# Polarization Effects in Higher-order Guiding-center Lagrangian Dynamics

Alain J. Brizard Email address for correspondence<EMAIL_ADDRESS>Department of Physics, Saint Michael’s College, Colchester, VT 05439, USA

(?; revised ?; accepted ?. - To be entered by editorial office)

###### Abstract

The extended guiding-center Lagrangian equations of motion are derived by Lie-
transform perturbation method under the assumption of time-dependent and
inhomogeneous electric and magnetic fields that satisfy the standard guiding-
center space-time orderings. Polarization effects are introduced into the
Lagrangian dynamics by the inclusion of the polarization drift velocity in the
guiding-center velocity and the appearance of finite-Larmor-radius corrections
in the guiding-center Hamiltonian and guiding-center Poisson bracket.

##  1 Introduction

Polarization effects have a rich history in plasma physics (Pfirsch, 1984;
Pfirsch & Morrison, 1985; Cary & Brizard, 2009). Their importance stems from
the assumption of quasineutrality in a strongly magnetized plasma and the
dielectric properties of a guiding-center plasma (Hinton & Robertson, 1984).
While these effects are traditionally associated with the presence of an
electric field in a magnetized plasma (Itoh & Itoh, 1996; Hazeltine & Meiss,
2003; Wang & Hahm, 2009; Joseph, 2021; Brizard, 2023a), they are also
associated with magnetic drifts (Kaufman, 1986; Brizard, 2013; Tronko &
Brizard, 2015).

Recently, second-order terms in guiding-center Hamiltonian theory (in the
absence of an electric field) were shown to be crucial (Brizard & Hodgeman,
2023) in assessing the validity of the guiding-center representation in
determining whether guiding-center orbits were numerically faithful to the
particle orbits in axisymmetric magnetic geometries, which partially confirmed
earlier numerical studies in axisymmetric tokamak plasmas (Belova et al.,
2003). In particular, it was shown that a second-order correction associated
with guiding-center polarization (Kaufman, 1986; Brizard, 2013; Tronko &
Brizard, 2015) was needed in order to obtain faithful guiding-center orbits.

Indeed, without the inclusion of second-order effects, it was shown that,
within a few bounce periods after leaving the same physical point in particle
phase space, a first-order guiding-center orbit deviated noticeably from its
associated particle orbit, while a second-order guiding-center orbit followed
the particle orbit to a high degree of precision (Brizard & Hodgeman, 2023).
In addition, as initially reported by Belova et al. (2003), the guiding-center
Hamiltonian formulation is a faithful representation of the particle toroidal
angular momentum (Tronko & Brizard, 2015; Brizard & Hodgeman, 2023), which is
an exact particle constant of motion in an axisymmetric magnetic field, only
if second-order effects are included.

###  1.1 Lagrangian dynamics in extended phase space

In the present work, we consider time-dependent and inhomogeneous electric and
magnetic fields (which still satisfy the guiding-center space-time orderings
$|\nabla|^{-1}\gg\rho_{\rm th}=v_{\rm th}/\Omega$ and $\partial/\partial
t\ll\Omega=eB/mc$) and we assume that the $E\times B$ velocity ${\bf u}_{\rm
E}={\bf E}\,\mbox{\boldmath$\times$}\,c\widehat{{\sf b}}/B$ is comparable to
the particle’s thermal velocity $v_{\rm th}$. Because of the explicit time
dependence of the electromagnetic fields, the Lagrangian charged-particle
dynamics takes place in an odd-dimensional space $({\bf q},{\bf p},t)$, where
the non-autonomous Hamiltonian $H({\bf q},{\bf p},t)$ is a function of the
canonical coordinates $({\bf q},{\bf p})$, from which the canonical Hamilton
equations $d{\bf q}/dt=\partial H/\partial{\bf p}$ and $d{\bf
p}/dt=-\,\partial H/\partial{\bf q}$ are derived, as well as time $t$, from
which we obtain the energy equation $dH/dt=\partial H/\partial t$ (i.e.,
energy is not conserved).

The use of an extended phase space is a well-known method in classical
mechanics (Lanczos, 1970) used to deal with a time-dependent Hamiltonian
system by transforming it into an autonomous Hamiltonian system in an even-
dimensional symplectic setting. Here, the canonical time-energy coordinates
$(t,w)$ are included in the extended phase-space coordinates $({\bf q},t;{\bf
p},w)$, where the space-time coordinates $({\bf q},t)$ are canonically
conjugate to the momentum-energy coordinates $({\bf p},w)$, with the extended
Hamilton equations $dw/ds=\partial{\mathcal{H}}/\partial t=\partial H/\partial
t$ and $dt/ds=-\,\partial{\mathcal{H}}/\partial w=1$, where
${\mathcal{H}}\equiv H({\bf q},{\bf p},t)-w$ is the extended Hamiltonian and a
particle orbit in extended phase space (parametrized by $s$) takes place on
the energy surface ${\mathcal{H}}=0$, i.e., $w=H({\bf q},{\bf p},t)$.

Using the dimensional ordering parameter $\epsilon$ associated with the
renormalized particle mass $m\rightarrow\epsilon\,m$ (Brizard, 1995), instead
of the standard ordering $e\rightarrow e/\epsilon$ (Kulsrud, 1983; Littlejohn,
1983), we begin with the extended phase-space particle Lagrangian one-form

$\gamma\;=\;\left(\frac{e}{c}\;{\bf A}\;+\;\epsilon\,{\bf
p}_{0}\right)\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf x}\;-\;w\;{\sf
d}t\;\equiv\;\gamma_{0}\;+\;\epsilon\,\gamma_{1},$ (1)

and the extended particle Hamiltonian

${\mathcal{H}}\;=\;e\,\Phi\;-\;w\;+\;\epsilon\,|{\bf
p}_{0}|^{2}/2m\;\equiv\;{\mathcal{H}}_{0}+\epsilon\,{\mathcal{H}}_{1},$ (2)

where ${\bf p}_{0}$ denotes the local particle kinetic momentum at position
${\bf x}$. In the present work, we consider the standard ordering (Kulsrud,
1983) for the parallel electric field: ${\bf E}={\bf
E}_{\bot}+\epsilon\,E_{\|}\,\widehat{{\sf b}}$. In contrast to Madsen (2010)
and Frei et al. (2020), who used the same mass ordering
($m\rightarrow\epsilon\,m$), we use extended (eight-dimensional) phase space
in Eqs. (1)-(2), where the energy coordinate $w$ is canonically conjugate to
time $t$ (Littlejohn, 1981; Cary & Brizard, 2009). This extended phase-space
formulation yields a simple form for the extended Poisson bracket [see Eq.
(60)], also adopted (without derivation) by Madsen (2010), which plays an
important role in the variational formulation of the guiding-center Vlasov-
Maxwell equations (Brizard, 2023b).

Here, the electric-field ordering implies that the local particle momentum

${\bf p}_{0}\;\equiv\;p_{\|0}\,\widehat{{\sf b}}({\bf x},t)\;+\;{\bf P}_{\rm
E}({\bf x},t)\;+\;{\bf q}_{\bot 0}(J_{0},\theta_{0};{\bf x},t)$ (3)

is decomposed into the gyroangle-independent parallel component
$p_{\|0}\equiv{\bf p}_{0}\,\mbox{\boldmath$\cdot$}\,\widehat{{\sf b}}$ and the
$E\times B$ momentum

${\bf P}_{\rm E}\;\equiv\;{\bf
E}\,\mbox{\boldmath$\times$}\,\frac{e\widehat{{\sf b}}}{\Omega}\;=\;m\,{\bf
u}_{\rm E},$ (4)

and the gyroangle-dependent perpendicular momentum ${\bf q}_{\bot
0}\equiv|{\bf q}_{\bot 0}|\,\widehat{\bot}$, respectively, where
$J_{0}\equiv|{\bf q}_{\bot 0}|^{2}/(2m\Omega)$ represents the lowest-order
gyroaction and the gyroangle derivative $\partial{\bf q}_{\bot
0}/\partial\theta_{0}\equiv{\bf q}_{\bot
0}\,\mbox{\boldmath$\times$}\,\widehat{{\sf b}}=-\,|{\bf q}_{\bot
0}|\,\widehat{\rho}$ introduces the rotating orthogonal unit-vector fields
$(\widehat{{\sf b}},\widehat{\bot},\widehat{\rho})$.

###  1.2 Purpose of the present work

The purpose of the present work is motivated by the need to derive higher-
order guiding-center equations that can accurately describe the magnetic
confinement of charged particles in regions with steep gradients (e.g., the
pedestal region of advanced tokamak plasmas). For many situations of practical
interest, the presence of a strong electric field is associated with strong
plasma flows with steep sheared rotation profiles for which second-order
effects (including finite-Larmor-radius effects) must be included in a self-
consistent guiding-center theory (Hahm, 1996; Chang et al., 2004; Lanthaler et
al., 2019; Frei et al., 2020).

Guiding-center equations of motion with second-order corrections in the
presence of time-independent electric and magnetic fields were derived using
Lie-transform perturbation method by Brizard (1995) and Hahm (1996), following
the earlier work of Littlejohn (1981). These perturbation methods were also
used by Miyato et al. (2009), Madsen (2010), and Frei et al. (2020), who
derived self-consistent guiding-center Vlasov-Maxwell equations that included
guiding-center polarization and magnetization effects. Not all second-order
effects were included in these models, however, and it is the purpose of the
present work to derive a more complete higher-order guiding-center Vlasov-
Maxwell theory, with a full representation of guiding-center polarization that
can be directly derived by the guiding-center push-forward method (Brizard,
2013; Tronko & Brizard, 2015).

