1. Introduction
Polarization effects have a rich history in plasma physics (Pfirsch Reference Pfirsch1984; Pfirsch & Morrison Reference Pfirsch and Morrison1985; Cary & Brizard Reference Cary and Brizard2009). Their importance stems from the assumption of quasineutrality in a strongly magnetized plasma and the dielectric properties of a guiding-centre plasma (Hinton & Robertson Reference Hinton and Robertson1984). While these effects are traditionally associated with the presence of an electric field in a magnetized plasma (Itoh & Itoh Reference Itoh and Itoh1996; Hazeltine & Meiss Reference Hazeltine and Meiss2003; Wang & Hahm Reference Wang and Hahm2009; Joseph Reference Joseph2021; Brizard Reference Brizard2023a), they are also associated with magnetic drifts (Kaufman Reference Kaufman1986; Brizard Reference Brizard2013; Tronko & Brizard Reference Tronko and Brizard2015).
Recently, second-order terms in guiding-centre Hamiltonian theory (in the absence of an electric field) were shown to be crucial (Brizard & Hodgeman Reference Brizard and Hodgeman2023) in assessing the validity of the guiding-centre representation in determining whether guiding-centre orbits were numerically faithful to the particle orbits in axisymmetric magnetic geometries, which partially confirmed earlier numerical studies in axisymmetric tokamak plasmas (Belova, Gorelenkov & Cheng Reference Belova, Gorelenkov and Cheng2003). In particular, it was shown that a second-order correction associated with guiding-centre polarization (Kaufman Reference Kaufman1986; Brizard Reference Brizard2013; Tronko & Brizard Reference Tronko and Brizard2015) was needed in order to obtain faithful guiding-centre 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-centre orbit deviated noticeably from its associated particle orbit, while a second-order guiding-centre orbit followed the particle orbit to a high degree of precision (Brizard & Hodgeman Reference Brizard and Hodgeman2023). In addition, as initially reported by Belova et al. (Reference Belova, Gorelenkov and Cheng2003), the guiding-centre Hamiltonian formulation is a faithful representation of the particle toroidal angular momentum (Tronko & Brizard Reference Tronko and Brizard2015; Brizard & Hodgeman Reference Brizard and Hodgeman2023), 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 the motion of a charged particle (of mass m and charge e) in time-dependent and inhomogeneous electric (E) and magnetic (B = B $\hat{\textsf {b}}$) fields (which still satisfy the guiding-centre space–time orderings $|\boldsymbol {\nabla }|^{-1} \gg \rho _{\rm th} = v_{\rm th}/\varOmega \def\mathsfbit#1{\boldsymbol{\mathsf{#1}}}$, i.e., characteristic spatial scales are long compared to the thermal gyroradius $\rho _{\rm th}$, and $\partial /\partial t \ll \varOmega = eB/mc$, i.e., time scales are long compared to the gyroperiod, where c denotes the speed of light) and we assume that the $E\times B$ velocity $\boldsymbol {u}_{E} = \boldsymbol {E}\boldsymbol {\times } c\hat {\textsf {b}}/B$ 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 $(\boldsymbol {q},\boldsymbol {p},t)$, where the non-autonomous Hamiltonian $H(\boldsymbol {q},\boldsymbol {p},t)$ is a function of the canonical coordinates $(\boldsymbol {q},\boldsymbol {p})$, from which the canonical Hamilton equations ${\rm d}\boldsymbol {q}/{\rm d} t = \partial H/\partial \boldsymbol {p}$ and ${\rm d}\boldsymbol {p}/{\rm d} t = -\partial H/\partial \boldsymbol {q}$ are derived, as well as time $t$, from which we obtain the energy equation ${\rm d} H/{\rm d} t = \partial H/\partial t$ (i.e. energy is not conserved).
The use of an extended phase space is a well-known method in classical mechanics (Lanczos Reference Lanczos1970) 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 $(\boldsymbol {q},t;\boldsymbol {p},w)$, where the space–time coordinates $(\boldsymbol {q},t)$ are canonically conjugate to the momentum–energy coordinates $(\boldsymbol {p},w)$, with the extended Hamilton equations ${\rm d} w/{\rm d} s = \partial \mathcal {H}/\partial t = \partial H/\partial t$ and ${\rm d} t/{\rm d} s = -\partial \mathcal {H}/\partial w = 1$, where $\mathcal {H} \equiv H(\boldsymbol {q},\boldsymbol {p},t) - w$ 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. $w = H(\boldsymbol {q},\boldsymbol {p},t)$.
