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The Maxwell-Vlasov equations in Euler-Poincaré form 8 9 9 Hernán Cendra 1 Control and Dynamical Systems n California Institute of Technology 107-81 a Pasadena, CA 91125 J ∗ 3 [email protected] 1 Darryl D. Holm † 1 Los Alamos National Laboratory v Los Alamos, NM 87545 6 1 [email protected] 0 1 Mark J. W. Hoyle 0 Department of Mathematics 8 9 University of California, Santa Cruz, CA 95064 / n [email protected] y Jerrold E. Marsden‡ d - Control and Dynamical Systems o California Institute of Technology 107-81 a h Pasadena, CA 91125 c : [email protected] v i X To appear in J. Math. Phys. r a PACS Numbers: 03.40.Gc, 47.10.+g, 03.40.-t, 03.40.-z Abstract Low’s well known action principle for the Maxwell-Vlasov equations of idealplasmadynamicswasoriginally expressedintermsofamixtureofEule- rian and Lagrangian variables. By imposing suitable constraints on the variations and analyzing invariance properties of theLagrangian, as onedoes for ∗Permanent address: Universidad Nacional del Sur, 8000 Bahia Blanca, Argentina. email: [email protected] †ResearchsupportedbyUSDOEandtheUniversityofCalifornia ‡ResearchpartiallysupportedbyDOEcontract DE–FG0395–ER25251 1 LAUR-97-3326 Maxwell-Vlasov system in Euler-Poincaré form 2 theEulerequationsfortherigidbodyandidealfluids,wefirsttransformthis action principle into purely Eulerian variables. Hamilton’s principle for the Eulerian description of Low’s action principle then casts the Maxwell-Vlasov equations into Euler-Poincaré form for right invariant motion on the diffeomorphism group of position-velocity phase space, R6. Legendre transforming theEulerianformofLow’sactionprincipleproducestheHamiltonianformula- tionoftheseequationsintheEuleriandescription. SinceitarisesfromEuler- Poincaré equations, this Hamiltonian formulation can be written in terms of a Poisson structure that contains the Lie-Poisson bracket on the dual of a semidirect product Lie algebra. Because of degeneracies in the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints. Another Maxwell-Vlasov Poisson structure is known, whose ingredients are theLie-PoissonbracketonthedualoftheLiealgebraofsymplectomorphisms of phase space and the Born-Infeld brackets for the Maxwell field. We dis- cuss the relationship between these two Hamiltonian formulations. We also discussthegeneralKelvin-NoethertheoremforEuler–Poincaréequationsand its meaning in theplasma context. Contents 1 Introduction 2 2 The Maxwell-Vlasov equations 5 3 The Euler-Poincaré equations, Semidirect Products, and Kelvin’s Theorem 6 4 An action for the Maxwell-Vlasov equations 8 5 The Maxwell-Vlasov system as Euler-Poincaré equations 11 6 The Generalized Legendre Transformation. 15 7 Hamiltonian formulation 19 8 Conclusion 24 9 Acknowledgments 25 1 Introduction Reduction of action principles. Due to their wide applicability, the Maxwell- Vlasov equations of ideal plasma dynamics have been studied extensively. Low [1958]wrotedownanactionprinciple fortheminpreparationforstudyingstability of plasma equilibria. Low’s action principle is expressed in terms of a mixture of Lagrangianparticle variables and Eulerian field variables. LAUR-97-3326 Maxwell-Vlasov system in Euler-Poincaré form 3 FollowingtheinitiativeofArnold[1966]anditslaterdevelopments(seeMarsden andRatiu [1994]for background),we startwith a purely Lagrangiandescriptionof the plasma and investigate the invariance properties of the corresponding action. Using this set up and recent developments in the theory of the Euler-Poincaré equations (Poincaré [1901b]) due to Holm, Marsden and Ratiu [1997], we are able to cast Low’s action principle into a purely Eulerian description. Inthispaper,westartwiththestandardformofHamilton’svariationalprinciple (in the Lagrangianrepresentation)and derive the new Eulerianaction principle by asystematicreductionprocess,muchasonedoesinthecorrespondingderivationof Poisson brackets in the Hamiltonian formulation of the Maxwell-Vlasov equations startingwiththestandardcanonicalbracketsandproceedingbysymmetryreduction (as in Marsden and Weinstein [1982]). In particular, the Eulerian action principle we obtaininthis wayis differentfromthe ones foundin Ye andMorrison[1992]by ad hoc procedures. We also mention that the method of reduction of variational principleswedevelopnaturallyjustifiesconstraintsonthevariationsofthesocalled “Lin constraint” form, well known in fluid mechanics. The methods of this paper are based on reduction of variational principles, that is, on Lagrangian reduction (see Cendra et al. [1986, 1987] and Marsden and Scheurle [1993a,b]). These methods have also been useful for systems with non- holonomic constraints. This has been demonstrated in the work of Bloch, Krish- naprasad,Marsden and Murray [1996],who derived the reduced Lagranged’Alem- bertequationsfornonholonomicsystems,whichalsohaveaconstrainedvariational structure. The methods of the present paper should enhance the applicability of the Lagrangianreduction techniques for even wider classes of continuum systems. Passage to the Hamiltonian formulation. The Hamiltonian structure and nonlinear stability properties of the equilibrium solutions for the Maxwell-Vlasov system have been thoroughly explored. Some of the key references are Iwínski and Turski [1976],Morrison [1980],Marsden and Weinstein [1982] and Holm, Marsden, RatiuandWeinstein[1985]. SeealsotheintroductionandbibliographyofMarsden, Weinstein et al. [1983] for a guide to the history and literature of this subject. In our approach, Lagrangian reduction leads to the Euler-Poincaré form of the equations, which is still in the Lagrangianformulation. Using this set up, one may pass from the Lagrangian to the Hamiltonian formulation of the Maxwell-Vlasov equationsbyLegendretransformingtheactionprincipleintheEuleriandescription at either the level of the group variables (the level that keeps track of the particle positions),oratthe levelofthe Lie algebravariables. Onemustbe cautiousin this procedure because the relevant Hamiltonian and Lagrangian are degenerate. We deal with this degeneracy by using a version of the Dirac theory of constraints. Legendre transforming at the group level leads to a canonical Hamiltonian formulation and the latter leads to a new Hamiltonian formulation of the Maxwell- VlasovequationsintermsofaPoissonstructurecontainingtheLie-Poissonbracket on the dual of a semidirect product Lie algebra. This new formulation leads us naturally to the starting point for Hamiltonian reduction used by Marsden and Weinstein [1982] (see also Morrison [1980]and Kaufman and Dewar [1984]). LAUR-97-3326 Maxwell-Vlasov system in Euler-Poincaré form 4 Stability and asymptotics. The new Hamiltonian formulation of the Maxwell- Vlasov system places these equations into a framework in which one can use the energy-momentum and energy-Casimir methods for studying nonlinear stability propertiesof their relativeequilibrium solutions. This is directly in line with Low’s intended program, since the study of stability was Low’s original motivation for writing his action principle. Sample references in this direction are Holm, Mars- den,WeinsteinandRatiu[1985],Morrison[1987],MorrisonandPfirsch[1990],Wan [1990], Batt and Rein [1993] and Batt, Morrison and Rein [1995]. Other historical references for the Lagrangian approach to the Maxwell-Vlasov equations include Sturrock [1958], Galloway and Kim [1971] and Dewar [1972]. The