Vlasov-type and Liouville-type equations, their

Izvestiya: Mathematics 81:3 505–541

Izvestiya RAN : Ser. Mat. 81:3 45–82

DOI: https://doi.org/10.1070/IM8444

Vlasov-type and Liouville-type equations, their microscopic, energetic and hydrodynamical consequences V. V. Vedenyapin, M. A. Negmatov, and N. N. Fimin Abstract. We give a derivation of the Vlasov–Maxwell and Vlasov– Poisson–Poisson equations from the Lagrangians of classical electrodynamics. The equations of electromagnetic hydrodynamics (EMHD) and electrostatics with gravitation are derived from them by means of a ‘hydrodynamical’ substitution. We obtain and compare the Lagrange identities for various types of Vlasov equations and EMHD equations. We discuss the advantages of writing the EMHD equations in Godunov’s double divergence form. We analyze stationary solutions of the Vlasov–Poisson–Poisson equation, which give rise to non-linear elliptic equations with various properties and various kinds of behaviour of the trajectories of particles as the mass passes through a critical value. We show that the classical equations can be derived from the Liouville equation by the Hamilton–Jacobi method and give an analogue of this procedure for the Vlasov equation as well as in the non-Hamiltonian case. Keywords: Liouville equation, Hamilton–Jacobi method, hydrodynamical substitution, Vlasov–Maxwell equation, Vlasov–Poisson–Poisson equation, Lagrange identity.

§ 1. Introduction The Vlasov–Maxwell equation is the main tool for describing plasma. It was introduced by Vlasov in 1938 [1] and is used increasingly often instead of the equations of magnetohydrodynamics for the numerical solution of complex plasma problems because of the growing resources of computers. The most important thing about this equation is probably its ‘superfundamental nature’, which is comparable in some sense with that of Liouville’s equation. Vlasov-type equations contain solutions of the N -body problem for every N . This makes them superfundamental, as observed by many authors. This was known to Vlasov himself, and Bogolyubov wrote in the introduction of [2]: “The Vlasov equation is the foundation of plasma This paper was written with the support of RFBR grant no. 16-02-00656 and RAS Presidium Programme no. 7 (N. N. Fimin) and with the financial support of the Ministry of Education and Science of the Russian Federation under the programme ‘5-100’ of raising the competitive ability of PFUR among leading scientific and educational centres in 2016–2020, as well as with support of the RAS DMS programme 1.3.1 for problems of computational mathematical physics (V. V. Vedenyapin). AMS 2010 Mathematics Subject Classification. 35Q83.

c 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

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V. V. Vedenyapin, M. A. Negmatov, and N. N. Fimin

physics. It seems very significant that the Vlasov equation has microscopic solutions corresponding to exact solutions of classical mechanics”. This property is used both for the derivation from Bogolyubov’s chain [3] and for approximation by these microscopic solutions [4], [5]. This approach cannot be used to derive the Vlasov–Maxwell equation, which certainly possesses such microscopic solutions, but the corresponding potential is the retarded Liénard–Wiechert potential [6], [7]. It is desirable to derive this equation from the exact Lagrangian aiming, first, to avoid microscopic solutions and, second, to understand the nature of approximation as well as the influence of relativistic expressions. Since the Vlasov–Maxwell equations are written differently in different textbooks, their exact derivation is highly desirable. To do this, we take the Lorentz Lagrangian of classical electrodynamics [6]–[8] as a starting point. For equations of magnetohydrodynamics (MHD) type, there are even more versions than for Vlasov-type equations. By an exact substitution, we derive some versions of equations of electromagnetic hydrodynamics type (or EMHD type, which is now a customary terminology for MHD-type equations in the presence of an electric field) from the Vlasov–Maxwell equations and automatically obtain the Lagrange identity for them. Here the famous Lagrange identity serves as a suitable test for comparing various forms of the equations. Such an identity was proved by Kozlov [9], [10] for Vlasov-type equations with two-particle interaction, and we study its form for various types of the Vlasov equation and MHD. Stationary solutions of the Vlasov–Poisson–Poisson equation are discussed. We derive this equation from the Lorentz Lagrangian of classical electrodynamics complemented by the Lagrangian of gravitational interaction. Reducing the stationary Vlasov equations by means of an energetic substitution gives rise to a system of non-linear elliptic equations in which nature of the motion of particles changes at the critical mass m2 = e2 /G, where G is the gravitational constant and e is the electron charge. We use the hydrodynamical substitution for the Vlasov equations (see [11], [12]) which reduces them to MHD-type equations [13]–[16]. Arnold [17], [18] proved a theorem on the structure of stationary solutions of the equations of an ideal incompressible fluid. This theorem is based on the presence of two commuting vector fields. Kozlov [19] extended this construction to compressible fluids. In what follows we investigate whether such constructions are possible for MHD and other consequences of the Vlasov–Poisson and Vlasov–Maxwell equations. It turns out that the hydrodynamical substitution is also consistent for the Liouville equation. This yields the shortest path to obtaining the Hamilton–Jacobi (HJ) equations from the corresponding Liouville equation. Moreover, this method naturally extends to the non-Hamiltonian case, and the resulting equations are considerably simpler than in the ordinary Hamiltonian case. This extension gives examples of systems of ordinary differential equations that are soluble in quadratures, including explicit examples of systems with finite or countable sets of limit cycles. Layout of the paper. § 2 indicates connections between the Vlasov equation and the N -body and continuum-body problems. § 3 is devoted to derivation of

Vlasov-type and Liouville-type equations

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the Vlasov–Maxwell equations from the Lagrangian of classical electrodynamics and a similar derivation of the Vlasov–Poisson–Poisson equation for charged gravitating particles. In § 4 we consider consequences of the Vlasov equations: the Lagrange identity, Godunov’s divergence form, and the existence of a critical mass. In § 5 we consider the results of hydrodynamical substitution into the Liouville and Vlasov equations, obtain the classical Hamilton–Jacobi equation, and generalize the Hamilton–Jacobi method to non-Hamiltonian systems. § 6 is devoted to studying the topological properties of stationary solutions of generalized hydrodynamical equations obtained from the Vlasov and Liouville equations. § 7 is a conclusion. In the appendix we prove a generalization of the Arnold–Kozlov lemma. § 2. Microscopic solutions of the Vlasov equation and the N -body problem The Vlasov equation describes long-range action (in contrast to the Boltzmann equation describing short-range action) in a system of interacting particles by introducing a self-consistent field. In its general form, it can be written as the following equation of shift along characteristics: ∂F ∂F ∂F + v, + f (F ), = 0, (1) ∂t ∂x ∂v where F (x, v, t) is the distribution function of particles with respect to the position x ∈ Rn and velocities v ∈ Rn at time t, and f (F ) is a certain vector depending on the distribution function (a ‘force’). The simplest type of dependence of the force f (F ) corresponds to pairwise interaction potentials K(x, y): Z f (F ) = −∇x K(x, y)F (y, v, t) dy dv. (2) Consider the following substitution into (1), (2) (the so-called ‘microscopic solutions’): N X F (x, v, t) = ρi δ(x − Xi (t))δ(v − Vi (t)), (3) i=1

where Xi (t) and Vi (t) are functions of time (the fields of coordinates and velocities corresponding to the ith particle), and ρi > 0 are the weights of particles. This substitution gives a solution if Xi , Vi satisfy the equations of motion of N bodies: ˙ i = Vi , X

˙i=− V

N X

∇1 K(Xi , Xj )ρj ,

j=1

where ∇1 is the gradient operator with respect to the first argument. Consider the ˙ 1 = V1 , V ˙ 1 = −ρ∇1 K(X1 , X1 ). case when N = 1. Then we obtain the system X The condition ∇1 K(X1 , X1 ) = 0 means that there is no self-action. Consider an analogous substitution of the following form, which yields the equations of motion of a continuum of bodies: Z F (x, v, t) = ρ(q)δ x − X(q, t) δ v − V(q, t) dq.