##  2 Guiding-center Lie-transform Perturbation Analysis

The derivation of the guiding-center equations of motion by Lie-transform
perturbation method is based on a phase-space transformation from the (local)
particle extended (eight-dimensional) phase-space coordinates
$z_{0}^{\alpha}=({\bf x},p_{\|0};J_{0},\theta_{0};w_{0},t)$, where the energy-
time canonical coordinates $(w_{0},t)$ are included, to the guiding-center
phase-space coordinates $Z^{\alpha}=({\bf X},P_{\|};J,\theta;W,t)$ generated
by the vector fields $({\sf G}_{1},{\sf G}_{2},\cdots)$:

$Z^{\alpha}\;=\;z_{0}^{\alpha}\;+\;\epsilon\,G_{1}^{\alpha}\;+\;\epsilon^{2}\,\left(G_{2}^{\alpha}+\frac{1}{2}\,{\sf
G}_{1}\cdot{\sf d}G_{1}^{\alpha}\right)\;+\;\cdots,$ (5)

and its inverse

$z_{0}^{\alpha}\;=\;Z^{\alpha}\;-\;\epsilon\,G_{1}^{\alpha}\;-\;\epsilon^{2}\,\left(G_{2}^{\alpha}-\frac{1}{2}\,{\sf
G}_{1}\cdot{\sf d}G_{1}^{\alpha}\right)\;+\;\cdots.$ (6)

In order for the particle time to be invariant under the guiding-center
transformation (5), we require that $G_{n}^{t}\equiv 0$ to all orders $n\geq
1$, i.e., the guiding-center time is identical to the particle time. From
these generating vectors fields, the pull-back and push-forward Lie-transform
operators ${\sf T}_{\rm
gc}=\exp(\epsilon\pounds_{1})\,\exp(\epsilon^{2}\pounds_{2})\cdots$ and ${\sf
T}_{\rm
gc}^{-1}=\cdots\exp(-\epsilon^{2}\pounds_{2})\,\exp(-\epsilon\pounds_{1})$ are
constructed in terms of Lie derivatives $(\pounds_{1},\pounds_{2},\cdots)$
generated by the vector fields $({\sf G}_{1},{\sf G}_{2},\cdots)$. More
details about the Lie-transform perturbation method used in guiding-center
theory can be found in Littlejohn (1982) (as well as the unpublished UCLA
report Variational Principles for Guiding Center Motion (PPG-611) written by
Littlejohn in 1982), while the notation used here is taken from Tronko &
Brizard (2015) and Brizard & Tronko (2016).

We now wish to derive the extended guiding-center phase-space Lagrangian one-
form

$\Gamma_{\rm gc}\;\equiv\;{\sf T}_{\rm gc}^{-1}\gamma+{\sf
d}\sigma=\Gamma_{0{\rm gc}}+\epsilon\,\Gamma_{1{\rm
gc}}+\epsilon^{2}\,\Gamma_{2{\rm gc}}+\cdots,$ (7)

where the gauge scalar field
$\sigma=\epsilon\,\sigma_{1}+\epsilon^{2}\sigma_{2}+\cdots$ is chosen at each
order in order to simplify the transformation, with

$\displaystyle\Gamma_{1{\rm gc}}$ $\displaystyle=$
$\displaystyle\gamma_{1}\;-\;\iota_{1}\cdot\mbox{\boldmath$\omega$}_{0}\;+\;{\sf
d}\sigma_{1},$ (8) $\displaystyle\Gamma_{2{\rm gc}}$ $\displaystyle\equiv$
$\displaystyle-\;\iota_{2}\cdot\mbox{\boldmath$\omega$}_{0}-\frac{1}{2}\;\iota_{1}\cdot\left(\mbox{\boldmath$\omega$}_{1}\;+\frac{}{}\mbox{\boldmath$\omega$}_{{\rm
gc}1}\right)+{\sf d}\sigma_{2},$ (9)

and the extended guiding-center Hamiltonian

${\mathcal{H}}_{\rm gc}\;\equiv\;{\sf T}_{\rm
gc}^{-1}{\mathcal{H}}\;=\;{\mathcal{H}}_{0{\rm
gc}}+\epsilon\,{\mathcal{H}}_{1{\rm gc}}+\epsilon^{2}\,{\mathcal{H}}_{2{\rm
gc}}+\cdots,$ (10)

where

$\displaystyle{\mathcal{H}}_{1{\rm gc}}$ $\displaystyle=$
$\displaystyle{\mathcal{H}}_{1}\;-\;{\sf G}_{1}\cdot{\sf d}{\mathcal{H}}_{0},$
(11) $\displaystyle{\mathcal{H}}_{2{\rm gc}}$ $\displaystyle=$
$\displaystyle-\,{\sf G}_{2}\cdot{\sf
d}{\mathcal{H}}_{0}\;-\;\frac{1}{2}\;{\sf G}_{1}\cdot{\sf
d}\left({\mathcal{H}}_{1}+{\mathcal{H}}_{1{\rm gc}}\right).$ (12)

Here, we use the formulas $\iota_{n}\cdot\mbox{\boldmath$\omega$}\equiv
G_{n}^{\alpha}\omega_{\alpha\beta}{\sf d}Z^{\beta}$ (for an arbitrary two-form
$\omega$) and ${\sf G}_{n}\cdot{\sf d}{\mathcal{K}}\equiv
G_{n}^{\alpha}\partial{\mathcal{K}}/\partial Z^{\alpha}$ (for an arbitrary
scalar field ${\mathcal{K}}$), where the summation rule is used. We note that
the guiding-center phase-space transformation considered in the present work
will contain all terms associated with first-order space-time derivatives of
the electric and magnetic fields, which will require us to consider some terms
at third order in $\epsilon$ in Eq. (7).

In order to construct the extended guiding-center Lagrangian one-form (7), we
will need to evaluate the contractions
$\iota_{n}\cdot\mbox{\boldmath$\omega$}_{0}$ generated by the vector fields
$({\sf G}_{1},{\sf G}_{2},\cdots)$ on the zeroth-order two-form:

$\displaystyle\mbox{\boldmath$\omega$}_{0}\;=\;{\sf d}\gamma_{0}$
$\displaystyle=$ $\displaystyle\frac{e}{c}\left(\frac{\partial A_{j}}{\partial
x^{i}}\;{\sf d}x^{i}\;+\;\frac{\partial A_{j}}{\partial t}\;{\sf
d}t\right)\;\wedge\;{\sf d}x^{j}\;-\;{\sf d}w\wedge{\sf d}t,$ (13)

so that we obtain the $n^{\rm th}$-order expression

$\iota_{n}\cdot\mbox{\boldmath$\omega$}_{0}=\frac{e}{c}\;{\bf
B}\,\mbox{\boldmath$\times$}\,G_{n}^{{\bf x}}\,\mbox{\boldmath$\cdot$}\,{\sf
d}{\bf X}-\left(\frac{e}{c}\,G_{n}^{{\bf
x}}\,\mbox{\boldmath$\cdot$}\,\frac{\partial\bf A}{\partial
t}+G_{n}^{w}\right)\;{\sf d}t,$ (14)

where $G_{n}^{{\bf x}}$ and $G_{n}^{w}$ denote the spatial and energy
components of the $n$th-order generating vector field ${\sf G}_{n}$.
Similarly, we will need to evaluate the contractions
$\iota_{n}\cdot\mbox{\boldmath$\omega$}_{1}$ generated by the vector fields
$({\sf G}_{1},{\sf G}_{2},\cdots)$ on the first-order two-form
$\mbox{\boldmath$\omega$}_{1}={\sf d}\gamma_{1}$, which yields the $(n+1)^{\rm
th}$-order expression

$\iota_{n}\cdot\mbox{\boldmath$\omega$}_{1}\equiv D_{n}({\bf
p}_{0})\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf X}-G_{n}^{\bf
x}\,\mbox{\boldmath$\cdot$}\,\left(\frac{\partial{\bf p}_{0}}{\partial
t}\;{\sf d}t+\widehat{{\sf b}}\;{\sf d}p_{\|0}+\frac{\partial{\bf q}_{\bot
0}}{\partial J_{0}}{\sf d}J_{0}+\frac{\partial{\bf q}_{\bot
0}}{\partial\theta_{0}}{\sf d}\theta_{0}\right),$ (15)

where ${\bf p}_{0}$ is defined in Eq. (3) and the spatial components are
expressed in terms of the operator (Tronko & Brizard, 2015)

$\displaystyle D_{n}({\bf C})$ $\displaystyle\equiv$
$\displaystyle\left(G_{n}^{p_{\|}}\,\frac{\partial{\bf C}}{\partial
p_{\|0}}+G_{n}^{J}\,\frac{\partial{\bf C}}{\partial
J_{0}}+G_{n}^{\theta}\,\frac{\partial{\bf
C}}{\partial\theta_{0}}\right)\;-\;G_{n}^{\bf
x}\,\mbox{\boldmath$\times$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf C},$ (16)

where ${\bf C}$ is an arbitrary vector function on guiding-center phase space.
In what follows, unless it is necessary, we will omit writing the subscript
$0$ on local particle phase-space coordinates (i.e., $p_{\|0}$ is written as
$p_{\|}$).