EulerianformulationofLow’sactionprinciple alsocasts itinto aformthat is amenable to asymptotic expansions and creation of approximate theories (such asguidingcentertheories)possessingthesamemathematicalstructurearisingfrom the Euler-Poincaré setting. See, for example, Holm [1996] for applications of this approachof Hamilton’s principle asymptotics in geophysicalfluid dynamics. Comments on the Maxwell-Vlasov system. The rest of this paper will be concernedwithvariationalprinciplesfortheMaxwell-Vlasovsystemofequationsfor thedynamicsofanidealplasma. Theseequationshavealonghistorydatingbackat leasttoJeans[1902],whousedtheminasimplerformknownasthePoisson-Vlasov system to study structure formation on stellar and galactic scales. Even before Jeans,Poincaré[1890,1901a]hadinvestigatedthe stabilityofequilibriumsolutions ofthePoisson-Vlasovsystemforthepurposeofdeterminingthestabilityconditions for steller configurations. The history of the efforts to establish stellar stability conditionsusingthePoisson-VlasovsystemissummarizedbyChandrasekhar[1977]. The Poisson-Vlasov system is also used to describe the self-consistent dynamics of an electrostatic collisionless plasma, whereas the Maxwell-Vlasov system is used to describe the dynamics of a collisionless plasma evolving self-consistently in an electromagnetic field. Organization of the paper. The paper is organized as follows. Section 2 in- troduces the Maxwell-Vlasov equations. In section 3 we state the Euler-Poincaré theoremforLagrangiansdependingonparametersalongwiththeassociatedKelvin- Noether theorem. This general theorem plays a key role in our analysis. Section 5 reformulates these equations in a purely Eulerian form and shows how they satisfy the Euler-Poincarétheorem. The following section reviews some aspects of the Legendre transformation for degenerate Lagrangians. Section 4 reprises Low’s ac- tionprinciple forthe Maxwell-Vlasovequations. Section7caststhe Euler-Poincaré formulation of the Maxwell-Vlasov equations into Hamiltonian form possessing a Poisson structure that contains a Lie-Poisson bracket. In Section 8 we summarize our conclusions. LAUR-97-3326 Maxwell-Vlasov system in Euler-Poincaré form 5 2 The Maxwell-Vlasov equations The Maxwell-Vlasov system of equations describes the single particle distribution for a set of charged particles of one species moving self-consistently in an electromagnetic field. In this description, the Boltzmann function f(x,v,t) is viewed as the instantaneous probability density function for the particle distribution, i.e., given a region Ω of phase space, the probability of finding a particle in that region is dxdvf(x,v,t), (2.1) Z Ω wherexandvarethecurrentpositionsandvelocitiesoftheplasmaparticles. Thus, if the phase-space domain Ω is the whole (x,v) space, the value of this integral at a certain time t is normalized to unity. As is customary, we assume that the particles of the plasma obey dynamical equations and that the plasma density f is advected as a scalar along the particle trajectories in phase space, i.e., ∂f +x˙ ·∇xf +v˙ ·∇vf =0. (2.2) ∂t In this equation, and in the sequel, an overdot refers to a time derivative along a phase space trajectory,and ∇x and ∇v denote the gradientoperators with respect topositionandvelocityrespectively. Forpressurelessmotionintheelectromagnetic field of the charged particle distribution, the acceleration of a particle is given by q ∂A x¨ =− ∇xΦ+ −v×(∇x×A) , (2.3) m(cid:20) ∂t (cid:21) where(q/m)denotesthechargetomassratioofanindividualparticle,Φistheelec- tric potential, and A is the magnetic vector potential. Substituting this expression for v˙ in equation (2.2) yields ∂f q ∂A +v·∇xf − ∇xΦ+ −v×(∇x×A) ·∇vf =0. (2.4) ∂t m(cid:20) ∂t (cid:21) This is the