Using the dimensional ordering parameter $\epsilon$ associated with the renormalized particle mass $m \rightarrow \epsilon m$ (Brizard Reference Brizard1995), instead of the standard ordering $e \rightarrow e/\epsilon$ (Kulsrud Reference Kulsrud, Rosenbluth and Sagdeev1983; Littlejohn Reference Littlejohn1983), we begin with the extended phase-space particle Lagrangian one-form
and the extended particle Hamiltonian
where $\boldsymbol {p}_{0}$ denotes the local particle kinetic momentum at position $\boldsymbol {x}$. In the present work, we consider the standard ordering (Kulsrud Reference Kulsrud, Rosenbluth and Sagdeev1983) for the parallel electric field: $\boldsymbol {E} = \boldsymbol {E}_{\bot } + \epsilon E_{\|}\hat {\textsf {b}}$. In contrast to Madsen (Reference Madsen2010) and Frei, Jorge & Ricci (Reference Frei, Jorge and Ricci2020), who used the same mass ordering ($m \rightarrow \epsilon m$), we use extended (eight-dimensional) phase space in (1.1)–(1.2), where the energy coordinate $w$ is canonically conjugate to time $t$ (Littlejohn Reference Littlejohn1981; Cary & Brizard Reference Cary and Brizard2009). This extended phase-space formulation yields a simple form for the extended Poisson bracket (see (4.5)), also adopted (without derivation) by Madsen (Reference Madsen2010), which plays an important role in the variational formulation of the guiding-centre Vlasov–Maxwell equations (Brizard Reference Brizard2023b).
Here, the electric-field ordering implies that the local particle momentum
is decomposed into the gyroangle-independent parallel component $p_{\|0} \equiv \boldsymbol {p}_{0}\boldsymbol {\cdot }\hat {\textsf {b}}$ and the $E\times B$ momentum
and the gyroangle-dependent perpendicular momentum $\boldsymbol {q}_{\bot 0} \equiv |\boldsymbol {q}_{\bot 0}|\hat {\bot }$, respectively, where $J_{0} \equiv |\boldsymbol {q}_{\bot 0}|^{2}/(2m\varOmega )$ represents the lowest-order gyroaction and the gyroangle derivative $\partial \boldsymbol {q}_{\bot 0}/\partial \theta _{0} \equiv \boldsymbol {q}_{\bot 0}\boldsymbol {\times }\hat {\textsf {b}} = -|\boldsymbol {q}_{\bot 0}|\hat {\rho }$ introduces the rotating orthogonal unit-vector fields $(\hat {\textsf {b}},\hat {\bot },\hat {\rho })$.
1.2. Purpose of the present work
The purpose of the present work is motivated by the need to derive higher-order guiding- centre 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-centre theory (Hahm Reference Hahm1996; Chang, Ku & Weitzner Reference Chang, Ku and Weitzner2004; Lanthaler et al. Reference Lanthaler, Graves, Pfefferlé and Copper2019; Frei et al. Reference Frei, Jorge and Ricci2020).
Guiding-centre 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 (Reference Brizard1995) and Hahm (Reference Hahm1996), following the earlier work of Littlejohn (Reference Littlejohn1981). These perturbation methods were also used by Miyato et al. (Reference Miyato, Scott, Strintzi and Tokuda2009), Madsen (Reference Madsen2010) and Frei et al. (Reference Frei, Jorge and Ricci2020), who derived self-consistent guiding-centre Vlasov–Maxwell equations that included guiding-centre 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-centre Vlasov–Maxwell theory, with a full representation of guiding-centre polarization that can be directly derived by the guiding-centre push-forward method (Brizard Reference Brizard2013; Tronko & Brizard Reference Tronko and Brizard2015).