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Here the expression on the right-hand side is naturally defined as a distribution: this is a linear functional acting by the formula Z Z ρ(q)δ x − X(q, t) δ v − V(q, t) dq, ϕ(x, v) = ϕ X(q), V(q) ρ(q) dq. This is a far-reaching generalization of the ‘single layer’ [20] for parametrically given surfaces q → X(q, t), V(q, t) . The parameter q can vary along any domain in the space Rk of any dimension (k ∈ N) or along a manifold. Which equations must hold for the functions X(q, t) and V(q, t) to guarantee that the continual substitution into (1) yields an identity? These equations are of the following form (see [13], [21], [22]): Z ˙ ˙ X(q, t) = V(q, t), V(q, t) = − ∇1 K X(q, t), X(q0 , t) ρ(q0 ) dq0 . (4) They can naturally be called continuum-body equations. Let q = X(0), V(0) be the initial coordinates of the points. When t > 0, the quantities q are called Lagrangian coordinates and the equations (4) enable one to pass to these coordinates. Such substitutions were already known to Vlasov. They show the fundamental nature of the equation (1): it contains not only N -body motion for any N , but even continuum-body motion. For quadratic functions K(x − y), the Vlasov equation admits an exact solution in Lagrangian coordinates (see [13], [9]). It would be interesting to perform continuum-body generalizations of solutions for the Calogero–Moser potentials (K = |x − y|−2 or K = ℘(x − y), where ℘(. . .) is the Weierstrass function), as was done by Calogero and his successors [23]. Although the integral in (2) diverges in this case, it can be regularized in the standard way [20]. § 3. Derivation of the Vlasov–Maxwell and Vlasov–Poisson–Poisson equations The term ‘Vlasov equation’ is nowadays commonly used with certain prefixes: there are Vlasov–Poisson equations (for gravitation, electrons and plasmas), the Vlasov–Maxwell equation, the Vlasov–Einstein equation, and new prefixes continue to appear because of their constant shortage (as can be seen from what was listed above) for understandable reasons. The equation introduced in Vlasov’s original paper [1] is now referred to as the Vlasov–Maxwell equation. Its introduction was very timely, aimed at a description of plasma, and its power was shown in the example of small vibrations of plasma. The number of versions even of this equation is now large (and still growing) so that a simple derivation from the classical Lagrangian becomes highly desirable. Yes, the Lagrangian is known (see, for example, [6]–[8]) and the derivation is simple (basically obvious and even trivial) and is essentially reported in [6], although not for the Vlasov–Maxwell equation but for its hydrodynamical consequences. The derivation gives a firm basis for: 1) classifying equations with this name, 2) estimating their validity, 3) clarifying the nature of the approximations adopted by various authors. We also note that


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the Bogolyubov-type derivation in terms of chains [3] or using the approaches of Braun–Hepp [21], Maslov [4] and Neunzert [24] in terms of microscopic solutions is very difficult and, moreover, not applicable to the Vlasov–Maxwell equations. There are several obstacles in the way of deriving the Vlasov–Maxwell equation from the many-body equations of Hamiltonian dynamics. First, the equation for particles is obtained using retarding with the Liénard–Wiechert potentials. Second, one must prove convergence as in [4], [21], [24] for the Vlasov–Poisson equations. However, this procedure gives convergence not to the Maxwell equations but to their hyperbolic consequences, where the wave operator is already inverted in terms of the fundamental solution, which coincides with the Liénard–Wiechert potentials. This does not yield the Vlasov–Maxwell system since the information on electromagnetic fields remains encapsulated in the Liénard–Wiechert potentials, and the Maxwell equations certainly cannot be obtained. Therefore we proceed in a different way. We write down the variational principle with the classical Lagrangian, among whose consequences are the equations of motion of charged particles and the Maxwell equations. Then we do not assume that the electromagnetic field acting on the charged particles is determined by the averaged densities of charge and electric current, but simply write down the Liouville equation for the resulting motion of a particle in these fields. The variational principle is known and classical (this is particularly valuable). The connection between the Lagrangian (which is referred to as the Lorentz–Schwarzschild Lagrangian in [8]) and the Vlasov–Maxwell equation is also known. We mention [25], where such a derivation is sketched. But we adjust this and similar derivations to classifying the Vlasov equations. Note that this derivation opens the shortest path to the Vlasov–Maxwell equation and, therefore, gives rise to the classification and the essentially correct equations. We deduce the Vlasov–Maxwell equation from the Lorentz–Schwarzschild Lagrangian of classical electrodynamics and obtain the Vlasov–Poisson–Poisson equation from the same Lagrangian restricted to electrostatics and complemented by the Lagrangian of gravitational interaction. The Vlasov–Maxwell system of equations describes the motion of particles in their own electromagnetic field. It can be obtained from the Lagrangian of classical electrodynamics. We begin with the ordinary action of an electromagnetic field [6], the Vlasov–Maxwell action SV M or the Lorentz action SL : SL = SV M = −

X α

X eα − c α

XZ qµ

0

mα c

XZ q

0

T

sX

gµν X˙ αµ (q, t)X˙ αν (q, t) dt

µν

T

Aµ (Xα (q, t), t)X˙ αµ (q, t) dt −

X 1 Z Fµν F µν d4 x = Sp + Spf + Sf , 16πc µν (5)

where Sp , Sf and Spf stand for the action of particles, fields and particle-fields respectively. Here α is a type of particles with mass mα and charge eα , q enumerates the particles within a given type, Xαµ (q, t) (µ = 0, 1, 2, 3; q = 1, . . . , Nα ; α = 1, . . . , r) are

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the 4-coordinates of the qth particle of type α, Aµ (x) is the potential, Fµν = (∂ν Aµ − ∂µ Aν ) are the electromagnetic fields, and gµν is the Minkowski metric gµν = diag(1, −1, −1, −1), that is, a diagonal matrix. Variations will be calculated by the following special method [6]. We first obtain the motion of a particle in the field: δ(Sp + Spf ) = 0, and then the motion of a field with a given motion of the particles: δ(Spf +Sp ) = 0. We also pass to the distribution functions of particles in order to obtain the desired system of equations. µ 1) The variation of Sp + Spf in the coordinates qP Xα (q, t) yields the equation of motion of charges in a field. We rewrite gµν X˙ µ (q, t)X˙ ν (q, t) in the µν

Minkowski metric (here and in what follows we use the Greek indices µ, ν = 0, 1, 2, 3 and the Latin indices i, j = 1, 2, 3 ): r Z X XZ x˙ 2 (q, t) 2 1 − α 2 dt = Lp dt, Sp = − mα c c α q where Lp is the Lagrangian of the particles. P Here x˙ 2 = |v|2 = x˙ 21 + x˙ 22 + x˙ 23 = −(x˙ 1 x˙ 1 + x˙ 2 x˙ 2 + x˙ 3 x˙ 3 ) = − i x˙ i x˙ i is the square of three-dimensional velocity. We keep in mind that x0 = ct and factor c2 out of the root. Varying this expression (and omitting the index α), we get X 1 Z XZ d x˙ i δ x˙ i x˙ i 2 p p δSp = mc dt = −m δxi dt. 2 2 /c2 2 /c2 c dt 1 − v 1 − v qi qi Varying Spf (and again omitting α), we get Z X eX i δ cA0 (x(q, t), t) + Ai (x(q, t), t)x˙ (q, t) dt δSpf = c q i Z eX d ∂A0 i X ∂Ai i j = c i δx + x˙ δx − Ai δxi dt. j c qi ∂x ∂x dt j Hence, using the condition δ(Sp + Spf ) = 0, we obtain the equation of motion of a charged particle in the field: dpαi 1 ∂Ai ∂A0 1X j = eα − − Fij x˙ α , (6) − dt c ∂t ∂xi c j pαi =

∂Lp mα x˙ iα p = , ∂xiα 1 − x˙ 2α /c2

Fij =

∂Ai ∂Aj − . j ∂x ∂xi

2) We now cease to follow [6] and pass to distribution functions. The equation for the distribution function is obtained as a shift equation along the trajectories of the resulting dynamical system for the motion of charges in the field. It is clearly convenient to take distribution functions of momenta, not of velocities. We first express the velocities in terms of the momenta: mvi pi = p 1 − v2 /c2

⇒

|p|2 =

m2 |v|2 . 1 − |v|2 /c2


Putting 1 − |v|2 /c2 = γ −2 , we obtain r |p|2 γ = 1+ 2 2 m c

and

vi =

511

pi . γm

Hence we find the following equation for the distribution function fα (x, p, t): X ∂fα ∂fα ∂fα eα ∂Ai ∂A0 X Fij vαj + vα , + − −c i − = 0. ∂t ∂x c ∂t ∂x ∂pi j i

(7)

−1/2 Here vαj = (pj /mα ) 1 + p2 /(m2α c2 ) , and we have used the identity X divp Fji v j = 0. j