Vlasov equation (also called the collisionless Boltzmann, or Jeans equation). The system is completed by the Maxwell equations with sources: ∂E ∇x·E=ρ, ∇x×B= +j, (2.5) ∂t where E and B are the electric and magnetic field variables respectively, ρ is the chargedensity andjis the currentdensity. These quantities areexpressedin terms of the Boltzmann function f and the Maxwell scalar and vector potentials Φ and A by: ∂A E=−∇xΦ− , B=∇x×A, ∂t ρ(x,t)=q dvf(x,v,t), j(x,t)=q dvvf(x,v,t). (2.6) Z Z LAUR-97-3326 Maxwell-Vlasov system in Euler-Poincaré form 6 By their definitions, E and B satisfy the kinematic Maxwell equations ∂B ∇x·B=0, ∇x×E=− . (2.7) ∂t Equations (2.4) - (2.7) comprise the Maxwell-Vlasov equations. When A is absent,the field is electrostatic andone obtains the Poisson-Vlasovequations. The Poisson-Vlasov system can also be used to describe a self gravitating collisionless fluid,andsoitformsamodelfortheevolutionofgalacticdynamics,see,e.g.,Binney and Tremaine [1987]. Notethattheintegralin(2.1)isindependentoftime(astheregionandthefunc- tion f evolve), since the vector field defining the motion of particles (see equation (2.3)) is divergence free with respect to the standard volume element on velocity phase space. Thus, one may interpret f either as a density or as a scalar. For our purposes later, we will need to be careful with the distinction, since the volume preserving nature of the flow of particles will be a consequence of our variational principle and will not be imposed at the outset. 3 The Euler-Poincaré equations, Semidirect Prod- ucts, and Kelvin’s Theorem The general Euler-Poincaré equations. Here we recall from Holm, Marsden and Ratiu [1997] the general form of the Euler-Poincaré equations and their asso- ciated Kelvin-Noether theorem. In the next section, we will immediately specialize these statements for a general invariance group G to the case of plasmas when G is the diffeomorphism group, Diff(TR3). We shall state the general theorem for rightactionsandrightinvariantLagrangians,whichisappropriatefortheMaxwell- Vlasov situation. The notation is as follows. • There is a right representation of the Lie groupG on the vector space V and G acts in the natural way from the right on TG×V∗: (v ,a)h=(v h,ah). g g • ρ : g → V is the linear map given by the corresponding right action of the v Lie algebra on V: ρ (ξ) =vξ, and ρ∗ :V∗ →g∗ is its dual. The g–action on v v g∗ and V∗ is defined to be minus the dual map of the g–action on g and V respectively and is denoted by µξ and aξ for ξ ∈ g, µ ∈ g∗, and a ∈ V∗. For v ∈V and a∈V∗, it will be convenient to write: ∗ v⋄a=ρ a i.e., hv⋄a,ξi=ha,vξi=−hv,aξi , v for all ξ ∈g. Note that v⋄a∈g∗. • Let Q be a manifold on which G acts trivally and assume that we have a function L:TG×TQ×V∗ →R which is right G–invariant. • In particular, if a ∈ V∗, define the Lagrangian L : TG×TQ → R by 0 a0 L (v ,u )=L(v ,u ,a ). ThenL isrightinvariantunderthelift toTG× a0 g q g q 0 a0 TQ of the right action of G on G×Q. a0 LAUR-97-3326 Maxwell-Vlasov system in Euler-Poincaré form 7 • Right G–invariance of L permits us to define l :g×TQ×V∗ →R by l(v g−1,u ,ag−1)=L(v ,u ,a). g q g q Conversely, this relation defines for any l : g×TQ×V∗ → R a right G– invariant function L:TG×TQ×V∗ →R. • For a curve g(t) ∈ G, let ξ(t) := g˙(t)g(t)−1 and define the curve a(t) as the unique solution of the linear differential equation with time dependent coefficients a˙(t) = −a(t)ξ(t) with initial condition a(0) = a . The solution 0 can be equivalently written as a(t)=a g(t)−1. 