2. Guiding-centre Lie-transform perturbation analysis
The derivation of the guiding-centre 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 $z_{0}^{\alpha } = (\boldsymbol {x}, p_{\|0}; J_{0},\theta _{0};w_{0},t)$, where the energy–time canonical coordinates $(w_{0},t)$ are included, to the guiding-centre phase-space coordinates $Z^{\alpha } = (\boldsymbol {X}, P_{\|}; J, \theta ;W,t)$ generated by the vector fields $(\boldsymbol{\mathsf{G}}_{1}, \boldsymbol{\mathsf{G}}_{2}, \ldots )$
and its inverse
In order for the particle time to be invariant under the guiding-centre transformation (2.1), we require that $G_{n}^{t} \equiv 0$ to all orders $n \geq 1$, i.e. the guiding-centre time is identical to the particle time. From these generating vectors fields, the pull-back and push-forward Lie-transform operators $\boldsymbol{\mathsf{T}}_{\rm gc} = \exp (\epsilon \pounds _{1})\exp (\epsilon ^{2}\pounds _{2})\cdots$ and $\boldsymbol{\mathsf{T}}_{\rm gc}^{-1} = \cdots \exp (-\epsilon ^{2}\pounds _{2})\exp (-\epsilon \pounds _{1})$ are constructed in terms of Lie derivatives $(\pounds _{1}, \pounds _{2}, \ldots )$ generated by the vector fields $(\boldsymbol{\mathsf{G}}_{1}, \boldsymbol{\mathsf{G}}_{2}, \ldots )$. More details about the Lie-transform perturbation method used in guiding-centre theory can be found in Littlejohn (Reference Littlejohn1982) (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 (Reference Tronko and Brizard2015) and Brizard & Tronko (Reference Brizard and Tronko2016).
We now wish to derive the extended guiding-centre phase-space Lagrangian one-form
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
and the extended guiding-centre Hamiltonian
where
Here, we use the formulas $\iota _{n}\boldsymbol {\cdot }\boldsymbol {\omega } \equiv G_{n}^{\alpha }\omega _{\alpha \beta }\,\textsf {d} Z^{\beta }$ (for an arbitrary two-form $\boldsymbol {\omega }$) and $\boldsymbol{\mathsf{G}}_{n}\boldsymbol {\cdot }\textsf {d}\mathcal {K} \equiv G_{n}^{\alpha }\partial \mathcal {K}/\partial Z^{\alpha }$ (for an arbitrary scalar field $\mathcal {K}$), where the summation rule is used. We note that the guiding-centre 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 (2.3).
In order to construct the extended guiding-centre Lagrangian one-form (2.3), we will need to evaluate the contractions $\iota _{n}\boldsymbol {\cdot }\boldsymbol {\omega }_{0}$ generated by the vector fields $(\boldsymbol{\mathsf{G}}_{1}, \boldsymbol{\mathsf{G}}_{2}, \ldots )$ on the zeroth-order two-form
so that we obtain the $n$th-order expression
where $G_{n}^{\boldsymbol {x}}$ and $G_{n}^{w}$ denote the spatial and energy components of the $n$th-order generating vector field $\boldsymbol{\mathsf{G}}_{n}$. Similarly, we will need to evaluate the contractions $\iota _{n}\boldsymbol {\cdot }\boldsymbol {\omega }_{1}$ generated by the vector fields $(\boldsymbol{\mathsf{G}}_{1}, \boldsymbol{\mathsf{G}}_{2}, \ldots )$ on the first-order two-form $\boldsymbol {\omega }_{1} = \textsf {d}\gamma _{1}$, which yields the $(n + 1)$th-order expression
where $\boldsymbol {p}_{0}$ is defined in (1.3) and the spatial components are expressed in terms of the operator (Tronko & Brizard Reference Tronko and Brizard2015)
where $\boldsymbol {C}$ is an arbitrary vector function on the guiding-centre 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 (2.3) and (2.6) is to construct a gyroangle-independent extended guiding-centre phase-space Lagrangian one-form $\varGamma _{\rm gc}$ and extended guiding-centre Hamiltonian $\mathcal {H}_{\rm gc}$ in terms of which guiding-centre equations are derived. The generic forms considered for the extended guiding-centre phase-space Lagrangian one-form (2.3) and the extended guiding-centre Hamiltonian (2.6) in the present work are
where the gyroangle-independent guiding-centre kinetic momentum $\boldsymbol {P}_{\rm gc}$ and guiding- centre kinetic energy $K_{\rm gc}$ are expressed as asymptotic series in powers of $\epsilon$, while the vector field $\boldsymbol {R} \equiv \boldsymbol {\nabla }\hat {\bot }\boldsymbol {\cdot }\hat {\rho }$ and the scalar field $\mathcal {S} \equiv (\partial \hat {\bot }/\partial t)\boldsymbol {\cdot }\hat {\rho }$ are required to preserve gyrogauge invariance. See appendix A of the recent paper by Brizard (Reference Brizard2023b) for an updated discussion on gyrogauge invariance introduced by Littlejohn (Reference Littlejohn1981, Reference Littlejohn1983, Reference Littlejohn1988). We note that the separation of the guiding-centre transformations of the extended guiding-centre phase-space Lagrangian one-form (2.3) and the extended guiding-centre Hamiltonian (2.6) might enable the application of computer algorithms previously used by Burby, Squire & Qin (Reference Burby, Squire and Qin2013), but this consideration falls outside the scope of the present work.