Note that pj has no subscript α while vαj does. In [1], [26] this equation was written for ions (i) and electrons (e) in the following form: ∂fe ∂fe 1 ∂fe + v, − e E + [v, B] , = 0, ∂t ∂x c ∂p ∂fi ∂fi 1 ∂fi + v, + ze E + [v, B] , = 0. ∂t ∂x c ∂p Here fi (x, p, t) is the distribution function for ions with respect to the positions and momenta at time t, fe (x, p, t) is the distribution function for electrons, ze is the ion charge, (−e) is the electron charge, and [v, B] is the vector product. The expression for v in terms of p does not appear in [26], but it is often taken to be the classical one, vαj = pj /mα . When the equations are written in this form, v must be different for electrons and ions. Hence one must distinguish between vi and ve instead of a single v and indicate their dependence on the momenta, vi (p) and ve (p) (otherwise the equations are not closed). 3) Equation for fields. We follow [6] but use distribution functions instead of densities. First rewrite Spf in terms of the distribution function by performing the following passages: Z Z X → dq → f (x, p, t) d3 x d3 p dt, q

whence Spf takes the form X Z Spf = eα Aµ (x, t)vαµ fα (x, p, t) d3 x d3 p dt. Then variation with respect to the potentials Aµ (x, t) yields X Z δSpf = eα δAµ (x, t)vαµ fα (x, p, t) d3 x d3 p dt, Z Z 1 1 δSf = δFµν F µν d3 x dt = δAµ ∂µ F µν d3 x dt. 16π 8π

512


Putting δ(Spf + Sf ) = 0, we obtain ∂µ F µν = −

Z 4π X eα vαµ fα (x, p, t) d3 p. c α

(8)

The system (7), (8) is known as the Vlasov–Maxwell system. Remark 1. To derive the Vlasov–Maxwell equations using microscopic solutions [4], [21] or using Bogolyubov’s scheme [3], one must begin with Hamiltonian systems with Liénard–Wiechert potentials (retarded potentials). For weak relativism, the corresponding Lagrangian is known as Darvin’s Lagrangian and is quite complicated; see [6], [7]. Moreover, the description of the plasma usually deals with boundary-value problems, which forces one to introduce Green’s functions instead of the pairwise interaction potential. Finally, in cases when these additional difficulties do not arise in the derivation using Bogolyubov’s chains (for example, this happens in the case of the gravitational Vlasov–Poisson equation), the classical derivation as in [6], [9] with passage to the distribution functions as in [13] seems to be more simple and direct, as will now be shown. We now derive the Vlasov–Poisson system of equations with gravitation in the non-relativistic case. The electrostatic Lagrangian is obtained from the general Lagrangian defined by the action (5), while the gravitational part is obtained by analogy with electrostatics. As a by-product, we shall verify the constants in the original Lagrangian. In the non-relativistic case, r x˙ 2 x˙ 2 1 − 2α ≈ 1 − α2 , c 2c whence we obtain a valid expression for the electrostatic Lagrangian: Sp = −

X

mα c2 +

α,q

X mα x˙ 2 (q, t) α

2

α,q

.

e The electrostatic particle-field action Spf is obtained from the general case by equating (the magnetic fields) Ai to zero: X XZ e Spf = − eα ϕ(X α (q, t), t) dt α

q

Z

Z

→

f (x, p, t) d x d p dt 3

3

transition dq → X Z →− eα ϕ(x, t)fα (x, p, t) d3 x d3 p dt. α g The gravitational particle-field action Spf is given by g Spf =−

X α

mα

XZ q

U (X α (q, t), t) dt → −

X α

Z mα

U (x, t)fα (x, p, t) d3 x d3 p dt.


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The field action in (5) can be written as Z 1 e Sf = (∇ϕ)2 − H 2 dx dt. 8π Discarding the second term, we obtain Z 1 e Sf = (∇ϕ)2 dx dt. 8π For gravitation, we obtain by analogy (with different signs because like charges repel while masses attract each other): Z 1 Sfg = − (∇U )2 dx dt. 8πG We obtain a common expression for the gravitational and electrostatic action in the form g e S = Sp + Spf + Spf + Sfe + Sfg X Z mα x˙ 2 (q, t) X XZ α = ϕ(X α (q, t), t) dt dt − eα 2 α,q α q Z Z X XZ 1 1 (∇ϕ)2 dx dt − (∇U )2 dx dt − U (X α (q, t), t) dt + mα 8π 8πG α q X Z mα x˙ 2 (q, t) X Z α = dt − eα ϕ(x, t)fα (x, p, t) dx dp dt 2 α,q α Z Z Z X 1 1 2 (∇ϕ) dx dt − (∇U )2 dx dt. − mα U (x, t)fα (x, p, t) dx dp dt + 8π 8πG α

Varying this expression along the same lines as above yields the following Vlasov– Poisson–Poisson system of equations for plasma with gravitation (varying the first expression with respect to particles, we obtain the equation of motions and then the Liouville equation, and varying the second expression with respect to ϕ and U , we obtain two Poisson equations: one for the electric potential ϕ and the other for the gravitational potential U ): ∂fα p ∂fα ∂U ∂ϕ ∂fα + , − mα + eα , = 0, (9) ∂t mα ∂x ∂x ∂x ∂p Z X ∆U = 4πG mα fα (x, p, t) dp, (10) α

∆ϕ = −4π

X

Z eα

fα (x, p, t) dp.

(11)

α

The names Vlasov–Poisson–Poisson indicate that the system (9)–(11) contains two distinct Poisson equations, (10) and (11).

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Stationary solutions of Vlasov’s equation are sought in the form of functions of integrals of motion [22], [27]–[31]. In the case (9)–(11) we suppose that the distribution functions fα are functions of energy (energetic substitution) of the following form: 2 p fα = gα + mα U + eα ϕ . 2mα Here gα is an arbitrary non-negative function of the total energy. Then equations (9) hold. As a result, we obtain a system of non-linear elliptic equations for the potentials U (x) and ϕ(x): ∆U = V (U, ϕ), where

V (U, ϕ) = 4πG

N X

Z mα

gα

α=1

p2 + mα U + eα ϕ dp, 2mα (12)

∆ϕ = Ψ(U, ϕ),

where

Ψ(U, ϕ) = −4π

N X

Z eα

gα

α=1

2

p + mα U + eα ϕ dp. 2mα (13)

We now study this system of equations. It is known that the well-posedness of the Dirichlet or Neumann problem for a non-linear elliptic equation ∆u = ψ(u), where ψ(u) is a real-valued function of u, depends on the sign of the derivative dψ/du: the problem is well posed if ψ 0 (u) > 0, as already mentioned in the book of Courant and Hilbert [32]. The sign of ψ 0 was verified in [33]–[35] for pure electrostatics, that is, in the case when there are no magnetic or gravitational fields (we have only (13), the equation (12) for the gravitational field being absent, ϕ ≡ 0). This calculation can be transferred without change to the case when ϕ 6= 0: p2 + mα U + eα ϕ |p|r−1 d|p| 2mα 2 2 X Z p p = −4πSr e2α gα0 + mα U + eα ϕ |p|r−2 (2mα ) d 2m 2m α α α Z ∞ X = −4πSr e2α (2mα ) |p|r−2 dgα

X ∂Ψ = −4πSr e2α ∂ϕ α

α

= −4πSr

X α

Z

gα0

0

∞ Z 2 r−2 eα (2mα ) |p| gα − 0

∞ r−2

gα d|p|

0

 2 p   g + m U + e ϕ α α α  X 2mα = 8πSr e2α mα × Z ∞   α (r − 2) gα |p|r−3 d|p|

if r = 2, if r > 2,

0

provided that gα decreases sufficiently rapidly as the argument tends to infinity, and r > 2. In the formula above, Sr is the area of the (r − 1)-dimensional sphere. When r = 1 one cannot integrate by parts because of divergence.