0 Theorem 3.1 The following are equivalent: i Hamilton’s variational principle holds: t2 δ L (g(t),g˙(t),q(t),q˙(t))dt=0, (3.1) Z a0 t1 for variations of g and q with fixed endpoints. ii (g(t),q(t)) satisfies the Euler–Lagrange equations for L on G×Q. a0 iii The constrained variational principle1 t2 δ l(ξ(t),q(t),q˙(t),a(t))dt =0, (3.2) Z t1 holds on g×Q, upon using variations of the form ∂η ∂η δξ = −ad η = −[ξ,η], δa=−aη, (3.3) ξ ∂t ∂t where η(t) ∈ g vanishes at the endpoints and δq(t) is unrestricted except for vanishing at the endpoints. iv The following system of Euler–Poincaré equations (with a parameter) coupled with Euler-Lagrange equations holds on g×TQ×V∗: ∂ δl δl δl ∗ =−ad + ⋄a, (3.4) ∂tδξ ξδξ δa and ∂ ∂l ∂l − =0. (3.5) ∂t∂q˙i ∂qi 1Strictly speaking this is not a variational principle because of the constraints imposed on the variations. Rather, this principle is more like the Lagrange d’Alembert principle used in nonholonomicmechanics. LAUR-97-3326 Maxwell-Vlasov system in Euler-Poincaré form 8 Thestrategyoftheproofissimple: onejustdeterminestheformofthevariations on the reduced space g×Q×V∗ that are induced by variations on the unreduced space TG × TQ and includes the relation of a(t) to a . One then carries the 0 variational principle to the quotient. See Holm, Marsden and Ratiu [1997] for details. Here we have included the extra factor of Q which is needed in the present application; this will be the space of potentials for the Maxwell field. This extra factor does not substantively alter the arguments. The Kelvin-Noether Theorem. We start with a Lagrangian L depending a0 on a parameter a ∈ V∗ as above and introduce a manifold C on which G acts. 0 We assume this is also a right action and suppose we have an equivariant map K:C×V∗ →g∗∗. In the case of continuum theories, the space C is chosen to be a loop space and hK(c,a),µi for c ∈ C and µ ∈ g∗ will be a circulation. This class of examples also shows why we do not want to identify the double dual g∗∗ with g. Define the Kelvin-Noether quantity I :C×g×TQ×V∗ →R by δl I(c,ξ,q,q˙,a)= K(c,a), (ξ,q,q˙,a) . (3.6) (cid:28) δξ (cid:29) Theorem 3.2 (Kelvin-Noether) Fixingc ∈C,letξ(t),q(t),q˙(t),a(t)satisfy the 0 Euler-Poincaré equations and define g(t) to be the solution of g˙(t) = ξ(t)g(t) and, say, g(0)=e. Let c(t)=g(t)−1c and I(t)=I(c(t),ξ(t),q(t),q˙(t),a(t)). Then 0 d δl I(t)= K(c(t),a(t)), ⋄a . (3.7) dt (cid:28) δa (cid:29) The proof of this theorem is relatively straightforward;werefer to Holm, Mars- den and Ratiu [1997]. We shall express the relation (3.7) explicitly for Maxwell- Vlasov plasmas at the end of section 7. 4 An action for the Maxwell-Vlasov equations Atypicalelement ofTR3 ∼=R3×R3 willbe denotedz=(x,v). We letπ :TR3 → s R3 and π : TR3 → R3 be the projections π (z) = x and π (z) = v onto the first v s v and second factors, respectively. Spaces of fields. We let Diff(TR3) denote the group of C∞- diffeomophisms from TR3 onto itself. An element ψ ∈ Diff(TR3) maps plasma particles having initial position and velocity (x ,v ) to their current position and velocity (x,v)= 0 0 ψ(x ,v ). This is the particle evolution map. We shall sometimes abbreviate 0 0 (x ,v ) = z , (x,v) = z, etc. The spatial components of ψ(x ,v ) are written as 0 0 0 0 0 x(x ,v )andthe velocitycomponents as v(x ,v ). We shallalsouse the following 0 0 0 0 notation: • V =C∞(R3,R) is the space of electric potentials Φ(x); LAUR-97-3326 Maxwell-Vlasov system in Euler-Poincaré form 9 • A is the space of magnetic potentials A(x); • F =C∞(TR3,R) is the space of plasma densities f(x,v); • F =C∞(TR3,R) is the space of plasma densities with compact support; 0 0 • D =C∞(R3,R) is a space of test functions, denoted ϕ(x). 0 0 Thetestfunctionsϕ(x)areusedtolocalizethevariationalprinciple. Thus,once one obtains Euler-Lagrangeequations depending on f and ϕ , if their validity can 0 0 be naturally extended for any f and ϕ , which will happen in our case, then we 0 0 shall consider those extended equations to be the Euler-Lagrange equations of the system. We will usually be interested in the Euler-Lagrange equations for f > 0 0 and ϕ =1. 