3. Symplectic polarization guiding-centre theory
In the present work, we will consider the symplectic polarization guiding-centre theory, where the guiding-centre kinetic momentum $\boldsymbol {P}_{\rm gc}$ retains the contribution from the $E\times B$ momentum (1.4), which will then introduce the polarization drift velocity explicitly in the guiding-centre equations of motion.
In the alternate Hamiltonian polarization guiding-centre theory, on the other hand, $E\times B$ momentum (1.4) is removed from the guiding-centre kinetic momentum and polarization effects enter solely through the guiding-centre 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 Reference Brizard and Hahm2007).
The reader interested in results of the guiding-centre Lie-transform perturbation analysis can skip § 3 and go to § 4, where we present the extended guiding-centre Hamiltonian structure as well as the (regular) guiding-centre Lagrangian, from which guiding-centre polarization and magnetization can be derived by functional derivatives with respect to the electric field $\boldsymbol {E}$ and magnetic field $\boldsymbol {B}$, 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-centre symplectic one-form (2.4), which is now explicitly written in the symplectic polarization representation as
where the first-order gauge scalar field $\sigma _{1}$ is not needed and the gyroangle-dependent terms on the right of (3.1) are eliminated by selecting
and
where the gyroangle-dependent vector $\boldsymbol {\rho }_{0}$ represents the lowest-order particle gyroradius.
With $G_{1}^{\boldsymbol {x}}$ defined by (3.2), the resulting first-order guiding-centre phase-space Lagrangian $\varGamma _{1{\rm gc}} = \boldsymbol {P}_{0}\boldsymbol {\cdot }\textsf {d}\boldsymbol {X}$ yields the $(n+1)$th-order contraction
where, using the operator (2.12), the spatial components in (3.4) are
which contain gyroangle-dependent and -independent contributions.
3.1.2. First-order Hamiltonian
The first-order guiding-centre Hamiltonian is determined from (2.7) as
where the components (3.2)–(3.3) were substituted, and the electric field is defined as $\boldsymbol {E} = -\boldsymbol {\nabla }\varPhi - c^{-1}\partial \boldsymbol {A}/\partial t$. By using the identity
the first-order guiding-centre Hamiltonian is automatically gyroangle independent
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-centre symplectic one-form (2.5), which is explicitly expressed as
where the second-order gauge scalar field $\sigma _{2}$ is not needed, and, using the definition (2.12) with $\boldsymbol {P}_{2} \equiv \boldsymbol {P}_{0} + \frac {1}{2}\boldsymbol {q}_{\bot }$, we find
where $g_{1}^{J} \equiv G_{1}^{J} - J\boldsymbol {\rho }_{0}\boldsymbol {\cdot }\boldsymbol {\nabla }\ln B$ and $g_{1}^{\theta } \equiv G_{1}^{\theta } + \boldsymbol {\rho }_{0}\boldsymbol {\cdot }\boldsymbol {R}$, while $\tau \equiv \hat {\textsf {b}}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {\times }\hat {\textsf {b}}$ and $\alpha _{1} \equiv \mathsfbit {a}_{1}:\boldsymbol {\nabla }\hat {\textsf {b}}$ is defined in terms of the gyroangle-dependent dyadic tensor $\boldsymbol{\mathsf{a}}_{1} \equiv -\frac {1}{2}(\hat {\bot }\hat {\rho } + \hat {\rho }\hat {\bot })$ (Tronko & Brizard Reference Tronko and Brizard2015). The first-order guiding-centre symplectic momentum $\boldsymbol {\varPi }_{1}$ in (3.9), 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 § 3.3) as well as the guiding-centre push-forward derivation of the guiding-centre polarization in the absence of a background electric field (Brizard Reference Brizard2013; Tronko & Brizard Reference Tronko and Brizard2015).
