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Magnetohydrodynamic motion of a two-fluid plasma

The two-fluid Maxwell system couples frictionless electron and ion fluids via Maxwell's equations. When the frequencies of light waves, Langmuir waves, and single-particle cyclotron motion are scaled to be asymptotically large, the two-fluid Maxwell system becomes a fast-slow dynamical system. This fast-slow system admits a formally-exact single-fluid closure that may be computed systematically with any desired order of accuracy through the use of a functional partial differential equation. In the leading order approximation, the closure reproduces magnetohydrodynamics (MHD). Higher order truncations of the closure give an infinite hierarchy of extended MHD models that allow for arbitrary mass ratio, as well as perturbative deviations from charge neutrality. The closure is interpreted geometrically as an invariant slow manifold in the infinite-dimensional two-fluid phase space, on which two-fluid motions are free of high-frequency oscillations. This perspective shows that the full closure inherits a Hamiltonian structure from two-fluid theory. By employing infinite-dimensional Lie transforms, the Poisson bracket for the all-orders closure may be obtained in closed form. Thus, conservative truncations of the single-fluid closure may be obtained by simply truncating the single-fluid Hamiltonian. Moreover, the closed-form expression for the all-orders bracket gives explicit expressions for a number of the full closure's conservation laws. Notably, the full closure, as well as any of its Hamiltonian truncations, admits a pair of independent circulation invariants.


I. INTRODUCTION
Ideal magnetohydrodynamics 1 (MHD) is a well-known reduced plasma model that treats a plasma as a single conducting fluid. Because real plasma is made up of a large collection of discrete particles, it is natural to wonder how such a single-fluid model could have any predictive capability. This challenging problem has been addressed on numerous occasions 2 , and most assessments conclude that MHD does a good job of predicting plasma equilibrium and stability at "large-scales." However, this answer is unsatisfactory for a variety of reasons.
Most importantly, it does not tell us why MHD works, only that it does. In contrast, there must be some physical mechanism that enables a many-particle-field system to exhibit MHD behavior. Previous investigations provide only vague suggestions of what this mechanism might be.
The purpose of this article is to study an important part of this mechanism. Before elaborating further, however, the sense in which a "part" of the mechanism can even be discussed is worthy of explanation. One possible way to exhibit a mechanism for emergent MHD behavior in a many-particle system is to first show and explain emergent multi-fluid behavior, and then explain how MHD emerges from multi-fluid dynamics. This approach naturally breaks the mechanism into two parts, and this article will discuss (the simpler) one, namely the submechanism by which MHD behavior emerges from the dynamics of multiple charged fluids. Of course, if there is no submechanism for multi-fluid dynamics to emerge in a many-particle model, then the discussion contained in this article would be neither novel nor useful. Therefore the assumption that multi-fluid dynamics can indeed be found within many-particle dynamics will be tacitly assumed henceforth.
Roughly speaking, the dynamical content that is missing from MHD consists of rapidly oscillating modes, including Langmuir waves and light waves. Therefore one tempting explanation for the emergence of MHD motion in a two-fluid plasma is the effective damping of these rapidly varying modes. Even though the ideal two-fluid-Maxwell system does not include collisional dissipation, this damping mechanism may still be present as a result of phase mixing 3 . A second possible explanation may be that the rapidly oscillating modes do not damp, but instead are effectively averaged out. If this explanation is valid, it would be especially interesting because it would suggest that there must be some kind of ponderomotive forcing 4 of the MHD state variables by the rapidly oscillating modes that has not been calculated previously. Finally, there is at least one other possible explanation. Perhaps there are special initial configurations of a two-fluid plasma that do not excite the rapidly oscillating modes at all. In other words, it may be that Langmuir waves and light waves are neither damped, nor averaged-out, but instead fail to be excited in the first place. While all of these possible explanations for emergent MHD behavior may be interesting, interrelated, or perhaps mutually independent, the ensuing discussion and analysis will focus on the third possible mechanism, which is convenient to refer to as "lazy high-frequency modes." An oversimple caricature of lazy high-frequency modes consists of two pendula placed in a room, one much longer than the other. The most general motion of these pendula (assuming small amplitude oscillations, for simplicity) involves each pendulum swaying at its respective characteristic frequency, and therefore involves a pair of disparate time scales.