The purpose of the Lie-transform expressions (7) and (10) is to construct a
gyroangle-independent extended guiding-center phase-space Lagrangian one-form
$\Gamma_{\rm gc}$ and extended guiding-center Hamiltonian ${\mathcal{H}}_{\rm
gc}$ in terms of which guiding-center equations are derived. The generic forms
considered for the extended guiding-center phase-space Lagrangian one-form (7)
and the extended guiding-center Hamiltonian (10) in the present work are

$\displaystyle\Gamma_{\rm gc}$ $\displaystyle\equiv$
$\displaystyle\frac{e}{c}\,{\bf A}\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf
X}\;-\;W\,{\sf d}t\;+\;\epsilon\,{\bf P}_{\rm
gc}\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf X}\;+\;\epsilon^{2}J\,\left({\sf
d}\theta-{\bf R}\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf X}-{\mathcal{S}}\,{\sf
d}t\right),$ (17) $\displaystyle{\mathcal{H}}_{\rm gc}$ $\displaystyle\equiv$
$\displaystyle e\,\Phi\;+\;\epsilon\,K_{\rm gc}\;-\;W,$ (18)

where the gyroangle-independent guiding-center kinetic momentum ${\bf P}_{\rm
gc}$ and guiding-center kinetic energy $K_{\rm gc}$ are expressed as
asymptotic series in powers of $\epsilon$, while the vector field ${\bf
R}\equiv\nabla\widehat{\bot}\,\mbox{\boldmath$\cdot$}\,\widehat{\rho}$ and the
scalar field ${\mathcal{S}}\equiv(\partial\widehat{\bot}/\partial
t)\,\mbox{\boldmath$\cdot$}\,\widehat{\rho}$ are required to preserve
gyrogauge invariance. See App. A of the recent paper by Brizard (2023b) for an
updated discussion on gyrogauge invariance introduced by Littlejohn (1981,
1983, 1988). We note that the separation of the guiding-center transformations
of the extended guiding-center phase-space Lagrangian one-form (7) and the
extended guiding-center Hamiltonian (10) might enable the application of
computer algorithms previously used by Burby et al. (2013), but this
consideration falls outside the scope of the present work.

##  3 Symplectic Polarization Guiding-center Theory

In the present work, we will consider the symplectic polarization guiding-
center theory, where the guiding-center kinetic momentum ${\bf P}_{\rm gc}$
retains the contribution from the $E\times B$ momentum (4), which will then
introduce the polarization drift velocity explicitly in the guiding-center
equations of motion.

In the alternate Hamiltonian polarization guiding-center theory, on the other
hand, the $E\times B$ momentum (4) is removed from the guiding-center kinetic
momentum and polarization effects enter solely through the guiding-center
Hamiltonian. The Hamiltonian case generally requires a different ordering for
the electric field (because it produces a polarization displacement that must
be compared with the characteristic lowest-order particle gyroradius) and will
be considered in a future publication.This dual representation is analogous to
the treatment of the perturbed magnetic field in nonlinear gyrokinetic theory
(Brizard & Hahm, 2007).

The reader interested in results of the guiding-center Lie-transform
perturbation analysis can skip Section 3 and go to Section 4, where we present
the extended guiding-center Hamiltonian structure as well as the (regular)
guiding-center Lagrangian, from which guiding-center polarization and
magnetization can be derived by functional derivatives with respect to the
electric field ${\bf E}$ and magnetic field ${\bf B}$, respectively.

###  3.1 First-order perturbation analysis

####  3.1.1 First-order symplectic structure

We begin our perturbation analysis by considering the first-order guiding-
center symplectic one-form (8), which is now explicitly written in the
symplectic polarization representation as

$\displaystyle\Gamma_{1{\rm gc}}$ $\displaystyle=$
$\displaystyle\left(P_{\|}\;\widehat{{\sf b}}\;+\;{\bf P}_{\rm E}\;+\;{\bf
q}_{\bot}\right)\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf
X}\;-\;\frac{e}{c}\;{\bf B}\,\mbox{\boldmath$\times$}\,G_{1}^{{\bf
x}}\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf
X}\;+\;\left(\frac{e}{c}\,G_{1}^{{\bf
x}}\,\mbox{\boldmath$\cdot$}\,\frac{\partial\bf A}{\partial
t}\;+\;G_{1}^{w}\right)\;{\sf d}t$ (19) $\displaystyle=$
$\displaystyle\left(P_{\|}\,\widehat{{\sf b}}\;+\;{\bf P}_{\rm
E}\right)\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf X}\;\equiv\;{\bf
P}_{0}\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf X},$

where the first-order gauge scalar field $\sigma_{1}$ is not needed and the
gyroangle-dependent terms on the right of Eq. (19) are eliminated by selecting

$G_{1}^{{\bf x}}\;=\;{\bf
q}_{\bot}\,\mbox{\boldmath$\times$}\,\frac{c\widehat{{\sf
b}}}{eB}\;=\;\frac{1}{m\Omega}\;\frac{\partial{\bf
q}_{\bot}}{\partial\theta}\;\equiv\;-\;\mbox{\boldmath$\rho$}_{0},$ (20)

and

$G_{1}^{w}\;=\;-\,\frac{e}{c}\,G_{1}^{{\bf
x}}\,\mbox{\boldmath$\cdot$}\,\frac{\partial\bf A}{\partial
t}\;=\;\frac{e}{c}\,\frac{\partial\bf A}{\partial
t}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\rho$}_{0},$ (21)

where the gyroangle-dependent vector $\mbox{\boldmath$\rho$}_{0}$ represents
the lowest-order particle gyroradius.

With $G_{1}^{{\bf x}}$ defined by Eq. (20), the resulting first-order guiding-
center phase-space Lagrangian $\Gamma_{1{\rm gc}}={\bf
P}_{0}\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf X}$ yields the $(n+1)^{\rm
th}$-order contraction

$\iota_{n}\cdot\mbox{\boldmath$\omega$}_{{\rm gc}1}\equiv D_{n}({\bf
P}_{0})\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf X}-G_{n}^{\bf
x}\,\mbox{\boldmath$\cdot$}\,\left(\widehat{{\sf b}}\;{\sf
d}P_{\|}+\frac{\partial{\bf P}_{0}}{\partial t}{\sf d}t\right),$ (22)

where, using the operator (16), the spatial components in Eq. (22) are

$\displaystyle D_{n}({\bf P}_{0})$ $\displaystyle=$ $\displaystyle
G_{n}^{p_{\|}}\;\widehat{{\sf b}}-G_{n}^{\bf
x}\,\mbox{\boldmath$\times$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf P}_{0},$
(23)

which contain gyroangle dependent and independent contributions.

####  3.1.2 First-order Hamiltonian

The first-order guiding-center Hamiltonian is determined from Eq. (11) as

$\displaystyle{\mathcal{H}}_{1{\rm gc}}$ $\displaystyle=$
$\displaystyle\mu\,B+\frac{|{\bf P}_{0}|^{2}}{2m}+{\bf P}_{\rm
E}\,\mbox{\boldmath$\cdot$}\,\frac{{\bf q}_{\bot}}{m}-e\,G_{1}^{\bf
x}\,\mbox{\boldmath$\cdot$}\,\nabla\Phi+G_{1}^{w}$ (24) $\displaystyle=$
$\displaystyle\mu\,B+\frac{|{\bf P}_{0}|^{2}}{2m}+{\bf P}_{\rm
E}\,\mbox{\boldmath$\cdot$}\,\frac{{\bf
q}_{\bot}}{m}-e\,\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,{\bf
E},$

where the components (20)-(21) were substituted, and the electric field is
defined as ${\bf E}=-\,\nabla\Phi-c^{-1}\partial{\bf A}/\partial t$. By using
the identity

${\bf P}_{\rm E}\,\mbox{\boldmath$\cdot$}\,{\bf q}_{\bot}/m\;=\;{\bf
E}\,\mbox{\boldmath$\times$}\,\frac{c\widehat{{\sf
b}}}{B}\,\mbox{\boldmath$\cdot$}\,{\bf
q}_{\bot}\;=\;e\,\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,{\bf
E},$ (25)

the first-order guiding-center Hamiltonian is automatically gyroangle-
independent

${\mathcal{H}}_{1{\rm gc}}\;=\;\mu\,B+|{\bf P}_{0}|^{2}/(2m)\;=\;\mu
B\;+\;P_{\|}^{2}/(2m)\;+\;(m/2)\;\left|{\bf
E}\,\mbox{\boldmath$\times$}\,c\widehat{{\sf b}}/B\right|^{2},$ (26)

which corresponds to the kinetic energy in the frame drifting with the
$E\times B$ velocity.

###  3.2 Second-order perturbation analysis

####  3.2.1 Second-order symplectic structure

We now proceed with the second-order guiding-center symplectic one-form (9),
which is explicitly expressed as

$\displaystyle\Gamma_{2{\rm gc}}$ $\displaystyle=$
$\displaystyle-\;\left[\frac{e}{c}{\bf
B}\,\mbox{\boldmath$\times$}\,G_{2}^{{\bf x}}\;+\;D_{1}({\bf
P}_{2})\right]\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf
X}\;+\;\frac{1}{2}\,G_{1}^{{\bf
x}}\,\mbox{\boldmath$\cdot$}\,\left(\frac{\partial{\bf q}_{\bot}}{\partial
J}\,{\sf d}J+\frac{\partial{\bf q}_{\bot}}{\partial\theta}\;{\sf
d}\theta\right)$ (27) $\displaystyle+\;\left(G_{1}^{\bf
x}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf P}_{2}}{\partial
t}\;+\;G_{2}^{w}\;+\;\frac{e}{c}\,G_{2}^{\bf
x}\,\mbox{\boldmath$\cdot$}\,\frac{\partial\bf A}{\partial t}\right)\;{\sf
d}t$ $\displaystyle\equiv$
$\displaystyle\mbox{\boldmath$\Pi$}_{1}\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf
X}\;+\;J\;\left({\sf d}\theta\;-\frac{}{}{\bf
R}\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf X}\;-\;{\mathcal{S}}\;{\sf
d}t\right),$

where the second-order gauge scalar field $\sigma_{2}$ is not needed, and,
using the definition (16) with ${\bf P}_{2}\equiv{\bf P}_{0}+\frac{1}{2}\,{\bf
q}_{\bot}$, we find

$\displaystyle D_{1}({\bf P}_{2})$ $\displaystyle=$ $\displaystyle D_{1}({\bf
P}_{0})\;+\;\frac{1}{2}\left(g_{1}^{J}\;\frac{\partial{\bf q}_{\bot}}{\partial
J}+g_{1}^{\theta}\;\frac{\partial{\bf
q}_{\bot}}{\partial\theta}\right)\;+\;J\left[{\bf
R}-\left(\frac{1}{2}\,\tau+\alpha_{1}\right)\widehat{{\sf b}}\right],$ (28)

where $g_{1}^{J}\equiv
G_{1}^{J}-J\,\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,\nabla\ln B$
and $g_{1}^{\theta}\equiv
G_{1}^{\theta}+\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,{\bf R}$,
while $\tau\equiv\widehat{{\sf
b}}\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,\widehat{{\sf
b}}$ and $\alpha_{1}\equiv{\sf a}_{1}:\nabla\widehat{{\sf b}}$ is defined in
terms of the gyroangle-dependent dyadic tensor ${\sf
a}_{1}\equiv-\,\frac{1}{2}(\widehat{\bot}\widehat{\rho}+\widehat{\rho}\widehat{\bot})$
(Tronko & Brizard, 2015). The first-order guiding-center symplectic momentum
$\mbox{\boldmath$\Pi$}_{1}$ in Eq. (27), which is assumed to be gyroangle-
independent, will be determined based on the consistency of the Lie-transform
perturbation analysis at the third order (see Sec. 3.3) as well as the
guiding-center push-forward derivation of the guiding-center polarization in
the absence of a background electric field (Brizard, 2013; Tronko & Brizard,
2015).