Substituting these expressions into (3.9), we obtain the vector equation
where $g_{1}^{p_{\|}} \equiv G_{1}^{p_{\|}} - J(\frac {1}{2}\tau + \alpha _{1})$, while we choose the second-order energy component
where we used
Next, from the parallel component of (3.11), we obtain the first-order component
where $\varPi _{1\|} \equiv \hat {\textsf {b}}\boldsymbol {\cdot }\boldsymbol {\varPi }_{1}$ is the parallel component of the first-order symplectic momentum $\boldsymbol {\varPi }_{1}$. From the perpendicular components of (3.11), on the other hand, we find
where $G_{2\|}^{\boldsymbol {x}} \equiv \hat {\textsf {b}}\boldsymbol {\cdot } G_{2}^{\boldsymbol {x}}$. The interpretation of the first-order guiding-centre symplectic momentum $\boldsymbol {\varPi }_{1}$ will be given in § 3.3.2.
3.2.2. Second-order Hamiltonian
The second-order guiding-centre Hamiltonian is determined from (2.8) as
where $G_{1}^{p_{\|}}$ is given by (3.14).
Next, we introduce the particle gyroradius
which is defined as the difference between the particle position $\boldsymbol {x}$ and the pull-back $\boldsymbol{\mathsf{T}}_{\rm gc}\boldsymbol {X}$ of the guiding-centre position $\boldsymbol {X}$, where $\boldsymbol {\rho }_{1} \equiv -G_{2}^{\boldsymbol {x}} + \frac {1}{2}\boldsymbol{\mathsf{G}}_{1}\boldsymbol {\cdot }\textsf {d}\boldsymbol {\rho }_{0}$ is the first-order particle gyroradius, where
Hence, using (3.15) and (3.18), we obtain the first-order particle gyroradius vector
where $4\alpha _{2} \equiv -\partial \alpha _{1}/\partial \theta$.
We now note that the gyroangle-dependent part $\tilde {G}_{1}^{J} \equiv G_{1}^{J} - \langle G_{1}^{J}\rangle$ can be defined such that the right side of (3.16) only contains terms that are gyroangle independent. Hence, we find
where $\boldsymbol{\mathsf{a}}_{2} \equiv \frac {1}{4}(\hat {\bot }\hat {\bot } - \hat {\rho }\hat {\rho }) = -\frac {1}{4}\partial \boldsymbol{\mathsf{a}}_{1}/\partial \theta$ and the gyroangle-dependent part of (3.14) is
The second-order guiding-centre Hamiltonian is, thus, defined as
where $\langle G_{1}^{p_{\|}}\rangle = \frac {1}{2}J\tau - \varPi _{1\|}$ and the gyroangle-averaged first-order particle gyroradius is
where $\langle \boldsymbol {\rho }_{0}\boldsymbol {\rho }_{0}\rangle = (J/m\varOmega )(\mathbb {I} - \hat {\textsf {b}}\hat {\textsf {b}})$. Hence, we now need expressions for $\langle G_{1}^{J}\rangle$ and $\boldsymbol {\varPi }_{1}$ in order to obtain an explicit expression for the second-order guiding-centre Hamiltonian (3.22), which are determined at third order in our perturbation analysis.
3.3. Third-order perturbation analysis
The third-order term in the guiding-centre phase-space Lagrangian one-form (2.3) is expressed as
In what follows, the gauge function $\sigma _{3}$ will play an important role in completing the guiding-centre phase-space transformation, while the third-order guiding-centre Hamiltonian will not be needed in the present guiding-centre formulation.
3.3.1. Third-order symplectic structure
The remaining components $(G_{2\|}^{\boldsymbol {x}}, \langle G_{1}^{J}\rangle, G_{1}^{\theta })$ and the first-order guiding-centre momentum $\boldsymbol {\varPi }_{1}$ will now be determined from the momentum components of the third-order guiding-centre symplectic one-form (3.24)
where $\boldsymbol {P}_{3} \equiv \frac {1}{2}\boldsymbol {P}_{0} + \frac {1}{3}\boldsymbol {q}_{\bot }$, so that
Since $\partial \boldsymbol {\rho }_{0}/\partial p_{\|} = 0$, the $p_{\|}$-component of (3.25) suggests that we define the new gauge function
where the last expression follows from (3.26). Using the new gauge function (3.27), the momentum components (3.25), therefore, become
where, using (3.26), we introduced the identities
so that we can also introduce yet another gauge function
in the $\theta$-component of (3.25). By requiring that the momentum components (3.28) vanish, we now obtain the definitions
Hence, the components $G_{2\|}^{\boldsymbol {x}}$ and $G_{1}^{\theta }$ are determined from the third-order gauge function $\bar {\sigma }_{3}$, which is determined from (3.30), while $\bar {\bar {\sigma }}_{3}$ is determined from the gyroangle-dependent part $\tilde {G}_{1}^{J} \equiv \partial \bar {\bar {\sigma }}_{3}/\partial \theta$ obtained from (3.20). Since these gyroangle-dependent components are not needed in what follows, however, they will not be derived here.