However, there are special "slow" motions of the system wherein the short pendulum is motionless, meaning that the fast time scale in the problem is not present. Note that these motions are characterized by special initial conditions that allow only the long pendulum to be displaced from its equilibrium location in phase space. Also note that the slow dynamics is governed by Newton's Second Law applied to just the long pendulum, which is a dynamical system whose dimension is less than that of the total system. Here the short pendulum is the analogue of the rapidly-varying modes in the two-fluid model, while the long pendulum represents MHD motion. Although the special "slow" initial conditions are obvious in this toy problem, the same cannot be said of two-fluid dynamics. There, all of the modes are coupled nonlinearly, and so it is not clear that slow initial conditions even exist, let alone possess a simple parameterization.
In order to argue that slow initial conditions for two-fluid dynamics do exist, this article will deduce three technical results: (a) an asymptotic expansion for the set of slow two-fluid initial conditions, (b) an asymptotic expansion of the reduced dynamical equations that govern slow dynamics, and (c) the variational and Hamiltonian structures underlying the slow dynamics, which are naturally inherited from the corresponding structures underlying two-fluid dynamics. Interestingly, the mathematical tools that will lead to these results are powerful enough to provide a simple, closed-form expression for the slow dynamics' Poisson bracket.
Modulo delicate issues related to convergence of the asymptotic expansions (see Section V for a discussion of this point), these results will show: (1) that there is a collection of slow initial conditions for the two-fluid system of equations that is parameterized by the MHD phase space, (2) that the reduced equations governing the slow dynamics are equivalent to an extended MHD model with low-order truncations that reproduce ideal MHD and Hall MHD, and (3) that the Hall MHD Poisson bracket governs the slow dynamics to all orders. Taken together, (1) and (2) imply that lazy high-frequency modes may indeed be a plausible mechanism for MHD-like motion of a two-fluid plasma. Moreover, (3) implies that the problem of developing dissipation-free approximations of this MHD-like motion reduces to finding an approximate expression for the slow dynamics' Hamiltonian functional. This is a desireable feature for an extended MHD theory to have, for instance, when using such a model to study collisionless reconnection.

PRINCIPLE
The asymptotically-scaled ideal two-fluid-Maxwell system is given by where n i is the ion number density, u σ is the velocity of species σ, p σ is the species-σ partial pressure, B is the divergence-free magnetic field, E is the electric field, m σ is the species-σ mass, Z i is the ionic atomic number, q e is (minus) the elementary unit of charge, and µ o , ǫ o are the usual MKS vacuum permeability and permittivity. I will assume that the spatial domain is the 3-torus; non-periodic boundary conditions will require a separate analysis.
Upon setting the mass ratio m e /m i = ν, it is also useful to introduce the scalar fields which represent the (squared) speed of Alfvén waves, the (squared) frequency of Langmuir oscillations, and the frequency of ion cyclotron motion.
In their order of appearance, the equations comprising the two-fluid Maxwell system express the conservation of momentum for ions and electrons, the conservation of ion number, the Ampére-Maxwell Law, and Faraday's Law. For the sake of simplicity, I have assumed a barotropic equation of state for both electrons and ions but most of the ensuing discussion would only be modified superficially upon adopting an equation of state that accounts for entropic dynamics. Notice that the electron number density does not appear explicitly in the two-fluid Maxwell system as it is written above. This is accomplished by using Gauss' Law to eliminate the electron number density in favor of the electric field and the ion number density. The two-fluid state is therefore given by the tuple of fields Z = (n i , u i , B, u e , E), and the two-fluid-Maxwell system may be regarded as a first-order ODE on Z-space, i.e. Z-space is the (infinite-dimensional) two-fluid-Maxwell phase space. Note that if the electron number density were not eliminated, the two-fluid Maxwell system would instead take the form of a differential-algebraic system on a slightly larger space. Because I will evetually use some ideas from dynamical systems theory to analyze two-fluid dynamics, this would be a technical inconvenience.
