilon}dz^{\mu}$
$\displaystyle=$ $\displaystyle-d\gamma_{\mu}\frac{\partial
z^{\mu}}{\partial\epsilon}+\frac{\partial\gamma_{\mu}}{\partial\epsilon}\frac{\partial
z^{\mu}}{\partial Z^{\nu}}dZ^{\nu}+d\left(\gamma_{\mu}\frac{\partial
z^{\mu}}{\partial\epsilon}\right)$ $\displaystyle\to$
$\displaystyle-\frac{\partial\gamma_{\mu}}{\partial Z^{\nu}}\frac{\partial
z^{\mu}}{\partial\epsilon}dZ^{\nu}+\frac{\partial\gamma_{\mu}}{\partial\epsilon}\frac{\partial
z^{\mu}}{\partial Z^{\nu}}dZ^{\nu}.$

Here, the total derivative has been dropped, since we consider the variational
principle. Using Eqs. (9) and (11), one has

$\displaystyle\frac{\partial}{\partial\epsilon}\left(\Gamma_{\mu}dZ^{\mu}\right)$
$\displaystyle=$ $\displaystyle\frac{\partial\gamma_{\mu}}{\partial
Z^{\nu}}\left(g^{\lambda}\frac{\partial z^{\mu}}{\partial
Z^{\lambda}}\right)dZ^{\nu}-g^{\lambda}\frac{\partial\gamma_{\mu}}{\partial
Z^{\lambda}}\frac{\partial z^{\mu}}{\partial Z^{\nu}}dZ^{\nu}$ (53)

Carrying out the expansion for $\partial{z^{\nu}}/\partial Z^{\nu}$ to
sufficient order, the transformation rule for one form in Eq. (39) is
recovered.

Similar to the derivation of Eq. (52), it can be shown that the conventional
result

$\displaystyle\frac{\partial}{\partial\epsilon}\left(\Gamma_{\mu}dZ^{\mu}\right)$
$\displaystyle=$ $\displaystyle
g^{\mu}\left(\frac{\partial\gamma_{\nu}}{\partial
Z^{\mu}}-\frac{\partial\gamma_{\mu}}{\partial Z^{\nu}}\right)dZ^{\nu}$

is actually obtained from the formula
$\Gamma_{\mu}dZ^{\mu}=\gamma_{\mu}dZ^{\mu}$, instead of
$\Gamma_{\mu}dZ^{\mu}=\gamma_{\mu}dz^{\mu}$ in Eq. (51). This is the
consequence of the non-consistent exchange between the limit ($\epsilon\to 0$)
and derivative.

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