Substituting these expressions into Eq. (27), we obtain the vector equation

$\displaystyle\mbox{\boldmath$\Pi$}_{1}$ $\displaystyle=$
$\displaystyle-\,\frac{e}{c}{\bf B}\,\mbox{\boldmath$\times$}\,G_{2}^{\bf
x}\;-\;g_{1}^{p_{\|}}\;\widehat{{\sf
b}}\;-\;\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\times$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{0}\;-\;\frac{1}{2}\left(g_{1}^{J}\;\frac{\partial{\bf q}_{\bot}}{\partial
J}+g_{1}^{\theta}\;\frac{\partial{\bf q}_{\bot}}{\partial\theta}\right),$ (29)

where $g_{1}^{p_{\|}}\equiv
G_{1}^{p_{\|}}\;-\;J\left(\frac{1}{2}\,\tau\;+\;\alpha_{1}\right)$, while we
choose the second-order energy component

$G_{2}^{w}\;=\;\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
P}_{0}}{\partial t}\;-\;\frac{e}{c}\frac{\partial\bf A}{\partial
t}\,\mbox{\boldmath$\cdot$}\,G_{2}^{\bf x},$ (30)

where we used

$G_{1}^{\bf x}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf P}_{2}}{\partial
t}\;=\;-\;\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
P}_{2}}{\partial
t}\;=\;-\;\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
P}_{0}}{\partial t}\;-\;J\,{\mathcal{S}}.$ (31)

Next, from the parallel component of Eq. (29), we obtain the first-order
component

$G_{1}^{p_{\|}}\;=\;\frac{\partial\mbox{\boldmath$\rho$}_{0}}{\partial\theta}\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{0}\;+\;J\left(\frac{1}{2}\,\tau\;+\;\alpha_{1}\right)\;-\;\Pi_{1\|},$ (32)

where $\Pi_{1\|}\equiv\widehat{{\sf
b}}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\Pi$}_{1}$ is the parallel
component of the first-order symplectic momentum $\mbox{\boldmath$\Pi$}_{1}$.
From the perpendicular components of Eq. (29), on the other hand, we find

$\displaystyle G_{2}^{{\bf x}}$ $\displaystyle=$ $\displaystyle G_{2\|}^{\bf
x}\;\widehat{{\sf b}}+\left(\frac{\widehat{{\sf
b}}}{m\Omega}\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{0}\right)\;\mbox{\boldmath$\rho$}_{0}+\frac{1}{2}\left(g_{1}^{J}\;\frac{\partial\mbox{\boldmath$\rho$}_{0}}{\partial
J}+g_{1}^{\theta}\;\frac{\partial\mbox{\boldmath$\rho$}_{0}}{\partial\theta}\right)-\mbox{\boldmath$\Pi$}_{1}\,\mbox{\boldmath$\times$}\,\frac{\widehat{{\sf
b}}}{m\Omega},$ (33)

where $G_{2\|}^{\bf x}\equiv\widehat{{\sf
b}}\,\mbox{\boldmath$\cdot$}\,G_{2}^{{\bf x}}$. The interpretation of the
first-order guiding-center symplectic momentum $\mbox{\boldmath$\Pi$}_{1}$
will be given in Sec. 3.3.2.

####  3.2.2 Second-order Hamiltonian

The second-order guiding-center Hamiltonian is determined from Eq. (12) as

$\displaystyle{\mathcal{H}}_{2{\rm gc}}$ $\displaystyle=$ $\displaystyle
e\,{\bf E}\,\mbox{\boldmath$\cdot$}\,\left(G_{2}^{\bf x}\;-\;\frac{1}{2}\,{\sf
G}_{1}\cdot{\sf
d}\mbox{\boldmath$\rho$}_{0}\right)\;+\;\frac{e}{2}\;\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,\nabla{\bf
E}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\rho$}_{0}\;-\;\frac{P_{\|}}{m}\,G_{1}^{p_{\|}}\;-\;\Omega\;G_{1}^{J}$
(34)
$\displaystyle+\;\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,\left(\mu\nabla
B\;+\;\nabla{\bf u}_{\rm E}\,\mbox{\boldmath$\cdot$}\,m\,{\bf u}_{\rm
E}\;+\;\frac{\partial{\bf P}_{0}}{\partial t}\right),$

where $G_{1}^{p_{\|}}$ is given by Eq. (32).

Next, we introduce the particle gyroradius

$\displaystyle\mbox{\boldmath$\rho$}\equiv{\bf x}-{\sf T}_{\rm gc}{\bf X}$
$\displaystyle=$
$\displaystyle\epsilon\,\mbox{\boldmath$\rho$}_{0}-\epsilon^{2}\left(G_{2}^{\bf
x}-\frac{1}{2}\,{\sf G}_{1}\cdot{\sf
d}\mbox{\boldmath$\rho$}_{0}\right)+\cdots\;=\;\epsilon\,\mbox{\boldmath$\rho$}_{0}\;+\;\epsilon^{2}\,\mbox{\boldmath$\rho$}_{1}\;+\;\cdots,$
(35)

which is defined as the difference between the particle position ${\bf x}$ and
the pull-back ${\sf T}_{\rm gc}{\bf X}$ of the guiding-center position ${\bf
X}$, where $\mbox{\boldmath$\rho$}_{1}\equiv-\,G_{2}^{\bf x}+\frac{1}{2}\,{\sf
G}_{1}\cdot{\sf d}\mbox{\boldmath$\rho$}_{0}$ is the first-order particle
gyroradius, where

$\displaystyle{\sf G}_{1}\cdot{\sf d}\mbox{\boldmath$\rho$}_{0}$
$\displaystyle=$ $\displaystyle
g_{1}^{J}\;\frac{\partial\mbox{\boldmath$\rho$}_{0}}{\partial
J}\;+\;g_{1}^{\theta}\;\frac{\partial\mbox{\boldmath$\rho$}_{0}}{\partial\theta}\;+\;\frac{J}{m\Omega}\left(\nabla\,\mbox{\boldmath$\cdot$}\,\widehat{{\sf
b}}\;-\;4\,\alpha_{2}\right)\,\widehat{{\sf b}}.$ (36)

Hence, using Eqs. (33) and (36), we obtain the first-order particle gyroradius
vector

$\displaystyle\mbox{\boldmath$\rho$}_{1}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\Pi$}_{1}\,\mbox{\boldmath$\times$}\,\frac{\widehat{{\sf
b}}}{m\Omega}-\left[G_{2\|}^{\bf
x}-\frac{J}{m\Omega}\left(\frac{1}{2}\,\nabla\,\mbox{\boldmath$\cdot$}\,\widehat{{\sf
b}}-2\,\alpha_{2}\right)\right]\,\widehat{{\sf b}}$ (37)
$\displaystyle+\left(\frac{1}{2}\,\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,\nabla\ln
B-\frac{\widehat{{\sf
b}}}{m\Omega}\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{0}\right)\;\mbox{\boldmath$\rho$}_{0},$

where $4\,\alpha_{2}\equiv-\,\partial\alpha_{1}/\partial\theta$.

We now note that the gyroangle-dependent part $\widetilde{G}_{1}^{J}\equiv
G_{1}^{J}-\langle G_{1}^{J}\rangle$ can be defined such that the right side of
Eq. (34) only contains terms that are gyroangle-independent. Hence, we find

$\displaystyle\Omega\,\widetilde{G}_{1}^{J}$ $\displaystyle=$
$\displaystyle-\,\frac{p_{\|}}{m}\;\widetilde{G}_{1}^{p_{\|}}\;-\;e\,\widetilde{\mbox{\boldmath$\rho$}}_{1}\,\mbox{\boldmath$\cdot$}\,{\bf
E}\;-\;\frac{2J}{m\Omega}\;e\,{\sf a}_{2}:\nabla{\bf E}$ (38)
$\displaystyle+\;\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,\left(\mu\nabla
B\;+\;\nabla{\bf P}_{\rm E}\,\mbox{\boldmath$\cdot$}\,{\bf u}_{\rm
E}\;+\;\frac{\partial{\bf P}_{0}}{\partial t}\right),$

where ${\sf
a}_{2}\equiv\frac{1}{4}\;(\widehat{\bot}\,\widehat{\bot}-\widehat{\rho}\,\widehat{\rho})=-\,\frac{1}{4}\partial{\sf
a}_{1}/\partial\theta$ and the gyroangle-dependent part of Eq. (32) is