The unscaled (ǫ = 1) two-fluid-Maxwell system, which is perhaps more familiar than the scaled version, may be transformed into the scaled two-fluid-Maxwell system by making the simple substitutions Formally, rescaling the elementary charge and vacuum permittivity is equivalent to working with dimensionless variables and then adopting the "drift-kinetic ordering" where v the is the electron thermal velocity, and ω, L represent the characteristic time and length scales for the slow dynamics. In this article, the dimensional MKS unit system will be adopted instead of the natural dimensionless unit system implied by the drift kinetic ordering. Physical expressions with the correct units may therefore always be recovered by setting ǫ = 1. Physically, the drift kinetic ordering implies that the observation time scale is much longer than the ion cyclotron period, while the observation length scale is much longer than the ion gyroradius. Thus, the usual assumptions underlying guiding center theory are valid in the drift kinetic ordering. The physical interpretation of the remainder of the drift kinetic ordering is: non-relativistic electrons ( v 2 the c 2 ∼ ǫ), arbitrary mass ratio and plasma beta (β ∼ me m i ∼ 1), bulk flow speed comparable to E × B-speed, and E × B-speed comparable to the electron thermal speed ( uo v the ∼ u E v the ∼ 1). This scaling will be leveraged in what follows to find asymptotic expansions for both the special slow two-fluid configurations as well as the reduced evolution law for the slow dynamics.
An important property of the scaled two-fluid-Maxwell system is that it may be derived from a phase space variational principle. The phase space Lagrangian is given by where the Hamiltonian functional is given by The functions U e (n e ) and U i (n i ) are the internal energy densities for electrons and ions respectively. They are determined up to unimportant additive constants by the thermodynamic identities When computing the Euler-Lagrange equations associated with L, variations are taken with δv e =ξ e + v e · ∇ξ e − ξ e · ∇v e .

CONDITIONS
Define a slow initial configuration of a two-fluid plasma as an initial condition for the scaled two-fluid-Maxwell system that is O(1) and whose time derivative is O(1). The condition on the time derivative is reasonable because the drift-kinetic ordering involves equating the dynamical time scale with a time scale that is long when compared with the cyclotron period. The question this section will address is "what are the slow initial configurations, and how do they evolve in time?" Satisfying answers will be obtained in an asymptotic sense.
Consider first two basic mathematical properties of the slow initial conditions.
property 1: If Z is a slow initial state, then the state Z(T ) obtained by letting Z evolve for T seconds according to the two-fluid-Maxwell system is also a slow initial state, regardless of the value of T . To see this, first observe that since the path Z(t) is free of rapid oscillations, so too is the path Z ′ (t) = Z(t + T ). But, by the uniqueness of solutions to ordinary differential equations (e.g. the two-fluid-Maxwell system), Z ′ (t) is precisely the solution of the two-fluid-Maxwell system with initial condition Z(T ). In other words, Z(T ), regarded as an initial condition, produces a slow evolution. This proves the claim.
remark: Property 1 is a statement pertaining to the whole set of slow initial conditions. In the context of dynamical systems theory 6 , the collection of slow initial conditions would be referred to as an invariant set. More generally, given a dynamical system on some phase space, an invariant set is a subset of phase space with the property that points in the subset stay inside the subset under dynamical evolution.