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We see purely mathematically that when we differentiate the right-hand side of (12) with respect to U , the sign will be the opposite and the boundary-value problem for gravitation (with electrostatic potential ϕ = 0) is ill posed. This is in accordance with physics since one never states boundary-value problems for gravitational potentials. Therefore boundary-value problems for (12), (13) are jointly ill posed and are not stated. However, in dimensions r > 2 of velocities or momenta, the boundary-value problem for (13) (with U = 0) turns out to be well posed and has a clear physical meaning. Moreover, our calculation shows the absence of non-trivial periodic solutions, that is, solutions which are not constant (ϕ ≡ const). On the other hand, periodic solutions do exist when r = 1. They are referred to as Bernstein– Green–Kruskal waves. Since only three-dimensional distributions with respect to momenta are physically realizable, these waves reflect the mathematical model of charged planes when the momentum space is one-dimensional.1 In the simplest case, when N = 1, the system (12), (13) takes the form Z 2 p ∆U = 4πGm g + mU + eϕ d3 p, (14) 2m Z 2 p ∆ϕ = −4πe g + mU + eϕ d3 p. 2m It can conveniently be rewritten in the form Z 2 p ∆(mU + eϕ) = (Gm2 − e2 ) g + mU + eϕ d3 p, 2m

(15)

∆(eU + Gmϕ) = 0. A similar calculation with differentiation shows that solubility conditions for the first equation depend essentially on the sign of the quantity B ≡ Gm2 − e2 ≷ 0. The boundary-value problem is well posed when Bp> 0, and otherwise there are global 2 solutions [13], p [33]–[35]. Thus the value m = e /G of the mass becomes critical. 2 When m > e /G, the gravitational forces exceed the electrostatic repelling forces, and vice versa. (If e is the electron charge, then this mass is m ≈ 1, 8 · 10−6 g.) 2 Taking the Maxwell distribution 2 |p| + mU + eϕ , fα = A exp −β 2m we obtain from (7) that ∆(mU + eϕ) = (Gm2 − e2 )B exp[−β(mU + eϕ)],

β > 0,

B > 0,

∆(eU + Gmϕ) = 0. 1 It

follows that BGK (Bernstein–Green–Kruskal) waves are absent in nature. √ critical mass is of the form m = me D0 , where D0 is the first Dirac large number, D0 = 4, 16 · 1042 , and me = 9, 1 · 10−28 g is the electron mass. 2 This

516


In the two-dimensional case, this equation has a large symmetry group (the conformal group), which enabled Liouville [36] to solve it. For our analysis of trajectories, it suffices to consider the one-dimensional case (in x, but not in p). Investigating this system, we obtain for B > 0 that the potential mU + eϕ is convex and the trajectories of the particles are bounded, otherwise the potential is concave and the trajectories of the particles diverge. For B = 0 the potentials mU , eϕ are linear functions. Remark 2. The origin of the Lagrangian corresponding to (5) was colourfully described by Pauli [8]: “Poincaré had already convinced himself of the invariance of Schwarzschild’s action integral with respect to the Lorentz group. Later, Born formulated the action principle very clearly, by writing it in four-dimensional notation. . . ”, with appropriate references. The equations (8) form the second pair of Maxwell’s equations, and the first pair follows from the equality Fµν = ∂ν Aµ − ∂µ Aν , which can equivalently be written in terms of differentiation of skew-symmetric tensors: Fµν dxµ ∧ dxν = 2d(Aµ dxµ ). The first pair of Maxwell’s equations is written in the form d(Fµν dxµ ∧ dxν ) = 0, that is, the exterior differential of this 2-form is equal to zero. Remark 3. It is desirable (and conceivably possible) to obtain the Vlasov– Yang–Mills equation in the same way, replacing the numbers by matrices in the four potentials Aµ . One can also derive the Vlasov–Einstein equations [22], [13]. Thus it is useful to classify Vlasov-type equations according to the Lagrangian, with further classification according to the dependence (relativistic or not) of the velocity on the momentum. It seems that every investigation of the Vlasov equation should begin with the Lagrangian in order to understand the nature of the approximations assumed. Remark 4. Stationary solutions of the Vlasov equation appeared earlier than the equation itself, and this also shows its fundamental nature. These solutions appeared as stationary solutions of self-consistent fields both in gravitation (the Emden–Fowler equation [27]) and for charged particles (von Laue, the Langmuir diode, Debye electrolytes [28], [13], [22]). In this sense, the fate of Vlasov’s equation is similar to that of Euler’s equation, whose stationary solutions were found earlier in the form of the Bernoulli integral using the same energetic substitutions, like twin brothers in fate. However, the Liouville and Vlasov equations are in some sense even more fundamental than Euler’s equation because Euler-type equations can be obtained from them by an exact substitution, as will be shown below. § 4. The Lagrange identity, EMHD equations and Godunov’s form We have shown that the full system of Vlasov–Maxwell equations is obtained by varying the electromagnetic actions (Lorentz actions) with a passage to distribution functions: X ∂fα ∂fα 1 ∂Ai ∂A0 X 1 ∂fα j + vα , + eα − − − F v = 0, (16) ij α i ∂t ∂x c ∂t ∂x c ∂pi i j


Z ∂F µν 4π X =− eα vαµ fα (x, p, t) dp, ∂xν c α Ei = −

Fµν =

1 ∂Ai ∂A0 , − c ∂t ∂xi

vα =

p , mα γα

517

∂Aµ ∂Aν − ∂xν ∂xµ

[vα , H]i = −

X

(µ, ν = 1, 2, 3, 4),

Fij vαj ,

j

s 1+

γα =

p2 . m2α c2

The first equation takes the form of a shift along the trajectories in the field of electric forces and the Lorentz magnetic force, and the second is Maxwell’s system of equations. By fixing the Lorentz gauge ∂Aν /∂xν = 0 in accordance with [6], one can transform Maxwell’s system of equations into the following form, which yields the Liénard–Wiechert potentials by inversion of the left-hand side: X 1 ∂2ϕ − ∆ϕ = 4πeα 2 2 c ∂t α

Z

X 1 ∂ 2 Ai − ∆Ai = 4πeα 2 2 c ∂t α

Z

fα (x, p, t) dp, vi fα (x, p, t) dp.

The Lagrange identity expresses the second derivative with respect to time of the moment of inertia in terms of the kinetic and potential energies. Following [9], we now show that the Lagrange identity can also be extended to the Vlasov–Maxwell system of equations. We define the moment of inertia of particles with respect to the origin: XZ I(t) = fα (x, p, t)x2 d3 p d3 x α

and define the functionals 1X T (t) = 2 α XZ Π=

Z

fα (x, p, t)vα2 d3 p d3 x,

1 eα x, E + [vα , H] fα d3 p d3 x γ m c α α α Z X eα − (p, x)(p, E)fα d3 p d3 x. 3 3 c2 γ m α α α

The Lagrange identity holds in the form I¨ = 4T − 2Π.

(17)

In the first summand on the right-hand side T is the kinetic energy, and Π is related to forces of an interesting nature, which will be studied in what follows.

518


Proof. Using (16), we write the time derivative of the moment of inertia I as ∂fα ∂fα 1 ∂fα + vα , + eα E + [vα , H], = 0, ∂t ∂x c ∂p X Z ∂fα XZ ∂fα I˙ = x2 dx dp = − vα , x2 dp dx ∂t ∂x α α XZ 1 ∂f α 2 , E + [vα , H] dp dx. x eα − ∂p c α We must eliminate the derivatives of the distribution function. By the Gauss– Ostrogradskii formula, the first integral is equal to XZ 2 (x, vα )fα d3 p d3 x, α

and the second vanishes since E is independent of p and divp [vα , H] = 0. In a similar vein, the second time derivative of I is equal to XZ ∂fα ¨ , vα d3 p d3 x I = −2 (vα , x) ∂x α XZ ∂fα 3 3 1 d p d x. −2 (vα , x)eα E + [vα , H] c ∂p α Integrating by parts, we rewrite the first integral in this expression as XZ 2 vα , vα fα d3 p d3 x = 4T. α

To transform the second integral, we calculate ∂vi , ∂pj

where

s

pi vi = , mα γα

δij pi ∂vi = − pj 3 3 2 ∂pj γα mα γα mα c

γα = ⇒

X i,j

1+ Fij

p2 , m2α c2

∂vj =0 ∂pi

as the convolution of symmetric and skew-symmetric tensors. Then we get X Z ∂vαj 1 ∂Ai ∂A0 1 j 3 3 −2 xj eα − − F v − ij α fα d p d x i ∂p c ∂t ∂x c i αij X Z δij p i pj 1 ∂Ai ∂A0 1 j 3 3 = −2 − 3 3 2 xj eα − − − F v ij α fα d p d x i γ m γ m c c ∂t ∂x c α α α α αij X Z eα 1 = −2 x, E + [vα , H] fα d3 p d3 x γα mα c α Z X pi p j +2 x e E f d3 p d3 x = −2Π 3 3 c2 j α i α γ m α α αij


since

P

i,j

519

pi Fij vαi = 0. The second summand can be transformed into the form XZ eα 2 (p, x)(p, E)fα d3 p d3 x. 3 m3 c2 γ α α α