0 The Lagrangian and the action. For each choice of the initial plasma distribution function f and the test function ϕ , we define the Lagrangian 0 0 1 L (ψ,ψ˙,Φ,Φ˙,A,A˙) = dx dv f (x ,v ) m|x˙(x ,v )|2 f0,ϕ0 Z 0 0 0 0 0 (cid:18)2 0 0 1 + m|x˙(x ,v )−v(x ,v )|2 (4.1) 0 0 0 0 2 + qx˙(x ,v )·A(x(x ,v ))−qΦ(x(x ,v )) 0 0 0 0 0 0 (cid:19) 1 ∂A + drϕ0(r) |∇rΦ+ (r)|2−|∇r×A(r)|2 . 2Z (cid:18) ∂t (cid:19) This Lagrangian is the natural generalization of that for an N-particle system, with terms corresponding to kinetic energy, electric and magnetic field energy, the usual magnetic coupling term with coupling constant q (the electric charge),and a constraintthatties the Eulerianfluid velocityv to x˙, the materialderivativeofthe Lagrangianparticletrajectory. Herexandv areLagrangianphasespacevariables, while A and Φ are Eulerian field variables. Thus, there should be no confusion createdby the slightabuse of notation in abbreviating ∂A/∂t and ∂Φ/∂t as Φ˙ and A˙, respectively, in the arguments of the Lagrangian. This Lagrangian is inspired by Low [1958]. However, we have added the term 1 m|x˙(x ,v )−v(x ,v )|2, 0 0 0 0 2 which allows v to be varied independently in the variational treatment. Consider the action S= dt L (ψ,ψ˙,Φ,Φ˙,A,A˙), Z f0,ϕ0 definedonthe familyofcurves(ψ(t),Φ(t),A(t)) satisfyingtheusualfixed-endpoint conditions (ψ(t ),Φ(t ),A(t )) = (ψ ,Φ ,A ), i = 1,2. One now applies the stan- i i i i i i dardtechniquesofthecalculusofvariations. Inparticular,integrationbypartscan LAUR-97-3326 Maxwell-Vlasov system in Euler-Poincaré form 10 be performed since f and ϕ have compact support. Moreover, once the Euler- 0 0 Lagrange equations have been obtained, their validity can be easily extended in a natural way for f >0 and ϕ =1. 0 0 Derivation of the equations. To write the equations of motion, we need some additional notation. Consider the evolution map ψ (x ,v ) = (x,v) so that ψ t 0 0 t relates the initial positions and velocities of fluid particles to their positions and velocities at time t. Let u be the corresponding vector field: ∂ ∂ u(x,v):=ψ˙ ◦ψ−1(x,v)=:x˙ +v˙ , t t ∂x ∂v so the components of u are (x˙,v˙). Recall that the transport of f as a scalar is 0 given by f(x,v,t)=f ◦ψ−1(x,v), which satisfies 0 t ∂f +u·∇zf =0, (4.2) ∂t where ∇z = (∇x,∇v) is the six dimensional gradient operator in (x,v) space. Let J be the Jacobian determinant of the mapping ψ ∈ Diff(TR3), that is, the ψ determinant of the Jacobian matrix ∂(x,v)/∂(x ,v ). 0 0 Define F(x,v,t) to be f , transported as a density: 0 F(x(x ,v ),v(x ,v ),t)J (x ,v )=f (x ,v ), 0 0 0 0 ψ 0 0 0 0 0 so that ∂F +∇z·(Fu)=0. (4.3) ∂t TakingvariationsinourLagrangian(4.1)andmakinguseoftheprecedingequation for F, we obtain the following equations (taking ϕ =1) 0 ∂A δx: mx¨+m(x¨−v˙)=−q∇xΦ−q +qx˙ ×(∇x×A), ∂t δv: x˙ −v=0, ∂A (4.4) δΦ: ∇x· ∇xΦ+ =−q dvF(x,v,t), (cid:18) ∂t (cid:19) Z ∂ ∂A δA: ∇x×(∇x×A)=− ∇xΦ+ +q dvvF(x,v,t). ∂t(cid:18) ∂t (cid:19) Z The second equation in (4.4) treats the Eulerian velocity v as a Lagrange multi- plier, and ties its value to the fluid velocity x˙, hence v˙ = x¨ as well. The first two variational equations in the set (4.4) provide the desired relation for particle acceleration and the last two equations are the Maxwell equations with source terms. Thus, Hamilton’s principle with Low’s action provides the equations for self-consistent particle motion in an electromagnetic field, as required, and the description is completed by substituting q ∂A v,− ∇xΦ+ −v×(∇x×A) (cid:18) m(cid:20) ∂t (cid:21)(cid:19)