property 2: As ǫ → 0, the set of slow initial conditions, S ǫ (which is a subset of Z-space), must contain In order to verify that this is true, suppose that Z ǫ = (n iǫ , u iǫ , B ǫ , u eǫ , E ǫ ) is a slow solution of the scaled two-fluid-Maxwell system. Because the solution is slow, the time derivative of Z ǫ must be O(1). Therefore the terms in Eqs. (1)-(5) that are multiplied by ǫ −1 must individually vanish as ǫ → 0. In particular, The only solution of this system of equations is E 0 = −u i0 × B 0 , u e0 = u i0 . Thus, Z 0 must be contained in the set S 0 . This verifies the claim.
remark: Physically speaking, the set S 0 consists of two-fluid states that are current neutral 7 (u e = u i ) and that satisfy the ideal Ohm's Law (E = −u i × B). Charge neutrality is also enforced because the electron number density Thus, property 2 already provides some evidence that the slow initial conditions in the two-fluid-Maxwell system must be related to MHD. Property 2 also suggests that slow solutions of the two-fluid-Maxwell system have the property that the electric field and the electron fluid velocity are slaved to the MHD state variables (n i , u i , B), i.e. the former are expressed as functions of the latter. Such slaving relations, which may also be thought of as defining a closure, are commonplace in the theory of slow manifold reduction 8 and geometric singular perturbation theory 9 , which forms the theoretical basis underlying the discussion in this section.
Taken together, properties 1 and 2 indicate that a reasonable approach to finding the slow initial conditions is to look for an invariant subset of Z-space, S ǫ , of the form where u * eǫ and E * ǫ are undetermined O(1) functions of the MHD state variables (n i , u i , B). Keeping in line with the remark below property 2, I will refer to u * eǫ and E * ǫ as the slaving functions of the slow manifold S ǫ . Admittedly, Eq. (23) is nothing more than an ansatz for the set of slow initial conditions. In particular, it is not at all obvious that S ǫ needs to exist.
Moreover, even if S ǫ does exist, that would not imply that S ǫ contains all of the slow initial conditions -in fact it is not obvious at this stage that S ǫ contains any slow initial conditions whatsoever! Nevertheless, (23) will prove to be a good ansatz for two reasons. First, it will turn out that the slaving functions have unique asymptotic expansions in powers of ǫ.
Second, it will be possible to show formally that the dynamics of two-fluid states in S ǫ are indeed slow.
By extending the argument supporting property 2, I will now derive a functional partial differential equation satisfied by the slaving functions in Eq. (23). Suppose that Z ǫ = (n iǫ , u iǫ , B ǫ , u eǫ , E ǫ ) is a solution of the scaled two-fluid-Maxwell system contained in S ǫ .
Then the electric field and the electron fluid velocity must satisfy the slaving relations and Z ǫ must be a solution of the scaled two-fluid-Maxwell equations (1)- (5). An interesting consequence of these two constraints is that the electron momentum equation and the Ampére-Maxwell Law may be written in a manner that does not involve any time derivatives. To see this, first note that the slaving relations imply the partial time derivatives of E ǫ and u eǫ are given by Here the symbol D denotes Fréchet derivative (see Appendix A if unfamiliar with the Fréchet derivative). Next use the ion continuity equation, the ion momentum equation, and Faraday's Law to eliminate the partial time derivatives from the right-hand-sides of Eqs. (26) and (27), and where n * eǫ = Z i n i + ǫ 2 ǫ o q −1 e ∇ · E * ǫ is shorthand notation for the electron number density. I will refer to this system of functional PDE as the invariance equations.