It is of importance to study the stability of solutions of the Vlasov–Maxwell equations (for example, in facilities like the ITER project or ‘Galatea’ [37], [38], [14]). The Lagrange identity (17) may be useful when its right-hand side happens to have a definite sign. The derivation shows that the second term in the functional Π in (17) is related to relativism. However, we use the Lagrange identities by comparing their different forms for the equations of magnetohydrodynamics (MHD): it is interesting to look not only at the MHD equations, but also at their consequences, the Lagrange identities. There are many forms of MHD-type equations: they are understood by various authors as completely different systems of equations (see, for example, [37], [16], [39]). In the derivation of MHD-type equations, it is customary [40] to complement the Vlasov–Maxwell equation with a collision integral and obtain hydrodynamictype equations (with non-zero temperature) using the Maxwell–Chapman–Enskog procedure. This is declared to be the derivation of MHD, although it yields only approximate equations. The derivation of equations with zero temperature by substituting a deltafunction is certainly known (see, for example, [2], [41], and also [4] for a connection with Lagrangian submanifolds). We adjust it for the purpose of classifying, sharpening and clarifying the relevant approximations. The resulting equations (19), (20) are exact consequences of the Vlasov–Maxwell equations, and the Lagrange identity (21) can be obtained from them. The case of Vlasov–Maxwell equations with non-relativistic dependence of the velocity on the momentum yields a very similar MHD system. Therefore we do not write it down, but give the Lagrange identities in the kinetic case (22) and in the MHD case (23). Our second aim is to compare the derivations of MHD equations with zero and non-zero temperatures (see equations (24) below, which are derived by a moment method instead of exact substitution). 4.1. The zero-temperature case. We first consider the zero-temperature case and obtain the corresponding equations as exact consequences of the Vlasov– Maxwell system of equations by means of the so-called hydrodynamic substitution (see § 5 for details): fα (x, p, t) = nα (x, t)δ(p − Pα (x, t)).

(18)

This is the limit of the Maxwell distribution as the temperature Tα → 0: nα (x, t) (p − Pα )2 0 fα (x, p, t) = exp − −−−−→ nα (x, t)δ(p − Pα (x, t)). Tα →0 2kB Tα mα (2kB πTα mα )3/2 Here kB is the Boltzmann constant and Tα is the temperature of the α-component of the mixture. This yields a multifluid form (the term ‘multifluid’ means that each component has its own velocity) of the equations of electromagnetic hydrodynamics

520


(or EMHD equations, which is now a customary term for the MHD equations in the presence of an electric field). These equations are of the form ∂nα + div vα (Pα )nα = 0, ∂t 3 1 ∂Pα X i ∂Pα + − eα E + [vα (Pα ), H] = 0, vα (Pα ) ∂t ∂xi c i=1 where vα = Pα /(γα mα ), γα = Maxwell equations

p 1 + P2α /(m2α c2 ) . They are complemented by the ∂H = 0, ∇H = 0, ∂t 1 ∂E 4π X =− eα nα vα (Pα ). ∇×H− c ∂t c α

∇×E− ∇E = 4π

X

eα nα ,

α

(19)

(20)

We stress that these equations are exact consequences of the Vlasov–Maxwell system. Therefore the Lagrange identity for the system (19), (20) is obtained by substituting (18) into (17): I¨2 = 4T2 − 2Π2 , (21) where I2 (t) =

XZ

nα (x, t)x2 d3 x,

α

T2 (t) =

1X 2 α XZ

Z

nα (x, t)vα2 (Pα (x, t)) d3 x,

eα 1 Π2 = nα (x, t) x, E + [vα (Pα ), H] d3 x γα mα c α Z X eα − n (x, t)(Pα , x)(Pα , E) d3 x. 3 3 c2 a γ m α α α The Vlasov–Maxwell system for non-relativistic particles is obtained from the action (5) with the first summand replaced by X mα X Z Sp = x˙ 2α (q, t) dt. 2 α q We obtain a Vlasov–Maxwell system in the form (16). It must be closed by the velocity relation vα (p) = p/mα . The corresponding EMHD equations are obtained by the same substitution (18) and differ from (19) only in the expressions of the velocities in terms of momenta: vα = Pα (x, t)/mα . The Lagrange identity for the system (16) with vα (p) = p/mα holds in the form I¨3 = 4T3 − 2Π3 ,

(22)


521

where I3 (t) =

XZ

fα (x, p, t)x2 d3 p d3 x,

α Z 1X 2 3 T3 (t) = fα (x, p, t)vα d p d3 x, 2 α X Z eα 1 x, E + [vα , H] fα d3 p d3 x. Π3 = mα c α

The Lagrange identity for the system (19) with vα (p) = p/mα holds in the form I¨4 = 4T4 − 2Π4 ,

(23)

where I4 (t) =

XZ

nα (x, t)x2 d3 x,

α

Z 1X 2 nα (x, t)vα (Pα (x, t)) d3 x, 2 α X Z nα (x, t) 1 Π4 = eα x, E + [vα (Pα ), H] d3 x. m c α α

T4 (t) =

We give a generalization of the Lagrange identity to the case when x2 is replaced by an arbitrary function ϕ(x): XZ I(t) = fα ϕ(x) d3 p d3 x. α

By the Vlasov–Maxwell system (16) we have the following expression for the second time derivative of this functional: XZ X Z eα ∂ϕ ∂ϕ 1 3 3 ¨ I= vαi vαj fα d p d x + , E + [vα , H] fα d3 p d3 x ∂x ∂x γ m ∂x c i j α α α α XZ eα ∂ϕ − (p, E)fα d3 p d3 x. p, 3 m3 c2 γ ∂x α α α 4.2. The non-zero-temperature case. While in the zero-temperature case we substituted the Dirac delta function (18) into the Vlasov–Maxwell equation, in the non-zero-temperature case we substitute the Maxwell distribution fα0 . The two-fluid and one-fluid MHD (or EMHD) with non-zero temperature have many modifications [37], [39], [16], [40]. For example, the system in [14], [37] is of the form ∂ρα + div(ρα Vα ) = 0, ρα = mα nα , where α = {i, e}, ∂t dVα 1 dsα d ∂ ρα = −∇Pα + eα nα E + [Vα , H] , =0 ≡ + Vα ∇ , dt c dt dt ∂t 1 Pα = Pα (ρα , Tα ), α = α (ρα , Tα ), Tα dsα = dα + Pα d , ρα

522


where Pα is the pressure and α is the internal energy, rot H =

4π j, c

X

1 ∂H + rot E = 0, c ∂t

eα nα = 0,

α

div H = 0,

j=

X

eα nα Vα .

α

Hydrodynamic-type equations are usually obtained from the system of kinetic equations by successively integrating and introducing the moments Z Z 1 pk fα (x, p, t) d3 p, nα = fα (x, p, t) d3 p, Pαk = nα Z 1 Dα = (p − Pα )2 fα (x, p, t) d3 p. nα Here nα (x, t) is the density of the number of particles of the αth type, Pαk (x, t) is the expectation of the momentum or the average momentum (the kth component, k = x, y, z) and Dα is the variance with respect to the momenta of all particles of each type (Dα is proportional to the energy of the chaotic motion). These momenta of the function fα satisfy the following system of equations: ∂nα ∂ + (nα vα ) = 0, ∂t ∂x

(24)

X ∂ ∂ 1 (nα Pα` ) + (nα Pαk Pα` + σαk` ) − nα eα E + [vα , H] = 0, ∂t ∂xk c ` Z k σαk` = (pk − Pαk )(p` − Pα` )fα (x, p, t) d3 p (the stress tensor), X ∂ ∂ (nα Dα ) + qαk = 0, ∂t ∂xk k

Z qαk =

pk (p − Pα )2 fα (x, p, t) d3 p (the heat flow vector). m

The first equation of this system is the continuity equation, the second is the equation of motion, and the third describes the change of energy of the chaotic motion. This is an exact system of equations, but it is not closed. To close it, one performs Maxwellization either from pair collisions (by adding the collision integral) or from interaction with the medium (by adding a linear collision integral). This means that the higher-order moments are determined by the lower-order ones by means of the Maxwell distribution nα (x, t) (p − Pα )2 fα0 (x, p, t) = exp − . (25) 2kB Tα mα (2kB πTα mα )3/2 It turns out that σαk` = δk` kB nα Tα ,

Dα = 3kB Tα .