3.3.2. Second-order guiding-centre Hamiltonian
From (3.31), we immediately conclude that $\langle G_{1}^{J}\rangle$ must be defined as
which was obtained in previous derivations (Brizard Reference Brizard1995; Madsen Reference Madsen2010; Frei et al. Reference Frei, Jorge and Ricci2020), so that the second-order guiding-centre Hamiltonian (3.22) becomes
where we used the identity
In previous works (Littlejohn Reference Littlejohn1981; Hahm Reference Hahm1996; Miyato et al. Reference Miyato, Scott, Strintzi and Tokuda2009; Madsen Reference Madsen2010; Frei et al. Reference Frei, Jorge and Ricci2020), the choice for the first-order symplectic momentum $\boldsymbol {\varPi }_{1} = -\frac {1}{2}J\tau \hat {\textsf {b}}$ was used to eliminate the Baños drift (Baños Reference Baños1967; Northrop & Rome Reference Northrop and Rome1978) from the guiding-centre velocity (i.e. $\partial \mathcal {H}_{2\textrm {gc}}/\partial P_{\|} = 0$), which is instead added to the definition of the guiding-centre parallel momentum (3.14). This choice, therefore, yields the second-order guiding-centre Hamiltonian $\mathcal {H}_{2\textrm {gc}} = (J\hat {\textsf {b}}/2)\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {\times }\boldsymbol {u}_{E}$.
A different choice adopted by Tronko & Brizard (Reference Tronko and Brizard2015) for the first-order symplectic momentum
on the other hand, was previously shown to yield an exact Lie-transform perturbation derivation of the standard guiding-centre polarization derived by Kaufman (Reference Kaufman1986) in the absence of an electric field. While this choice still eliminates the Baños drift from the guiding-centre velocity, it also yields an expression for the second-order guiding-centre Hamiltonian (3.35) that exactly represents the guiding-centre finite-Larmor-radius (FLR) correction to the electrostatic potential energy $e\varPhi$ (Brizard Reference Brizard2023a):
Finally, we note that recent numerical studies of particle and guiding-centre orbits in axisymmetric magnetic fields (Brizard & Hodgeman Reference Brizard and Hodgeman2023) have shown that guiding-centre orbits are faithful (i.e. remain close) to particle orbits only if second-order effects, including the correction (3.37), are included, which confirms earlier results by Belova et al. (Reference Belova, Gorelenkov and Cheng2003).
4. Guiding-centre Hamiltonian dynamics
In this section, we summarize the results of the Lie-transform perturbation analysis of the guiding-centre Lagrangian dynamics presented in § 3, and we remove the explicit $\epsilon$ scaling by restoring the physical mass $\epsilon m \rightarrow m$. Hence, we find the guiding-centre phase-space extended one-form
where the guiding-centre symplectic momentum
includes the higher-order polarization correction (3.37). The extended guiding-centre Hamiltonian, on the other hand, is expressed as
where the guiding-centre kinetic energy in the drifting frame is
which includes the FLR correction (3.38) to the electrostatic potential energy $e\varPhi$. We note that the presence of the gyrogauge fields $(\mathcal {S},\boldsymbol {R})$ in (4.1) guarantees gyrogauge invariance of the guiding-centre equations of motion derived from them.
4.1. Extended guiding-centre Poisson bracket
The extended guiding-centre Poisson bracket $\{\,,\,\}_\textrm {gc}$ is obtained by, first, constructing an 8$\times$8 matrix out of the components of the extended guiding-centre Lagrange two-form $\boldsymbol {\omega }_\textrm {gc} = \textsf {d}\varGamma _\textrm {gc}$ and, then, inverting this matrix to obtain the extended guiding-centre Poisson matrix, whose components are the fundamental brackets $\{Z^{\alpha }, Z^{\beta }\}_\textrm {gc}$. From these components, we obtain the extended guiding-centre Poisson bracket
where
and the guiding-centre Jacobian is $\mathcal {J}_\textrm {gc} = (e/c)B_{\|}^{*} \equiv (e/c)\hat {\textsf {b}}\boldsymbol {\cdot }\boldsymbol {B}^{*}$. In addition, we have introduced the definitions
where $\boldsymbol {R}^{*} \equiv \boldsymbol {R} + \frac {1}{2}\boldsymbol {\nabla }\boldsymbol {\times }\hat {\textsf {b}}$. We note that the Poisson bracket (4.5) can be expressed in divergence form as
and that it automatically satisfies the Jacobi identity
since the extended guiding-centre Lagrange two-form $\boldsymbol {\omega }_\textrm {gc} = \textsf {d}\varGamma _\textrm {gc}$ is exact (i.e. $\textsf {d}\boldsymbol {\omega }_\textrm {gc} = \textsf {d}^{2}\varGamma _\textrm {gc} = 0$) provided $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {B}^{*} = 0$.