In general the invariance equations, which are nonlinear and involve both functional and ordinary derivatives, are hopelessly difficult to solve. However, if they admit a solution that is smooth in ǫ and O(1) as ǫ → 0, then this solution has the unique asymptotic expansion where the coefficients u * ek , E * k are most readily obtained by substituting the asymptotic expansions into the invariance equations and then solving order by order. For instance, the leading-order invariance equations (O(ǫ −1 ) as written) are which have the unique solution representing current neutrality and ideal Ohm's law. In general, the n'th order invariance equation determines uniquely the n'th order terms in the asymptotic expansions (32) and (33). In particular, the O(1) invariance equations lead to while the O(ǫ) invariance equations gives where is shorthand notation for the charge density and displacement current given by the ideal MHD model. The higher-order coefficients rapidly become very complicated, but they may be efficiently computed if desired by solving the invariance equations iteratively using a computer algebra system. Now that the slaving functions have been determined, the time evolution of two-fluid states that are contained in the slow manifold S ǫ is easy to determine. Suppose Z(t) ∈ S ǫ is two-fluid trajectory contained in the slow manifold. Combining the ion momentum equation, the ion continuity equation, Faraday's Law, and the slaving relations then implies which clearly gives a closed system of equations that determine the time evolution of the MHD state (n i , u i , B). Now, with the time evolution of the MHD state determined, the time evolution of the entire two-fluid state is also determined because two fluid states contained in . It is therefore sensible to refer to the system (44)-(46) as the slow two-fluid equations. Observe that the slaving function E * ǫ appears in the slow two-fluid equations while u * eǫ does not.
The following facts pertaining to two-fluid configurations that begin in the slow manifold S ǫ may now be inferred.
fact 1: Two-fluid states that begin on the slow manifold S ǫ remain on the slow manifold and evolve on the slow (O(1)) timescale. This fact follows from the construction of S ǫ , which guaranteed S ǫ is an invariant set, and from the expressions for the first few terms in the asymptotic expansion for E * ǫ . Indeed, by substituting the first two terms in the asymptotic expansion of E * ǫ into Eq. (44), it is simple to verify that the leading-order truncation of the slow two-fluid equations is given by This shows that the time derivatives Interestingly, the first correction to the leading-order slow two-fluid equations is which is a generalization of Hall MHD 10,11 . The reason for the additional terms relative to ordinary Hall MHD is the following. In the ordinary Hall theory, deviations from charge neutrality, the displacement current, and electron inertia are completely ignored. Formally There is one set of properties possessed by the full slow two-fluid system that can be understood in a rather complete sense. These special properties pertain to the Hamiltonian structure of the slow dynamics. This section will show that the Poisson bracket governing slow two-fluid dynamics may be obtained in closed form. Using this expression, it will be possible to deduce further closed form expressions for some of the conservation laws possessed by the all-orders slow two-fluid system. In addition, this result will lead to a convenient and practically useful method for obtaining conservative truncations of the slow two-fluid system. Because the data consisting of the Poisson bracket and the Hamiltonian functional completely determines the slow two-fluid equations, the problem of truncating while preserving the conservative properties of the slow dynamics is reduced to truncating the Hamiltonian.
In order to uncover the slow two-fluid system's Hamiltonian structure, it is easiest to start from the phase space variational principle governing the full two-fluid-Maxwell system, which is embodied by the phase space Lagrangian (13). Given a solution of the two-fluid Maxwell system, the phase space variational principle states that an arbitrary variation of the action S = L dt around that solution vanishes, δS = 0. In particular, if the solution is contained within the slow manifold S ǫ , any variation of the action that does not leave the slow manifold vanishes. In other words, the action S * obtained by restricting S to curves that are contained in S ǫ has as critical points solutions of the slow two-fluid equations. This implies that the slow two-fluid system inherits a phase space variational principle from the full two-fluid Maxwell system. This slow two-fluid variational principle represents the first crucial step toward obtaining a closed for expression for the slow two-fluid Poisson bracket.
Explicitly, the slow two-fluid action is given by S * = L * dt, where the slow two-fluid Lagrangian is, in accordance with the previous paragraph, given by and the slow two-fluid Hamiltonian is given by Notice that in these expressions the velocity u e has been replaced with the slaving function u * eǫ while the velocity v e has not. The reason for this is that the constraint imposed by restricting to the slow manifold S ǫ involves only Eulerian quantities. The Lagrangian configuration maps for both electrons and ions are completely unconstrained on the slow manifold. Of course, one may worry that if the velocity variable u e is constrained, then there might have to be a corresponding non-holonomic constraint on the electron configuration map. This is faulty reasoning because here we are working with a phase space Lagrangian.