523

We now write these equations more briefly in Godunov’s form. To do this, we define the Godunov function Z α α G (βµ ) = fα0 (βµα ) d3 p, µ = 0, 1, 2, 3, 4, (26) fα0 (βµα ) = exp β0α + β1α p1 + β2α p2 + β3α p3 + β4α p2 . Here βµα are new variables (Godunov’s variables), replacing the density nα , the temperature Tα and the average momentum Pα in the Maxwell distribution fα0 . Comparing ~ 2 β 2 + β22 + β32 β exp β4 p + , fα0 (βµα ) = exp β0 − 1 4β4 2β4

~ = (β1 , β2 , β3 ), β

with the expression (25), we obtain a connection between βµ and the ordinary thermodynamical variables: β4α = −

1 , 2kB Tα mα

β1α =

β0α = ln nα −

Pα1 , kB Tα mα

β2α =

Pα2 , kB Tα mα

β3α =

Pα3 , kTα mα

3 Pα2 ln(2πkB Tα mα ) − . 2 2kB Tα mα

We define a vector Kµα = 0, F1α nα , F2α nα , F3α nα , −2Fiα Gα βi ,

Fα = eα E + [vα , H] ,

i = 1, 2, 3.

Then the equations with non-zero temperature in the zero approximation of the Chapmen—Enskog method [22], [26] can be written in the following Godunov form [42], [43]: ∂Gα βµ ∂t

+

∂Gα βµ βi ∂xi

+ Kµα = 0,

where

Gα βµ ≡

∂Gα . ∂βµ

(27)

The generalized Lagrange identity takes the following form in this case: XZ I(t) = fα0 (βµα )φ(x) d3 p d3 x, α

I¨ =

XZ α

∂φ Gα Gα d 3 x − ∂xi ∂xj βi βj

Z

∂φ α α 3 G Fi d x. ∂xi

The authors of some courses (see, for example, [44]) use the equations for the common temperature and common momentum because the momentum and kinetic energy of each component are not collision-invariant for cross-section collision integrals. Then, instead of the 3n equations in the system (24) or (27), there are n + 3 + 1 equations.

524


Godunov’s form is useful for several reasons. All thermodynamical relations can be expressed in terms of the Gibbs potential, and the function G is a generalization of this potential to the non-equilibrium case. Here we use G to write macroscopic equations of motion, and it would be interesting to compare Gibbs’ thermodynamics with Godunov’s double divergence thermodynamics. A second reason is that the equations (27) in this form are automatically hyperbolic. And finally, this is the form of equations to which Godunov’s well-known numerical method is applicable. Godunov mentioned the analogy between (27) and the Gibbs non-equilibrium method in a conversation with one of the authors (V. V.) of this survey. Since no reaction followed, on the next occasion Godunov extracted relevant places from the survey [43]. Then the corresponding conclusion was made both for the Maxwell distribution and the Fermi–Dirac and Bose–Einstein distributions [13], [22]. In the case of the Maxwell distribution, it was observed there that Godunov’s form contains even triple (instead of double) divergence, and the equations are expressed in terms of only one potential G. This makes it even closer to the Gibbs method. Another advantage of Godunov’s form of the equations derived by the kinetic approach [43], [13], [22] is the natural character of the resulting variables. Indeed, the variables β in (26) result from writing the Maxwell distribution in two different forms (25) and (27) and comparing them. When Godunov derived such a form for other equations outside the kinetic approach (see, for example, [45] in the case of elasticity theory), the choice of variables was surprisingly arbitrary. Such queries also appeared during Godunov’s talks. Therefore a derivation of these equations by the kinetic approach would be very useful. Thus our attempt to classify the Vlasov and MHD equation gave us the Vlasov– Maxwell equations with relativistic or non-relativistic dependence of the velocity. The same applies to the MHD equations, the system of MHD equations with zero or non-zero temperature (the first is obtained by an exact substitution of the Dirac delta function into the Vlasov equation, and the second by the moment method and closure by means of the Maxwell distribution), one-fluid and twofluid MHD, and MHD equations in Godunov’s variables. § 5. Hydrodynamical substitution for the Liouville and Vlasov equations and a generalization of the Hamilton–Jacobi method In this section we consider applications of the hydrodynamical substitution in more detail. The hydrodynamical substitution first appeared in [41] and was used in the theory of Vlasov’s equation (see, for example, [1], [11], [13], [12]). Recently, it was also applied to Liouville’s equation and Hamiltonian mechanics [46], [47]. The simplest derivation of the Hamilton–Jacobi (HJ) equation was sketched in the works of Arzhanykh [48], Dolmatov [49] and Kozlov [50]–[52], and the hydrodynamical substitution connected this derivation with Liouville’s equation in a simple way [46], [47]. We first give this direct derivation of the Hamilton–Jacobi equation and then consider the possibility of hydrodynamical substitution in the nonHamiltonian situation.


525

The Hamiltonian canonical equations dx ∂H(x, p) = , dt ∂p

dp ∂H(x, p) =− dt ∂x

(28)

give rise to Liouville’s equation on the distribution function f (x, p, t) with respect to the momenta p ∈ Rn and spatial coordinates x ∈ Rn : n ∂f X ∂f ∂H ∂H ∂f + = 0. (29) − ∂t ∂xi ∂pi ∂xi ∂pi i=1 We use the substitution f (x, p, t) = ρ(x, t)δ(p − P(x, t)), where δ( · ) is the Dirac function, and ρ(x, t) and P(x, t) stand for the density of particles and the momentum of particles at a point x and time t. The equations for ρ and P can be deduced directly: ∂f ∂ρ ∂P = δ(p − P) − ρ ∇δ, , ∂t ∂t ∂t ∂f ∂ρ ∂P ∂f ∂δ = δ − ρ ∇δ, , =ρ . ∂xi ∂xi ∂xi ∂pi ∂pi Collecting the factors at the δ-function, we obtain n ∂ρ ∂H ∂ρ X ∂H + + ρ div = 0. ∂t i=1 ∂pi pi =Pi ∂xi ∂pi pi =Pi Here we must substitute p = P in the second summand after differentiation. For brevity we write V(x, P, t) ≡ (∂H/∂p) p=P . This yields an equation coinciding with the classical continuity equation: n

∂ρ X ∂ + (ρVi ) = 0, ∂t i=1 ∂xi so that the introduced quantity V(x, P, t) can be regarded as a ‘generalized velocity’ in accordance with its physical meaning. Equating the coefficients of the derivatives of the delta-function, we obtain the second equation, n ∂P X ∂P + Vi = F, ∂t ∂xi i=1 where F(x, t) = − ∂H(x, p)/∂x p=P(x,t) are components of the generalized force. Thus we arrive at the following system of equations of generalized hydrodynamics (it can be called the reduced Euler system, or briefly the RES) being an exact consequence of the Liouville equation: ∂ρ + div(ρV) = 0, ∂t

n

∂P X ∂P + Vi = F, ∂t ∂xi i=1

(30)

where the generalized velocity V and the generalized force F were defined above.

526


There is an alternative method for deriving (30) by means of equations for momenta [46], [47]. Then the second equation is obtained in a modified form: n ∂(ρP) X ∂ + (ρPVi ) = ρF. ∂t ∂xi i=1

(31)

The last equation differs from the second equation in the RES (30), but can be reduced to it using the continuity equation. It was shown in [50]–[52] how to obtain the HJ equations from (30). First write the second equation (30) in Gromeka–Lamb form [50], [53], [54]: n

n

n

∂P X ∂P X ∂Pi ∂H X ∂H ∂Pi + Vi =− − . − (x, P) Vi ∂t ∂x ∂x ∂x ∂p ∂x i i i=1 i=1 i=1

(32)

The expression (∂Pi /∂xj − ∂Pj /∂xi ) dxi ∧ dxj is the differential of Pi dxi , and the Gromeka form shows that the equation for the curl Rij = ∂Pi / ∂xj − ∂Pj / ∂xi has a solution R ≡ 0: ∂R + rot[R, V] = 0. (33) ∂t We take into account that the second and third terms on the left-hand side of (32) are components of the vector product [R, V], and the right-hand side is the gradient of a function: n dH(x, P) ∂H(x, P) X ∂H(x, P) ∂Pi = + dxj ∂xj ∂pi ∂xj i=1 (the composite rot ◦ grad is identically equal to 0). Conversely, if the curl of P is equal to zero, then P is the gradient of some function (in a simply connected domain): Pi = ∂S/∂xi . The equation (33) shows that this property is preserved for all time. We find from (32) that ∂S ∂ ∂S + H x, = 0. ∂x ∂t ∂x Hence ∂S/∂t + H(x, ∂S/∂x) Z = f (t). Again following [50]–[52] and making the e change of variable S = S + f (t) dt, we obtain a pure HJ equation, ∂S ∂S + H x, = 0. ∂t ∂x

(34)

Thus we obtain a direct derivation of the Hamilton–Jacobi equation from Liouville’s equation by means of the hydrodynamical substitution. Before [46], [47], the hydrodynamical substitution was apparently not used, and we transferred it from Vlasov’s equation. We now use the analogy between the Liouville and Vlasov equations again in order to obtain an analogue of the Hamilton–Jacobi equation for the Vlasov equation and generalize the Arnold–Kozlov argument to the Vlasov– Poisson equation.