Next, we note that the operators (4.7) and (4.8) contain the gyrogauge-invariant combinations $\partial /\partial t + \mathcal {S}\partial /\partial \theta$ and $\boldsymbol {\nabla } + \boldsymbol {R}\partial /\partial \theta$, while (4.6) and (4.8) include the gyrogauge-independent vector fields (Ye & Kaufman Reference Ye and Kaufman1992; Brizard Reference Brizard2023b)
where $\epsilon _{ijk}$ denotes the completely antisymmetric Levi-Civita tensor.
4.2. Guiding-centre Hamilton equations
The guiding-centre Hamilton equations $\dot {Z}^{\alpha } \equiv \{ Z^{\alpha }, \mathcal {H}_\textrm {gc}\}_\textrm {gc}$ include the guiding-centre velocity
where $\hat {\textsf {b}}\boldsymbol {\cdot }\dot {\boldsymbol {X}} = P_{\|}/m$ defines the parallel guiding-centre velocity, and the guiding-centre parallel force
where the modified electric field is represented as
and the gyroangle angular velocity
We note that the reduced guiding-centre equations of motion (4.12)–(4.13) satisfy the guiding-centre Liouville equation
where
and
where we made use of the modified Faraday's law $\partial \boldsymbol {B}^{*}/\partial t = -c\boldsymbol {\nabla }\boldsymbol {\times }\boldsymbol {E}^{*}$.
4.3. Eulerian variations of the guiding-centre Lagrangian
The results of the Lie-transform perturbation analysis carried out in § 3.3 can also be used to construct the following (regular) guiding-centre Lagrangian:
where $\boldsymbol {\varPi }_\textrm {gc}$ and $K_\textrm {gc}$ are defined in (4.2) and (4.4), respectively. The guiding-centre Euler–Lagrange equations are derived from this Lagrangian as
which are associated with virtual displacements $\delta \boldsymbol {X}$ and $\delta P_{\|}$, respectively. From these equations, we easily recover the guiding-centre Hamilton equations (4.12) and (4.13). Likewise, the guiding-centre Euler–Lagrange equation associated with the virtual displacement $\delta J$ yields (4.15), 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-centre phase-space coordinates, the guiding-centre Lagrangian (4.19) can also be varied with respect to the electric and magnetic fields $(\delta \boldsymbol {E}, \delta \boldsymbol {B})$, which respectively yield expressions for the guiding-centre polarization and magnetization (Brizard Reference Brizard2008). Here, the Eulerian variation of the guiding-centre Lagrangian (4.19) is expressed as (Brizard Reference Brizard2023b)
where the FLR corrections, which are ignored in (4.22), are calculated to first order in a recent paper (Brizard Reference Brizard2023b). Here, the guiding-centre electric-dipole moment $\boldsymbol {{\rm \pi} }_\textrm {gc}$ is defined as (Pfirsch Reference Pfirsch1984; Pfirsch & Morrison Reference Pfirsch and Morrison1985)
while the guiding-centre magnetic dipole moment $\boldsymbol {\mu }_\textrm {gc} + \boldsymbol {{\rm \pi} }_\textrm {gc}\boldsymbol {\times }\boldsymbol {P}_{0}/(mc)$ is defined as the sum of the intrinsic magnetic dipole moment
which includes the gyrogauge correction associated with $\delta \mathcal {S} + \dot {\boldsymbol {X}}\boldsymbol {\cdot }\delta \boldsymbol {R}$ (Brizard Reference Brizard2023b), and the moving electric-dipole contribution (Jackson Reference Jackson1975), expressed in terms of the lowest-order guiding-centre momentum $\boldsymbol {P}_{0} = P_{\|}\hat {\textsf {b}} + \boldsymbol {P}_{E}$. By ignoring these FLR corrections, the guiding-centre polarization and magnetization are defined as moments of the guiding-centre electric and magnetic dipole moments
where the guiding-centre phase-space density $F_\textrm {gc} \equiv \mathcal {J}_\textrm {gc}F$ includes the guiding-centre Jacobian $\mathcal {J}_\textrm {gc}$ and the notation $\int _{P}$ includes an integral over guiding-centre 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 Reference Kaufman1986) for the guiding-centre electric-dipole moment $\boldsymbol {{\rm \pi} }_\textrm {gc} = (e\hat {\textsf {b}}/\varOmega )\boldsymbol {\times }\dot {\boldsymbol {X}}$ is derived by Lie-transform perturbation method (Tronko & Brizard Reference Tronko and Brizard2015) only if the first-order polarization correction (3.37) is used.