$\widetilde{G}_{1}^{p_{\|}}\;\equiv\;G_{1}^{p_{\|}}\;-\;\langle
G_{1}^{p_{\|}}\rangle\;=\;\frac{\partial\mbox{\boldmath$\rho$}_{0}}{\partial\theta}\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{0}\;+\;J\;\alpha_{1}.$ (39)

The second-order guiding-center Hamiltonian is, thus, defined as

$\displaystyle{\mathcal{H}}_{2{\rm gc}}$ $\displaystyle=$
$\displaystyle\frac{P_{\|}}{m}\;\left(\Pi_{1\|}-\frac{1}{2}\,J\,\tau\right)\;-\;\Omega\,\langle
G_{1}^{J}\rangle\;+\;\frac{J\,c}{2B}\;\left(\mathbb{I}-\widehat{{\sf
b}}\widehat{{\sf b}}\right):\nabla{\bf
E}\;-\;\langle\mbox{\boldmath$\rho$}_{1}\rangle\,\mbox{\boldmath$\cdot$}\,e\,{\bf
E}$ $\displaystyle=$
$\displaystyle\frac{P_{\|}}{m}\;\left(\Pi_{1\|}-\frac{1}{2}\,J\,\tau\right)-\Omega\,\langle
G_{1}^{J}\rangle+\nabla\,\mbox{\boldmath$\cdot$}\,\left(\frac{e}{2}\,\langle\mbox{\boldmath$\rho$}_{0}\mbox{\boldmath$\rho$}_{0}\rangle\,\mbox{\boldmath$\cdot$}\,{\bf
E}\right)+\left(\mbox{\boldmath$\Pi$}_{1}-\frac{1}{2}\,J\,\nabla\,\mbox{\boldmath$\times$}\,\widehat{{\sf
b}}\right)\,\mbox{\boldmath$\cdot$}\,{\bf u}_{\rm E},$

where $\langle G_{1}^{p_{\|}}\rangle=\frac{1}{2}\,J\,\tau-\Pi_{1\|}$ and the
gyroangle-averaged first-order particle gyroradius is

$\displaystyle\langle\mbox{\boldmath$\rho$}_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{J}{2\,m\Omega}\left[\left(\mathbb{I}-\widehat{{\sf
b}}\widehat{{\sf b}}\right)\mbox{\boldmath$\cdot$}\nabla\ln
B+\left(\nabla\mbox{\boldmath$\cdot$}\widehat{{\sf b}}\right)\widehat{{\sf
b}}\right]+\mbox{\boldmath$\Pi$}_{1}\mbox{\boldmath$\times$}\frac{\widehat{{\sf
b}}}{m\Omega}$ (41) $\displaystyle\equiv$
$\displaystyle-\nabla\mbox{\boldmath$\cdot$}\left(\frac{1}{2}\langle\mbox{\boldmath$\rho$}_{0}\mbox{\boldmath$\rho$}_{0}\rangle\right)+\left(\mbox{\boldmath$\Pi$}_{1}-\frac{J}{2}\nabla\mbox{\boldmath$\times$}\widehat{{\sf
b}}\right)\mbox{\boldmath$\times$}\frac{\widehat{{\sf b}}}{m\Omega}$

where
$\langle\mbox{\boldmath$\rho$}_{0}\mbox{\boldmath$\rho$}_{0}\rangle=(J/m\Omega)\;(\mathbb{I}-\widehat{{\sf
b}}\widehat{{\sf b}})$. Hence, we now need expressions for $\langle
G_{1}^{J}\rangle$ and $\mbox{\boldmath$\Pi$}_{1}$ in order to obtain an
explicit expression for the second-order guiding-center Hamiltonian (3.2.2),
which are determined at third order in our perturbation analysis.

###  3.3 Third-order perturbation analysis

The third-order term in the guiding-center phase-space Lagrangian one-form (7)
is expressed as

$\displaystyle\Gamma_{3{\rm gc}}$ $\displaystyle=$
$\displaystyle-\iota_{3}\cdot\mbox{\boldmath$\omega$}_{0}\;-\;\iota_{2}\cdot\mbox{\boldmath$\omega$}_{{\rm
gc}1}\;+\;\frac{\iota_{1}}{3}\cdot{\sf
d}\left(\iota_{1}\cdot\mbox{\boldmath$\omega$}_{1}+\frac{\iota_{1}}{2}\cdot\mbox{\boldmath$\omega$}_{{\rm
gc}1}\right)\;+\;{\sf d}\sigma_{3}.$ (42)

In what follows, the gauge function $\sigma_{3}$ will play an important role
in completing the guiding-center phase-space transformation, while the third-
order guiding-center Hamiltonian will not be needed in the present guiding-
center formulation.

####  3.3.1 Third-order symplectic structure

The remaining components $(G_{2\|}^{\bf x},\langle
G_{1}^{J}\rangle,G_{1}^{\theta})$ and the first-order guiding-center momentum
$\mbox{\boldmath$\Pi$}_{1}$ will now be determined from the momentum
components of the third-order guiding-center symplectic one-form (42)

$\displaystyle\Gamma_{3{\bf p}}$ $\displaystyle\equiv$
$\displaystyle\left[G_{2\|}^{\bf x}\;+\;\frac{\partial D_{1}({\bf
P}_{3})}{\partial
p_{\|}}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\rho$}_{0}\;+\;\frac{\partial\sigma_{3}}{\partial
p_{\|}}\right]\;{\sf
d}p_{\|}\;+\;\left[\frac{2}{3}\,G_{1}^{\theta}+\frac{\partial D_{1}({\bf
P}_{3})}{\partial
J}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\rho$}_{0}+\frac{\partial\sigma_{3}}{\partial
J}\right]\;{\sf d}J$ (43)
$\displaystyle+\;\left[-\,\frac{2}{3}\,G_{1}^{J}+\frac{\partial D_{1}({\bf
P}_{3})}{\partial\theta}\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\rho$}_{0}+\frac{\partial\sigma_{3}}{\partial\theta}\right]\;{\sf
d}\theta,$

where ${\bf P}_{3}\equiv\frac{1}{2}\,{\bf P}_{0}+\frac{1}{3}\,{\bf q}_{\bot}$,
so that

$\displaystyle D_{1}({\bf P}_{3})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\;G_{1}^{p_{\|}}\,\widehat{{\sf
b}}\;+\;\frac{1}{3}\left(G_{1}^{J}\;\frac{\partial{\bf q}_{\bot}}{\partial
J}+G_{1}^{\theta}\;\frac{\partial{\bf
q}_{\bot}}{\partial\theta}\right)\;+\;\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\times$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{3}.$ (44)

Since $\partial\mbox{\boldmath$\rho$}_{0}/\partial p_{\|}=0$, the
$p_{\|}$-component of Eq. (43) suggests that we define the new gauge function

$\overline{\sigma}_{3}\;\equiv\;\sigma_{3}\;+\;D_{1}({\bf
P}_{3})\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\rho$}_{0}\;=\;\sigma_{3}\;-\;\frac{2}{3}\;J\,G_{1}^{\theta},$
(45)

where the last expression follows from Eq. (44). Using the new gauge function
(45), the momentum components (43), therefore, become

$\displaystyle\Gamma_{3{\bf p}}$ $\displaystyle=$
$\displaystyle\left(G_{2\|}^{\bf
x}\;+\;\frac{\partial\overline{\sigma}_{3}}{\partial p_{\|}}\right)\;{\sf
d}p_{\|}\;+\;\left(G_{1}^{\theta}\;+\;\frac{\partial\overline{\sigma}_{3}}{\partial
J}\right)\;{\sf d}J$ (46)
$\displaystyle+\;\left(\frac{\partial\overline{\overline{\sigma}}_{3}}{\partial\theta}\;-\;G_{1}^{J}\;+\;\frac{J\widehat{{\sf
b}}}{m\Omega}\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{0}\right)\;{\sf d}\theta,$

where, using Eq. (44), we introduced the identities

$\displaystyle D_{1}({\bf
P}_{3})\,\mbox{\boldmath$\cdot$}\,\frac{\partial\mbox{\boldmath$\rho$}_{0}}{\partial
J}$ $\displaystyle\equiv$ $\displaystyle-\;\frac{1}{3}\,G_{1}^{\theta},$
$\displaystyle D_{1}({\bf
P}_{3})\,\mbox{\boldmath$\cdot$}\,\frac{\partial\mbox{\boldmath$\rho$}_{0}}{\partial\theta}$
$\displaystyle\equiv$
$\displaystyle\frac{1}{3}\,G_{1}^{J}\;+\;\frac{2J\widehat{{\sf
b}}}{m\Omega}\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{3},$

so that we can also introduce yet another gauge function

$\overline{\overline{\sigma}}_{3}\;\equiv\;\overline{\sigma}_{3}\;-\;\frac{1}{3}\left(2\,J\;\mbox{\boldmath$\rho$}_{0}\,\mbox{\boldmath$\cdot$}\,{\bf
R}\;+\;J\;\frac{\partial\mbox{\boldmath$\rho$}_{0}}{\partial\theta}\,\mbox{\boldmath$\cdot$}\,\nabla\ln
B\right)$ (47)

in the $\theta$-component of Eq. (43). By requiring that the momentum
components (46) vanish, we now obtain the definitions

$\displaystyle G_{1}^{J}$ $\displaystyle\equiv$
$\displaystyle-\;\frac{J\widehat{{\sf
b}}}{m\Omega}\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{0}\;+\;\frac{\partial\overline{\overline{\sigma}}_{3}}{\partial\theta},$
(48) $\displaystyle G_{2\|}^{\bf x}$ $\displaystyle\equiv$
$\displaystyle-\;\frac{\partial\overline{\sigma}_{3}}{\partial p_{\|}},$ (49)
$\displaystyle G_{1}^{\theta}$ $\displaystyle\equiv$
$\displaystyle-\;\frac{\partial\overline{\sigma}_{3}}{\partial J}.$ (50)

Hence, the components $G_{2\|}^{\bf x}$ and $G_{1}^{\theta}$ are determined
from the third-order gauge function $\overline{\sigma}_{3}$, which is
determined from Eq. (47), while $\overline{\overline{\sigma}}_{3}$ is
determined from the gyroangle-dependent part
$\widetilde{G}_{1}^{J}\equiv\partial\overline{\overline{\sigma}}_{3}/\partial\theta$
obtained from Eq. (38). Since these gyroangle-dependent components are not
needed in what follows, however, they will not be derived here.