527

The following hydrodynamical substitution is known [15], [40], [41] to be consistent: fα (x, p, t) = nα (x, t)δ p − Pα (x, t) . It yields exact solutions of the Vlasov–Poisson–Poisson equations (9)–(11) if nα and Pα satisfy the following system of hydrodynamical equations: 1 ∂nα + div(nα Pα ) = 0, ∂t mα ∂Pα X 1 ∂ϕ ∂Pα ∂U + + eα , Pαi = −mα ∂t mα ∂xi ∂x ∂x i X X ∆U = 4πG mα nα (x, t), ∆ϕ = −4π eα nα (x, t). α

(35)

α

If we rewrite the second equation in Gromeka form and alternate the expression ∂Pαi / ∂xj , 1 ∂Pα X 1 ∂Pαi ∂Pα + = −mα ∇U + eα ∇ϕ − Pαi − ∇(P2α ), ∂t m ∂x ∂x 2m α i α i

(36)

then as expected, the right-hand side will be equal to the full gradient of the α ‘Bernoulli integral’. Write Rα for the matrix Rij (the curl of the momentum Pα ): α Rij =

∂Pαi ∂Pαj − . ∂xj ∂xi

Taking the curl of both sides of (36), we obtain the equation ∂Rα + rot (Rα × Pα ) = 0. ∂t In the stationary case we have rot (Rα × Pα ) = 0. As a corollary of this equation, it was shown by Arnold [17], [18] and Kozlov [19] that if the continuity equation div(nα Pα ) = 0 holds, then the vector fields Rα /nα and Pα commute: α R , Pα = 0. nα In our case, the continuity equation is the first equation in (35). Following [17]–[19], we conclude from the relations obtained for electrostatic and gravitational plasmatic configurations that the surfaces spanned by Pα (x, t) and Rα / nα (x, t) are topologically either tori or cylinders, or they are planes. In view of the importance of the Arnold–Kozlov lemma, we verify it in another way in the appendix.

528


Put Pα = ∂S/∂x in the equations (35). We obtain the system of equations ∂nα X 1 ∂ ∂Sα + nα = 0, ∂t mα ∂xi ∂xi i

(37)

∂S 1 + (∇Sα , ∇Sα ) + mα U (x) − eα ϕ(x) = 0, ∂t 2mα X X ∆U = 4πG mα nα , ∆ϕ = −4π eα nα . α

α

Thus, for the Vlasov–Poisson–Poisson system of equations one can indeed obtain a RES containing equations of Hamilton–Jacobi type. However, this is impossible for the Vlasov–Maxwell system since the Lorentz force is non-gradient. We have obtained the shortest and most direct derivation of the Hamilton–Jacobi equations by means of a hydrodynamical substitution, and now we generalize it to the non-Hamiltonian case. The second equation in (30) can be regarded as the equation of motion of an n-dimensional surface in a 2n-dimensional space, and we extend this equation to arbitrary dimensions. Consider a general (non-Hamiltonian) autonomous system of first-order ordinary differential equations in an n-dimensional space: dx = v(x), dt

x ∈ Rn ,

v(x) ∈ Rn ,

t ∈ R1 .

(38)

We again introduce the distribution function f (x, t) of the points in the ndimensional phase space at time t. Its evolution is described by the generalized Liouville equation ∂f + divn (vf ) = 0. ∂t

(39)

To describe the motion of an m-dimensional surface (1 6 m 6 n − 1), we write the vector x as an ordered pair (q, p)T , q ∈ Rm , p ∈ Rn−m (in other words, we split the phase space into a Cartesian sum of phase sets determined by the new n−m variables: Rxn = Rm ). Rewrite the system (38) in these variables: q ⊕ Rp dq = w(q, p), dt

dp = g(q, p), dt

(40)

where w(q, p) (resp. g(q, p)) are the first m (resp. last n − m) components of the vector-valued function v(q, p) in (38), that is, (w, g)T = v. We seek solutions of the Liouville equation (39) using the hydrodynamical sub stitution f (q, p, t) = ρ(q, t)δ p − P(q, t) . Here p = P(q, t) is the equation of an m-dimensional surface at time t and ρ(q, t) is the density of the points on this surface. Substituting this representation of f (q, p, t) into (39), we obtain


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the continuity equation and the equation of motion of the surface (see [46], [47]): m

∂ρ X ∂ + (ρWk ) = 0, ∂t ∂qk

(41)

k=1

m

∂P X ∂P + = G. Wk ∂t ∂qk

(42)

k=1

Here W(q, t) = w q, P(q, t) , G(q, t) = g q, P(q, t) . Thus we obtain the equation (42) of motion of m-dimensional surfaces in an n-dimensional space by means of an arbitrary system of n non-linear equations (38) in the Euler coordinates.3 In the simplest case m = 1, the stationary system (42) describes the trajectories of motion. When P does not depend on time, this system is obtained by dividing the last n − 1 equations in (38) by the first. This procedure for finding stationary trajectories is well known and shows the consistency of our calculations. Analyzing the HJ method in order to extend it to non-Hamiltonian situation, we conclude that there are three constituents of the method: 1) a change of variables, 2) a family of exactly soluble equations and 3) the HJ equation. In the classical Hamiltonian case, examples of the changes of variables are the spherical, elliptic and parabolic coordinate systems and canonical transformations (see, for example, [56]), and the exactly soluble systems are those integrable in the sense of Liouville. In the general non-Hamiltonian case, linear equations or equations with separated variables can be taken as exactly soluble ones, and canonical changes can be replaced by arbitrary ones. But what is the analogue of the HJ equation for the function S in the general case? Consider the following example. Example. Let H = p2 /(2M ) + U (x) be a Hamiltonian of one-dimensional motion. We put v1 = ∂H/ ∂p = p/M , v2 = −∂H/ ∂x = −U 0 (x). The classical HJ method (in the stationary case) yields the equation 2 1 dS + U 0 (x) = E. 2M dx At the same time, the stationary one-dimensional (m = 1) equation (42) takes the form 2 1 dP 1 dU 1 dP =− , whence + U (x) = E1 , M dx P dx 2M dx so that the role of the classical eikonal function S(x) is now played by the momentum variable P (x). We shall see that the ‘generalized HJ method’ is applicable to Hamiltonian and non-Hamiltonian systems, and the equations of the generalized HJ method obtained from (42) may be even simpler than the classical ones. 3 What can be said about the class of equations of the form (42) resulting from all the equations (38)? This class of quasi-linear equations coincides with the class of systems with identical principal part (see [32], [55]).

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Thus one can conclude that the equations (42) or their stationary consequences are analogues of the HJ equation in the non-Hamiltonian case. Here we shall use the simplest consequence: the equation (42) in the stationary case for m = 1 is the result of dividing the first n − 1 equations of the system (38) by the last. These n − 1 equations are well known since they describe the trajectory when time is eliminated. Consider a system (38) consisting of two equations x˙ = v1 (x, y),

y˙ = v2 (x, y).