5. Summary
In the present work, a set of higher-order guiding-centre Hamilton equations was derived by Lie-transform perturbation method for the case of time-dependent electric and magnetic fields that satisfy the standard guiding-centre space–time orderings. The second-order corrections in the guiding-centre Hamiltonian represented FLR corrections of the lowest-order electrostatic potential energy $e\varPhi$. Additional second-order corrections in the guiding-centre Lagrangian (4.19) included gyrogauge-invariance contributions to the guiding-centre Hamiltonian and Poisson bracket as well as corrections leading to the correct guiding-centre polarization.
When we turn our attention to the self-consistent interactions of the charged-particle guiding centres and the electromagnetic fields associated with plasma confinement, we need to derive a set of higher-order guiding-centre Vlasov–Maxwell equations. Work presented elsewhere (Brizard Reference Brizard2023b) considered the variational formulation of the higher-order guiding-centre Vlasov–Maxwell equations derived from the guiding-centre Lagrangian (4.19) and the Maxwell Lagrangian density.
According to this variational principle, using the guiding-centre Liouville equation (4.16), the guiding-centre Vlasov equation for the guiding-centre phase-space density $F_\textrm {gc} \equiv \mathcal {J}_\textrm {gc}F$ is written in divergence form as
while the Maxwell equations with particle sources are expressed as
where the guiding-centre charge and current densities are $\varrho _\textrm {gc} = \int _{P} eF_\textrm {gc}$ and $\boldsymbol {J}_\textrm {gc} = \int _{P} e\dot {\boldsymbol {X}}F_\textrm {gc}$. Here, the guiding-centre polarization charge density $\varrho _\textrm {pol} \equiv -\boldsymbol {\nabla }\boldsymbol {\cdot } \boldsymbol {\mathcal {P}}_\textrm {gc}$ and current density $\boldsymbol {J}_\textrm {pol} \equiv \partial \boldsymbol {\mathcal {P}}_\textrm {gc}/\partial t$ are derived from the guiding-centre polarization (4.25), while the guiding-centre magnetization current density $\boldsymbol {J}_\textrm {mag} \equiv c\boldsymbol {\nabla }\boldsymbol {\times }\boldsymbol {\mathcal {M}}_\textrm {gc}$ is derived from the guiding-centre magnetization (4.26). The remaining source-free Maxwell equations are Faraday's law $\partial \boldsymbol {B}/\partial t = -c\boldsymbol {\nabla }\boldsymbol {\times }\boldsymbol {E}$ and Gauss's law $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {B} = 0$. We note that the guiding-centre Vlasov–Maxwell variational principle also guarantees the existence of exact energy-momentum conservation laws, derived by the Noether method (Brizard Reference Brizard2008). Our recent work (Brizard Reference Brizard2023b) has considered the set of higher-order guiding-centre Vlasov–Maxwell equations obtained by explicitly imposing the quasineutrality constraint $\varrho _\textrm {gc} = \boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {\mathcal {P}}_\textrm {gc}$.
Future work will explore the Hamiltonian structure of the guiding-centre Vlasov–Maxwell equations, when guiding-centre polarization and magnetization are included, which will generalize our previous work (Brizard Reference Brizard2021), and its development is motivated by the desire to construct structure-preserving numerical algorithms (Morrison Reference Morrison2017) using an important set of reduced Vlasov–Maxwell equations.
Acknowledgements
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 Reference Kaufman1978) led to the landmark papers by Cary (Reference Cary1981) and Cary & Littlejohn (Reference Cary and Littlejohn1983), culminating with the review papers by Brizard & Hahm (Reference Brizard and Hahm2007) and Cary & Brizard (Reference Cary and Brizard2009).
Editor Paolo Ricci thanks the referees for their advice in evaluating this article.
Funding
The present work was supported by the National Science Foundation grant PHY-2206302.
Declaration of interests
The author report no conflict of interest.