The paths that we vary in the action S do not generally satisfy u e = v e , even though this relationship must hold along any solution of the Euler-Lagrange equations.
Using the phase space Lagrangian L * it is possible in principle to identify the Hamiltonian structure underlying the slow two-fluid dynamics. In fact, the Poisson bracket associated with L * is encoded in the part of L * that is linear in the velocities (v i , v e ,Ȧ,u i ) -the so-called "symplectic part" of L * (see Section II.C in Ref. 12). Littlejohn 13,14 gives a lucid discussion of this point, albeit in a finite-dimensional context, in his seminal work on the Hamiltonian formulation of guiding center motion. However, the situtation is more complicated than it seems to be at first glance. Because the symplectic part of L * involves the slaving functions E * ǫ , u * eǫ , the Poisson bracket associated with L * is very complicated. In fact, as a result of the slaving functions being known only as formal infinite series in ǫ, the Poisson bracket must also be a formal infinite series in ǫ. Thus, while the Hamiltonian structure for the slow dynamics is just in reach, the sought-after closed-form expression for the Poisson bracket still lies waiting on the other side of a theoretical chasm that has yet to be crossed.
A closed-form expression for the slow dynamics' Poisson bracket may be obtained by employing some ideas from symplectic geometry. The specific ideas that are necessary are described well by Littlejohn in Ref. 15. From Littlejohn's work, a procedure may be extracted that greatly simplifies the slow system's Poisson bracket. The specific details of this procedure, being perhaps more abstract than the present discussion requires, are contained in Appendix B. However, the essential idea behind the procedure is simple to describe and to understand. The goal is merely to find a near-identity transformation of the slow manifold S ǫ such that the linear terms in the transformed Lagrangian truncate at a finite order in ǫ. If such a transformation can be found, it would lead to an easily-computable closed-form expression for the Poisson bracket. The only price of going in this direction, aside from the labor involved, would be a somewhat more complicated Hamiltonian. As explained in the Appendix, existence of a transformation with the desired properties is ensured by a straightforward application of a geometric tool known as Moser's trick (see Ch.

in Ref. 16).
One transformation that leads to an O(1) truncation of the symplectic part of L * is given by where and where the transformed Hamiltonian is given byH Here the convenient shorthand notations u ⊥ = (1 − bb) · u and m = (1 + νZ i )m i have been introduced, as well as the so-called diamagnetic drift velocities It is important to emphasize here that even though the displayed expressions for the transformation (n i , u i , A) → (n, u, A) contain only finitely-many terms, the full transformation contains infinitely-many terms. As explained in Appendix B, all of these terms may be calculated in a systematic manner, and they ensure that the symplectic part of the transformed phase space Lagrangian is displayed entirely in Eq. (61); higher-order corrections to the symplectic part are zero in the "nice" coordinate system on S ǫ . It is also worth mentioning that the transformation calculated here is not the only one that leads to a simplified symplectic part in the phase space Lagrangian.
The Poisson bracket associated withL * may be found using standard techniques. In particular, the computation may be done by inverting the Lagrange tensor 17 associated with the symplectic part ofL * as is done for various kinetic systems in Ref. 18. Proceeding in this manner is useful because it ensures that the Poisson bracket satisfies the Jacobi identity; direct verification of the Jacobi identity as in Ref. 19 is not necessary. I will merely report the result of this calculation here. Given two arbitrary functionals G(n, u, B) and H(n, u, B), their Poisson bracket is given by Because this expression does not involve any infinite series, it represents the main result that was meant to be obtained in this section.