####  3.3.2 Second-order guiding-center Hamiltonian

From Eq. (48), we immediately conclude that $\langle G_{1}^{J}\rangle$ must be
defined as

$\langle G_{1}^{J}\rangle\;\equiv\;-\;\frac{J\widehat{{\sf
b}}}{m\Omega}\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
P}_{0},$ (51)

which was obtained in previous derivations (Brizard, 1995; Madsen, 2010; Frei
et al., 2020), so that the second-order guiding-center Hamiltonian (3.2.2)
becomes

${\mathcal{H}}_{2{\rm
gc}}\;=\;-\;\nabla\,\mbox{\boldmath$\cdot$}\,\left(\frac{e}{2}\,\langle\mbox{\boldmath$\rho$}_{0}\mbox{\boldmath$\rho$}_{0}\rangle\,\mbox{\boldmath$\cdot$}\,{\bf
E}\right)\;+\;\left(\mbox{\boldmath$\Pi$}_{1}+\frac{1}{2}\,J\,\nabla\,\mbox{\boldmath$\times$}\,\widehat{{\sf
b}}\right)\,\mbox{\boldmath$\cdot$}\,\frac{{\bf P}_{0}}{m},$ (52)

where we used the identity

$\frac{J}{2m\Omega}\,\left(\mathbb{I}-\widehat{{\sf b}}\widehat{{\sf
b}}\right)\,\mbox{\boldmath$\cdot$}\,e\,{\bf
E}\;\equiv\;\frac{e}{2}\,\langle\mbox{\boldmath$\rho$}_{0}\mbox{\boldmath$\rho$}_{0}\rangle\,\mbox{\boldmath$\cdot$}\,{\bf
E}\;\equiv\;\frac{1}{2}\,J\,\widehat{{\sf b}}\,\mbox{\boldmath$\times$}\,{\bf
u}_{\rm E}.$ (53)

In previous works (Littlejohn, 1981; Hahm, 1996; Miyato et al., 2009; Madsen,
2010; Frei et al., 2020), the choice for the first-order symplectic momentum
$\mbox{\boldmath$\Pi$}_{1}=-\frac{1}{2}\,J\tau\,\widehat{{\sf b}}$ was used to
eliminate the Baños drift (Baños, 1967; Northrop & Rome, 1978) from the
guiding-center velocity (i.e., $\partial{\mathcal{H}}_{2{\rm gc}}/\partial
P_{\|}=0$), which is instead added to the definition of the guiding-center
parallel momentum (32). This choice, therefore, yields the second-order
guiding-center Hamiltonian ${\mathcal{H}}_{2{\rm gc}}=(J\widehat{{\sf
b}}/2)\,\mbox{\boldmath$\cdot$}\,\nabla\,\mbox{\boldmath$\times$}\,{\bf
u}_{\rm E}$.

A different choice adopted by Tronko & Brizard (2015) for the first-order
symplectic momentum

$\mbox{\boldmath$\Pi$}_{1{\rm
pol}}\;\equiv\;-\,\frac{1}{2}\,J\;\nabla\,\mbox{\boldmath$\times$}\,\widehat{{\sf
b}},$ (54)

on the other hand, was previously shown to yield an exact Lie-transform
perturbation derivation of the standard guiding-center polarization derived by
Kaufman (1986) in the absence of an electric field. While this choice still
eliminates the Baños drift from the guiding-center velocity, it also yields an
expression for the second-order guiding-center Hamiltonian (52) that exactly
represents the guiding-center finite-Larmor-radius (FLR) correction to the
electrostatic potential energy $e\,\Phi$ (Brizard, 2023a):

${\mathcal{H}}_{2{\rm
gc}}\;=\;-\;\nabla\,\mbox{\boldmath$\cdot$}\,\left(\frac{e}{2}\,\langle\mbox{\boldmath$\rho$}_{0}\mbox{\boldmath$\rho$}_{0}\rangle\,\mbox{\boldmath$\cdot$}\,{\bf
E}\right).$ (55)

Finally, we note that recent numerical studies of particle and guiding-center
orbits in axisymmetric magnetic fields (Brizard & Hodgeman, 2023) have shown
that guiding-center orbits are faithful (i.e., remain close) to particle
orbits only if second-order effects, including the correction (54), are
included, which confirms earlier results by Belova et al. (2003).

##  4 Guiding-center Hamiltonian Dynamics

In this Section, we summarize the results of the Lie-transform perturbation
analysis of the guiding-center Lagrangian dynamics presented in Sec. 3, and we
remove the explicit $\epsilon$ scaling by restoring the physical mass
$\epsilon\,m\rightarrow m$. Hence, we find the guiding-center phase-space
extended one-form

$\displaystyle\Gamma_{\rm gc}$ $\displaystyle=$
$\displaystyle\left(\frac{e}{c}\;{\bf A}\;+\;\mbox{\boldmath$\Pi$}_{\rm
gc}\right)\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf X}\;+\;J\;\left({\sf
d}\theta\;-\frac{}{}{\bf R}\,\mbox{\boldmath$\cdot$}\,{\sf d}{\bf
X}\;-\;{\mathcal{S}}\,{\sf d}t\right)\;-\;W\,{\sf d}t,$ (56)

where the guiding-center symplectic momentum

$\mbox{\boldmath$\Pi$}_{\rm gc}\;=\;P_{\|}\,\widehat{{\sf b}}\;+\;{\bf P}_{\rm
E}\;-\;\frac{J}{2}\,\nabla\,\mbox{\boldmath$\times$}\,\widehat{{\sf b}}$ (57)

includes the higher-order polarization correction (54). The extended guiding-
center Hamiltonian, on the other hand, is expressed as

${\mathcal{H}}_{\rm gc}\;=\;e\,\Phi\;+\;K_{\rm gc}\;-\;W,$ (58)

where the guiding-center kinetic energy in the drifting frame is

$K_{\rm gc}\;=\;\mu\,B\;+\;\frac{P_{\|}^{2}}{2m}\;+\;\frac{m}{2}\,|{\bf
u}_{\rm
E}|^{2}\;-\;\nabla\,\mbox{\boldmath$\cdot$}\,\left(\frac{J\widehat{{\sf
b}}}{2}\,\mbox{\boldmath$\times$}\,{\bf u}_{\rm E}\right),$ (59)

which includes the FLR correction (55) to the electrostatic potential energy
$e\,\Phi$. We note that the presence of the gyrogauge fields
$({\mathcal{S}},{\bf R})$ in Eq. (56) guarantees gyrogauge invariance of the
guiding-center equations of motion derived from them.

###  4.1 Extended guiding-center Poisson bracket

The extended guiding-center Poisson bracket $\\{\;,\;\\}_{\rm gc}$ is obtained
by, first, constructing an 8$\times$8 matrix out of the components of the
extended guiding-center Lagrange two-form $\mbox{\boldmath$\omega$}_{\rm
gc}={\sf d}\Gamma_{\rm gc}$ and, then, invert this matrix to obtain the
extended guiding-center Poisson matrix, whose components are the fundamental
brackets $\\{Z^{\alpha},Z^{\beta}\\}_{\rm gc}$. From these components, we
obtain the extended guiding-center Poisson bracket

$\displaystyle\\{{\mathcal{F}},{\mathcal{G}}\\}_{\rm gc}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial\mathcal{F}}{\partial
W}\,\frac{\partial^{*}{\mathcal{G}}}{\partial
t}-\frac{\partial^{*}{\mathcal{G}}}{\partial
t}\,\frac{\partial\mathcal{G}}{\partial W}\right)\;+\;\frac{{\bf
B}^{*}}{B_{\|}^{*}}\,\mbox{\boldmath$\cdot$}\,\left(\nabla^{*}{\mathcal{F}}\,\frac{\partial\mathcal{G}}{\partial
P_{\|}}-\frac{\partial\mathcal{F}}{\partial
P_{\|}}\,\nabla^{*}{\mathcal{G}}\right)$ (60)
$\displaystyle-\;\frac{c\widehat{{\sf
b}}}{eB_{\|}^{*}}\,\mbox{\boldmath$\cdot$}\,\nabla^{*}{\mathcal{F}}\,\mbox{\boldmath$\times$}\,\nabla^{*}{\mathcal{G}}\;+\;\left(\frac{\partial\mathcal{F}}{\partial\theta}\,\frac{\partial\mathcal{G}}{\partial
J}-\frac{\partial\mathcal{F}}{\partial
J}\,\frac{\partial\mathcal{G}}{\partial\theta}\right),$

where

$\frac{e}{c}\,{\bf B}^{*}\;=\;\frac{e}{c}\,{\bf
B}\;+\;\nabla\,\mbox{\boldmath$\times$}\,\mbox{\boldmath$\Pi$}_{\rm
gc}\;-\;J\,\nabla\,\mbox{\boldmath$\times$}\,{\bf R},$ (61)

and the guiding-center Jacobian is ${\mathcal{J}}_{\rm
gc}=(e/c)\,B_{\|}^{*}\equiv(e/c)\,\widehat{{\sf
b}}\,\mbox{\boldmath$\cdot$}\,{\bf B}^{*}$. In addition, we have introduced
the definitions

$\displaystyle\frac{\partial^{*}}{\partial t}$ $\displaystyle\equiv$
$\displaystyle\frac{\partial}{\partial
t}\;+\;{\mathcal{S}}\;\frac{\partial}{\partial\theta},$ (62)
$\displaystyle\nabla^{*}$ $\displaystyle\equiv$ $\displaystyle\nabla\;+\;{\bf
R}^{*}\;\frac{\partial}{\partial\theta}\;-\;\left(\frac{e}{c}\frac{\partial{\bf
A}^{*}}{\partial t}+J\,\nabla{\mathcal{S}}\right)\frac{\partial}{\partial W},$
(63)

where ${\bf R}^{*}\equiv{\bf
R}+\frac{1}{2}\,\nabla\,\mbox{\boldmath$\times$}\,\widehat{{\sf b}}$. We note
that the Poisson bracket (60) can be expressed in divergence form as

$\\{{\mathcal{F}},{\mathcal{G}}\\}_{\rm
gc}\;=\;\frac{1}{B_{\|}^{*}}\frac{\partial}{\partial
Z^{\alpha}}\left(B_{\|}^{*}\frac{}{}{\mathcal{F}}\;\left\\{Z^{\alpha},\;{\mathcal{G}}\right\\}_{\rm
gc}\right),$ (64)

and that it automatically satisfies the Jacobi identity

$\left\\{{\mathcal{F}},\frac{}{}\\{{\mathcal{G}},{\mathcal{K}}\\}_{\rm
gc}\right\\}_{\rm
gc}+\left\\{{\mathcal{G}},\frac{}{}\\{{\mathcal{K}},{\mathcal{F}}\\}_{\rm
gc}\right\\}_{\rm
gc}+\left\\{{\mathcal{K}},\frac{}{}\\{{\mathcal{F}},{\mathcal{G}}\\}_{\rm
gc}\right\\}_{\rm gc}=0$ (65)

since the extended guiding-center Lagrange two-form
$\mbox{\boldmath$\omega$}_{\rm gc}={\sf d}\Gamma_{\rm gc}$ is exact (i.e.,
${\sf d}\mbox{\boldmath$\omega$}_{\rm gc}={\sf d}^{2}\Gamma_{\rm gc}=0$)
provided $\nabla\,\mbox{\boldmath$\cdot$}\,{\bf B}^{*}=0$.

Next, we note that the operators (62) and (63) contain the gyrogauge-invariant
combinations $\partial/\partial t+{\mathcal{S}}\,\partial/\partial\theta$ and
$\nabla+{\bf R}\;\partial/\partial\theta$, while Eqs. (61) and (63) include
the gyrogauge-independent vector fields (Ye & Kaufman, 1992; Brizard, 2023b)

$\left(\nabla\,\mbox{\boldmath$\times$}\,{\bf
R},\nabla{\mathcal{S}}\;-\;\frac{\partial\bf R}{\partial
t}\right)\;=\;\left(-\,\frac{1}{2}\,\epsilon_{ijk}\,b^{i}\;\nabla
b^{j}\,\mbox{\boldmath$\times$}\,\nabla b^{k},-\,\nabla\widehat{{\sf
b}}\,\mbox{\boldmath$\times$}\,\widehat{{\sf
b}}\,\mbox{\boldmath$\cdot$}\,\frac{\partial\widehat{{\sf b}}}{\partial
t}\right),$ (66)

where $\epsilon_{ijk}$ denotes the completely-antisymmetric Levi-Civita
tensor.

###  4.2 Guiding-center Hamilton equations

The guiding-center Hamilton equations
$\dot{Z}^{\alpha}\equiv\\{Z^{\alpha},{\mathcal{H}}_{\rm gc}\\}_{\rm gc}$
include the guiding-center velocity

$\dot{\bf X}\;=\;\frac{P_{\|}}{m}\;\frac{{\bf B}^{*}}{B_{\|}^{*}}\;+\;{\bf
E}^{*}\,\mbox{\boldmath$\times$}\,\frac{c\widehat{{\sf b}}}{B_{\|}^{*}},$ (67)

where $\widehat{{\sf b}}\,\mbox{\boldmath$\cdot$}\,\dot{\bf X}=P_{\|}/m$
defines the parallel guiding-center velocity, and the guiding-center parallel
force

$\dot{P}_{\|}\;=\;e\,{\bf E}^{*}\,\mbox{\boldmath$\cdot$}\,\frac{{\bf
B}^{*}}{B_{\|}^{*}},$ (68)

where the modified electric field is represented as

$\displaystyle e\,{\bf E}^{*}$ $\displaystyle=$ $\displaystyle e\,{\bf
E}-\frac{\partial\mbox{\boldmath$\Pi$}_{\rm gc}}{\partial t}-\nabla K_{\rm
gc}+J\left(\frac{\partial\bf R}{\partial t}-\nabla{\mathcal{S}}\right),$ (69)

and the gyroangle angular velocity

$\displaystyle\dot{\theta}$ $\displaystyle\equiv$ $\displaystyle\frac{\partial
K_{\rm gc}}{\partial J}\;+\;{\mathcal{S}}\;+\;\dot{\bf
X}\,\mbox{\boldmath$\cdot$}\,{\bf
R}^{*}=\Omega\;-\;\nabla\,\mbox{\boldmath$\cdot$}\,\left(\frac{\widehat{{\sf
b}}}{2}\,\mbox{\boldmath$\times$}\,{\bf u}_{\rm
E}\right)\;+\;{\mathcal{S}}\;+\;\dot{\bf X}\,\mbox{\boldmath$\cdot$}\,{\bf
R}^{*}.$ (70)

We note that the reduced guiding-center equations of motion (67)-(68) satisfy
the guiding-center Liouville equation

$\frac{\partial B_{\|}^{*}}{\partial
t}\;=\;-\;\nabla\,\mbox{\boldmath$\cdot$}\,\left(B_{\|}^{*}\;\dot{\bf
X}\right)\;-\;\frac{\partial}{\partial
P_{\|}}\left(B_{\|}^{*}\;\dot{P}_{\|}\right),$ (71)

where

$\displaystyle\nabla\,\mbox{\boldmath$\cdot$}\,\left(B_{\|}^{*}\;\dot{\bf
X}\right)$ $\displaystyle=$
$\displaystyle\nabla\,\mbox{\boldmath$\times$}\,{\bf
E}^{*}\,\mbox{\boldmath$\cdot$}\,c\widehat{{\sf b}}-e\,{\bf
E}^{*}\,\mbox{\boldmath$\cdot$}\,\frac{c}{e}\nabla\,\mbox{\boldmath$\times$}\,\widehat{{\sf
b}}=-\,\widehat{{\sf b}}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf
B}^{*}}{\partial t}-e\,{\bf
E}^{*}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf B}^{*}}{\partial P_{\|}}$

and

$\displaystyle\frac{\partial}{\partial
P_{\|}}\left(B_{\|}^{*}\;\dot{P}_{\|}\right)$ $\displaystyle=$ $\displaystyle
e\,{\bf E}^{*}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf B}^{*}}{\partial
P_{\|}}+{\bf B}^{*}\,\mbox{\boldmath$\cdot$}\,e\frac{\partial{\bf
E}^{*}}{\partial P_{\|}}=e\,{\bf
E}^{*}\,\mbox{\boldmath$\cdot$}\,\frac{\partial{\bf B}^{*}}{\partial
P_{\|}}-{\bf B}^{*}\,\mbox{\boldmath$\cdot$}\,\frac{\partial\widehat{{\sf
b}}}{\partial t},$

where we made use of the modified Faraday’s law $\partial{\bf B}^{*}/\partial
t=-c\,\nabla\,\mbox{\boldmath$\times$}\,{\bf E}^{*}$.

###  4.3 Eulerian variations of the guiding-center Lagrangian

The results of the Lie-transform perturbation analysis carried out in Section
3.3 can also be used to construct the following (regular) guiding-center
Lagrangian

$\displaystyle L_{\rm gc}$ $\displaystyle=$
$\displaystyle\left(\frac{e}{c}\,{\bf A}+\mbox{\boldmath$\Pi$}_{\rm
gc}-J\,{\bf R}\right)\,\mbox{\boldmath$\cdot$}\,\dot{\bf
X}\;+\;J\,\dot{\theta}\;-\;\left(e\,\Phi\;+\frac{}{}K_{\rm
gc}\;+\;J\,{\mathcal{S}}\right)$ (72) $\displaystyle\equiv$
$\displaystyle(e/c)\,{\bf A}^{*}\,\mbox{\boldmath$\cdot$}\,\dot{\bf
X}\;+\;J\,\dot{\theta}-H_{\rm gc},$

where $\mbox{\boldmath$\Pi$}_{\rm gc}$ and $K_{\rm gc}$ are defined in Eqs.
(57) and (59), respectively. The guiding-center Euler-Lagrange equations are
derived from this Lagrangian as

$\dot{P}_{\|}\,\widehat{{\sf b}}\;=\;e\,{\bf E}^{*}\;+\;(e/c)\,\dot{\bf
X}\,\mbox{\boldmath$\times$}\,{\bf B}^{*},$ (73) $\widehat{{\sf
b}}\,\mbox{\boldmath$\cdot$}\,\dot{\bf X}\;=\;P_{\|}/m,$ (74)

which are associated with virtual displacements $\delta{\bf X}$ and $\delta
P_{\|}$, respectively. From these equations, we easily recover the guiding-
center Hamilton equations (67) and (68). Likewise, the guiding-center Euler-
Lagrange equation associated with the virtual displacement $\delta J$ yields
Eq. (70), while the virtual displacement $\delta\theta$ yields $\dot{J}=0$ as
a result of Noether’s Theorem.