With a closed-form expression for the slow system's Poisson bracket in hand, it is now possible to deduce some general properties of the slow two-fluid system that are independent of truncation order. First and foremost, it is now clear that the slow dynamics possess Hamiltonian truncations with any desired level of accuracy. Such Hamiltonian truncations are obtained by using the full Poisson bracket in Eq. (67) while retaining only finitely-many terms in the expansion of the transformed HamiltonianH * =H * 0 + ǫH * 1 + ǫ 2H * 2 + . . . . These truncations of the slow two-fluid system are superior to naive truncations performed at the level of the equations of motion because they ensure that artificial dissipation is not introduced by truncation. An obvious manifestation of this fact is that any Hamiltonian truncation of the slow dynamics will automatically conserve the truncated Hamiltonian exactly.
This follows from the antisymmetry of the Poisson bracket. All Hamiltonian truncations of the slow two-fluid system also possess less obvious conservation laws. Notably, they all possess a pair of circulation invariants, which I will now describe.
For the sake of describing the circulation invariants precisely, let H =H * 0 + ǫH * 1 + ǫ 2H * 2 + · · · +H * n denote the n'th order truncation of the transformed Hamiltonian, where n is an arbitrary non-negative integer. The first circulation invariant, which is associated with motion of the ions, is given by Here ℓ i is a closed loop that moves with the velocity As the notation suggests, the velocity v * i is approximately equal to the ion fluid velocity. It is well-known that the barotropic two-fluid Maxwell system has an analogous circulation invariant, which is equivalent to the circulation of the ion canonical momentum. Actually, the invariant C i is the same quantity, merely restricted to the slow manifold. The second circulation invariant, which is associated with motion of electrons, is given by Here ℓ e is an arbitrary (not necessarily closed) curve that moves with the velocity v * e = 1 mn Just like C i , C e may be interpreted as an invariant inherited from the two-fluid-Maxwell system.

V. DISCUSSION
This article has put forth a new conceptual framework for understanding MHD and its relationship with ideal two-fluid theory. The key insight that underlies this new perspective is that the two-fluid theory, when appropriately scaled, admits a formally-exact single-fluid closure. At leading order, the closure reproduces ideal MHD. At higher orders, the closure leads to new or modified exteneded MHD models. Notably, the latter allow for arbitrary mass ratio as well as perturbative deviations from exact charge neutrality. after making the identifications The reason that the d i in Ref. 19 Of course, there is no guarantee that DF (ψ) exists. If it does, it is unique and it is said The most basic property satisfied by the Fréchet derivative is the chain rule. If F : W → V and G : V → U, then and it is now necessary to be mindful that DG is evaluated at F (ψ) and DF is evaluated at ψ. A useful consequence of the Chain rule is that the Fréchet derivative may be computed The basic idea that enables a systematic computation of the transformation used in Section IV is best understood in terms of differential forms. Essentially the same idea is described in the proof of Darboux's theorem in Ref. 16. Let Θ = θ + δθ be a 1-form on a manifold M and suppose that ω = −dθ is non-degenerate. Non-degeneracy means that the mapping X → ι X ω, where X is a vector field on M, is injective. Suppose further that δθ ≪ θ. Under these hypotheses, it is possible to find a near-identity transformation Φ of where the right-hand-side represents a time-ordered exponential of the vector field G t . Because G t must be small, Eq. (B4) provides an asymptotic expansion for the transformed Hamiltonian.
Next, expand the 1-form Θ in Eq. (B5) in a power series Θ = ǫ −1 Θ −1 + Θ 0 + ǫΘ 1 + ǫ 2 Θ 2 + . . . , and apply the transformation Φ. Modulo unimportant exact differentials, the 1-form Θ transforms intoΘ = ǫ −1Θ −1 +Θ 0 + ǫΘ 1 + ǫ 2Θ 2 + . . . , whereΘ The vector fields G k , which are usually referred to as Lie generators, are finally determined by demanding that the transformed 1-form truncates at finite-order in ǫ. There is a huge amount of freedom in this step. The prescription for finding the particular transformation that lead to the simple transformed 1-form found in Eq. (61) is given by These equations may be solved order-by-order in order to determine each of the G k . The first two Lie generators turn out to be G e 1 = 0 (B21) and G e 2 = −