In addition to variations with respect to guiding-center phase-space
coordinates, the guiding-center Lagrangian (72) can also be varied with
respect to the electric and magnetic fields $(\delta{\bf E},\delta{\bf B})$,
which respectively yield expressions for the guiding-center polarization and
magnetization (Brizard, 2008). Here, the Eulerian variation of the guiding-
center Lagrangian (72) is expressed as (Brizard, 2023b)

$\displaystyle\delta L_{\rm gc}$ $\displaystyle\equiv$
$\displaystyle\left(\frac{e}{c}\delta{\bf
A}\;+\;\delta\mbox{\boldmath$\Pi$}_{\rm
gc}\right)\,\mbox{\boldmath$\cdot$}\,\dot{\bf
X}\;-\;\left(e\,\delta\Phi\;+\frac{}{}\delta K_{\rm
gc}\right)\;-\;J\left(\delta{\mathcal{S}}\;+\frac{}{}\dot{\bf
X}\,\mbox{\boldmath$\cdot$}\,\delta{\bf R}\right)$ (75) $\displaystyle=$
$\displaystyle\left(\frac{e}{c}\delta{\bf
A}\,\mbox{\boldmath$\cdot$}\,\dot{\bf
X}-e\,\delta\Phi\right)+\mbox{\boldmath$\pi$}_{\rm
gc}\,\mbox{\boldmath$\cdot$}\,\delta{\bf E}+\left(\mbox{\boldmath$\mu$}_{\rm
gc}+\mbox{\boldmath$\pi$}_{\rm gc}\,\mbox{\boldmath$\times$}\,\frac{{\bf
P}_{0}}{mc}\right)\,\mbox{\boldmath$\cdot$}\,\delta{\bf B}\;+\;({\rm FLR}),$

where the FLR corrections, which are ignored in Eq. (75), are calculated to
first order in a recent paper (Brizard, 2023b). Here, the guiding-center
electric dipole moment $\mbox{\boldmath$\pi$}_{\rm gc}$ is defined as
(Pfirsch, 1984; Pfirsch & Morrison, 1985)

$\mbox{\boldmath$\pi$}_{\rm gc}\;\equiv\;\frac{e\widehat{{\sf
b}}}{\Omega}\,\mbox{\boldmath$\times$}\,\left(\dot{\bf X}\;-\frac{}{}{\bf
u}_{\rm E}\right),$ (76)

while the guiding-center magnetic dipole moment $\mbox{\boldmath$\mu$}_{\rm
gc}+\mbox{\boldmath$\pi$}_{\rm gc}\,\mbox{\boldmath$\times$}\,{\bf
P}_{0}/(mc)$ is defined as the sum of the intrinsic magnetic dipole moment

$\mbox{\boldmath$\mu$}_{\rm gc}\;\equiv\;\mu\left(-\,\widehat{{\sf
b}}\;+\;\frac{1}{\Omega}\,\frac{d\widehat{{\sf
b}}}{dt}\,\mbox{\boldmath$\times$}\,\widehat{{\sf b}}\right),$ (77)

which includes the gyrogauge correction associated with
$\delta{\mathcal{S}}+\dot{\bf X}\,\mbox{\boldmath$\cdot$}\,\delta{\bf R}$
(Brizard, 2023b), and the moving electric-dipole contribution (Jackson, 1975),
expressed in terms of the lowest-order guiding-center momentum ${\bf
P}_{0}=P_{\|}\,\widehat{{\sf b}}+{\bf P}_{\rm E}$. By ignoring these FLR
corrections, the guiding-center polarization and magnetization are defined as
moments of the guiding-center electric and magnetic dipole moments

$\displaystyle\mbox{\boldmath$\mathcal{P}$}_{\rm gc}$ $\displaystyle=$
$\displaystyle\int_{P}F_{\rm gc}\;\mbox{\boldmath$\pi$}_{\rm gc},$ (78)
$\displaystyle\mbox{\boldmath$\mathcal{M}$}_{\rm gc}$ $\displaystyle=$
$\displaystyle\int_{P}F_{\rm gc}\;\left(\mbox{\boldmath$\mu$}_{\rm
gc}+\mbox{\boldmath$\pi$}_{\rm gc}\,\mbox{\boldmath$\times$}\,\frac{{\bf
P}_{0}}{mc}\right),$ (79)

where the guiding-center phase-space density $F_{\rm
gc}\equiv{\mathcal{J}}_{\rm gc}\,F$ includes the guiding-center Jacobian
${\mathcal{J}}_{\rm gc}$ and the notation $\int_{P}$ includes an integral over
guiding-center momentum space as well as a sum over particle species.

Finally, we note that, in the absence of an electric field, the classical
expression (Kaufman, 1986) for the guiding-center electric dipole moment
$\mbox{\boldmath$\pi$}_{\rm gc}=(e\widehat{{\sf
b}}/\Omega)\,\mbox{\boldmath$\times$}\,\dot{\bf X}$ is derived by Lie-
transform perturbation method (Tronko & Brizard, 2015) only if the first-order
polarization correction (54) is used.

##  5 Summary

In the present work, a set of higher-order guiding-center Hamilton equations
was derived by Lie-transform perturbation method for the case of time-
dependent electric and magnetic fields that satisfy the standard guiding-
center space-time orderings. The second-order corrections in the guiding-
center Hamiltonian represented finite-Larmor-radius corrections of the lowest-
order electrostatic potential energy $e\,\Phi$. Additional second-order
corrections in the guiding-center Lagrangian (72) included gyrogauge-
invariance contributions to the guiding-center Hamiltonian and Poisson bracket
as well as corrections leading to the correct guiding-center polarization.

When we turn our attention to the self-consistent interactions of the charged-
particle guiding-centers and the electromagnetic fields associated with plasma
confinement, we need to derive a set of higher-order guiding-center Vlasov-
Maxwell equations. Work presented elsewhere (Brizard, 2023b) considered the
variational formulation of the higher-order guiding-center Vlasov-Maxwell
equations derived from the guiding-center Lagrangian (72) and the Maxwell
Lagrangian density.

According to this variational principle, using the guiding-center Liouville
equation (71), the guiding-center Vlasov equation for the guiding-center
phase-space density $F_{\rm gc}\equiv{\mathcal{J}}_{\rm gc}\,F$ is written in
divergence form as

$\frac{\partial F_{\rm gc}}{\partial
t}\;+\;\nabla\,\mbox{\boldmath$\cdot$}\,\left(\dot{\bf X}\frac{}{}F_{\rm
gc}\right)\;+\;\frac{\partial}{\partial
P_{\|}}\left(\dot{P}_{\|}\frac{}{}F_{\rm gc}\right)\;=\;0,$ (80)

while the Maxwell equations with particle sources are expressed as

$\displaystyle\nabla\,\mbox{\boldmath$\cdot$}\,{\bf E}$ $\displaystyle=$
$\displaystyle 4\pi\,\varrho\;\equiv\;4\pi\left(\varrho_{\rm
gc}\;-\frac{}{}\nabla\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\mathcal{P}$}_{\rm
gc}\right),$ (81) $\displaystyle\nabla\,\mbox{\boldmath$\times$}\,{\bf
B}-\frac{1}{c}\,\frac{\partial\bf E}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{c}\,{\bf J}\;\equiv\;\frac{4\pi}{c}\left({\bf
J}_{\rm gc}\;+\;\frac{\partial\mbox{\boldmath$\mathcal{P}$}_{\rm gc}}{\partial
t}\;+\;c\,\nabla\,\mbox{\boldmath$\times$}\,\mbox{\boldmath$\mathcal{M}$}_{\rm
gc}\right),$ (82)

where the guiding-center charge and current densities are $\varrho_{\rm
gc}=\int_{P}e\,F_{\rm gc}$ and ${\bf J}_{\rm gc}=\int_{P}e\,\dot{\bf
X}\;F_{\rm gc}$. Here, the guiding-center polarization charge density
$\varrho_{\rm
pol}\equiv-\nabla\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\mathcal{P}$}_{\rm
gc}$ and current density ${\bf J}_{\rm
pol}\equiv\partial\mbox{\boldmath$\mathcal{P}$}_{\rm gc}/\partial t$ are
derived from the guiding-center polarization (78), while the guiding-center
magnetization current density ${\bf J}_{\rm mag}\equiv
c\,\nabla\,\mbox{\boldmath$\times$}\,\mbox{\boldmath$\mathcal{M}$}_{\rm gc}$
is derived from the guiding-center magnetization (79). The remaining source-
free Maxwell equations are Faraday’s law $\partial{\bf B}/\partial
t=-c\,\nabla\,\mbox{\boldmath$\times$}\,{\bf E}$ and Gauss’s law
$\nabla\,\mbox{\boldmath$\cdot$}\,{\bf B}=0$. We note that the guiding-center
Vlasov-Maxwell variational principle also guarantees the existence of exact
energy-momentum conservation laws, derived by the Noether method (Brizard,
2008). Our recent work (Brizard, 2023b) has considered the set of higher-order
guiding-center Vlasov-Maxwell equations obtained by explicitly imposing the
quasineutrality constraint $\varrho_{\rm
gc}=\nabla\,\mbox{\boldmath$\cdot$}\,\mbox{\boldmath$\mathcal{P}$}_{\rm gc}$.

Future work will explore the Hamiltonian structure of the guiding-center
Vlasov-Maxwell equations, when guiding-center polarization and magnetization
are included, which will generalize our previous work (Brizard, 2021), and its
development is motivated by the desire to construct structure-preserving
numerical algorithms (Morrison, 2017) using an important set of reduced
Vlasov-Maxwell equations.

Acknowledgments

The Author wishes to thank the referees for their insightful comments, which
led to the final manuscript. The Author would also like to dedicate this paper
to the memory of Allan N. Kaufman (1927-2022), whose pioneering efforts in
introducing Lie-transform perturbation methods in plasma physics (Kaufman,
1978) led to the landmark papers by Cary (1981) and Cary & Littlejohn (1983),
culminating with the review papers by Brizard & Hahm (2007) and Cary & Brizard
(2009).

Funding

The present work was supported by the National Science Foundation grant
PHY-2206302.

Declaration of Interests

The Author reports no conflict of interest.

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