(43)

The equation (42) for this system reduces (in the stationary case) to the well-known consequence for finding the trajectories: v1 (x, y) dy = , dx v2 (x, y) and this is the only candidate for the role of the HJ equation in the non-Hamiltonian situation in the two-dimensional case. Let α(x, y), β(x, y) be the new variables (that is, the ‘new’ variables are singlevalued functions of the ‘old’ ones). Then we have dα = αx0 v1 + αy0 v2 ≡ Ω1 (α, β), dt and, after division,

dβ = βx0 v1 + βy0 v2 ≡ Ω2 (α, β) dt

αx0 + αy0 v2 /v1 dα Ω1 = 0 = . dβ βx + βy0 v2 /v1 Ω2

(44)

(45)

Wishing to separate the variables α and β, we equate the right-hand side Ω1 / Ω2 of this equation to an expression ω1 (α)ω2 (β) (where ω1 (. . .) and ω2 (. . .) are arbitrary functions of their arguments) and obtain a linear equation for v2 /v1 . Solving it, we have α0 − (Ω1 /Ω2 )βx0 v2 = x0 . v1 −αy + (Ω1 /Ω2 )βy0 Thus we obtain a general form for systems separated in the new variables α, β: Ω1 0 Ω1 0 βy − αy0 , v2 = χ(x, y) αx0 − βx , (46) v1 = χ(x, y) Ω2 Ω2 where χ(x, y) 6≡ 0 is an arbitrary function. The structure of the equations (46) is rather close to that of the Hamilton canonical equations, which can be obtained (in the simplest case χ = 1) by passing to the limit as ω1 ω2 → 0 or ω1 ω2 → ∞: x˙ = −αy0 ,

y˙ = αx0

or

x˙ = βy0 ,

y˙ = −βx0 ,

(47)

where the role of the Hamilton function is played by the ‘new’ variables α or β respectively.


Example. In the polar coordinate system (CS), r = and the equations (44) take the form

531

p x2 + y 2 , ϕ = arctan(y/x),

dr = v1 (r cos ϕ, r sin ϕ) cos ϕ + v2 (r cos ϕ, r sin ϕ) sin ϕ, dt dϕ 1 = −v1 (r cos ϕ, r sin ϕ) sin ϕ + v2 (r cos ϕ, r sin ϕ) cos ϕ , dt r whence

tan ϕ − ω1 (r)ω2 (ϕ) v1 (r cos ϕ, r sin ϕ) = . v2 (r cos ϕ, r sin ϕ) −1 − tan ϕω1 (r)ω2 (ϕ)

This separation condition for the polar coordinate system can be rewritten in terms of the original Cartesian coordinates: y y v1 (x, y) = χ(x, y) − ω1 (x2 + y 2 )ω2 , x x y y v2 (x, y) = χ(x, y) −1 − ω1 (x2 + y 2 )ω2 . x x We now consider some corollaries of the transformation formula obtained above. a) The separation condition can be illustrated by the example of Poincaré’s two-dimensional system [57]. Putting χ(x, y) = −x, ω1 (x, y) = 1−r2 , ω2 (x, y) ≡ 1, we have dx dy = −y + x(1 − x2 − y 2 ), = x + y(1 − x2 − y 2 ). dt dt We pass to the polar CS: dr = r(1 − r2 ), dt (with manifestly separated variables), where

dϕ = −1 dt

tan ϕ + 1 − r2 v1 = . v2 −1 + tan ϕ(1 − r2 ) The phase trajectory of the system is a stable limit cycle. b) One can obtain a picture of a dynamical system having several phase cycles. For example, take ω1 (r) = (r2 − 1)(r2 − 4), ω2 ≡ −1. Then the original system can be rewritten as dr dϕ = −r(r2 − 1)(r2 − 4), = −1, dt dt and the system has two limit cycles. c) Putting χ = −x, ω1 = ω1 (r2 ), ω2 = −1, we have the system of equations dx = −y + xω1 (r2 ), dt Passing to polar coordinates, we obtain dr = −rω1 (r2 ), dt

dy = x + yω1 (r2 ). dt dϕ = −1. dt

532


In this case one can easily construct examples of dynamical systems with an infinite (countable) set of limit cycles. For example, take ω1 (r2 ) = sin(r2 ). The phase portrait of this system on the plane (x, y) consists of concentric orbits which are alternately stable and unstable limit cycles. Another example is ω2 = −1, ω1 (r2 ) = exp(−r2 ) sin(r−2 ), where one can observe an unbounded concentration of limit cycles as r → 0, while the system (43) is of class C ∞ . For a system of three arbitrary (generally non-linear) ODE x˙ = v1 (x, y, z),

y˙ = v2 (x, y, z),

z˙ = v3 (x, y, z)

we proceed in the same way as above. Passing to the new coordinates α(x, y, z), β(x, y, z), γ(x, y, z) yields a system of linear algebraic equations with respect to v1 , v2 , v3 : α˙ = αx0 v1 + αy0 v2 + αz0 v3 = (∇α, v) = Ω1 , The answer is written as a 0 −1 Ω1 v = −χ Ω2 Ω 3

(∇β, v) = Ω2 ,

(∇γ, v) = Ω3 .

ratio of determinants ex ey ez T αx0 αy0 αz0 , χ = det ∇α, ∇β, ∇γ , βx0 βy0 βz0 γx0 γy0 γz0

or, expanding the second factor with respect to the first column, e ex ey ez ey ez Ω (α, β, γ) Ω1 (α, β, γ) x0 2 α 0 α 0 α 0 v= βx βy0 βz0 − z y x χ χ γ 0 γ 0 γ 0 γ0 γ0 γ0 z z y y x x ex ey e z Ω3 (α, β, γ) 0 + αx αy0 αz0 , χ β 0 β 0 β 0 x y z where the functions Ωi=1,2,3 (α, β, γ) are the right-hand sides of the autonomous differential equations being studied, written in the new variables (α, β, γ). Notice that this expression can easily be generalized to the n-dimensional case:

0 1 Ω1 v=− χ . . . Ω n

e1 ∂1 α1 ... ∂1 αn

e ∇α1 ... en n ... 1X . . . ∂n α1 k+1 , = (−1) Ω k ... . . . χ [∇αk ] k=1 . . . ∂n αn ... ∇α n

where the row [∇αk ] in the square brackets is omitted. We have completely solved the ‘Hamilton–Jacobi problem’ in the non-Hamiltonian situation: given a change of variables α1 (x), . . . , αn (x), find all systems that are


533

soluble (for example, by separation of variables) in quadratures. To do this, the following system must be taken as ‘exactly soluble’: dαi Ωi (α1 , . . . , αk ) = . dα1 Ω1 (α, . . . , αk )

(48)

This general possibility of obtaining exactly soluble equations by a change of coordinates depends on the nature of this change (for example, whether the new system of coordinates is orthogonal [58]–[60]), and the resources of this classical approach seem far from being exhausted. Another possible direction of generalization is to extend the non-equilibrium Gibbs method (in Godunov’s form) to the case of general systems of non-linear equations. To do this, one must write down the analogues of the equations (41), (42) replacing the δ-function by the Maxwell distribution or some other physically motivated distribution. § 6. The topological properties of stationary solutions of hydrodynamical consequences of Vlasov’s equation We have already mentioned that the topological structure of stationary solutions (see § 5) of the Vlasov–Maxwell equations does not follow directly from the Arnold–Kozlov scheme because the force term is non-gradient (and the relation between the velocities and momenta of particles may be essentially non-linear, for example, relativistic) in contrast to the case of the Vlasov–Poisson–Poisson equations. We now describe a modification needed to overcome these difficulties. To do this, we rewrite the Vlasov–Maxwell equations (16) replacing the 4potentials Fµ by the magnetic and electric field-strength vectors H and E in order to simplify the calculations: ∂fα ∂fα 1 ∂fα + vα (p), − eα E + [vα (p) × H], = 0, ∂t ∂x c ∂p Z 1 ∂E 4π X rot H − =− eα vα (p)fα (x, p, t) d3 p, c ∂t c α X Z div E = 4π eα fα (x, p, t) d3 p, α

∂H rot E = , ∂t

div H = 0.

Here vα (p) =

p s mα

1

.

p2 1+ 2 2 mα c

Substituting fα (x, p, t) = nα (x, t) δ(p − Pα (x, t))

534


into the system above, we repeat for the reader’s convenience the EMHD equations [15], [40], [42], [41], which are identical to (19): ∂nα + div(nα vα (Pα )) = 0, ∂t 3 ∂Pα X i 1 ∂Pα + − eα E + [vα (Pα ) × H] = 0, vα (Pα ) ∂t ∂xi c i=1 rot E = rot H −

∂H , ∂t

(49)

div H = 0,

1 ∂E 4π X =− eα nα vα (Pα ), c ∂t c α

div E = 4π

X

eα nα .

α

We transform the second equation in (49) into Gromeka form by subtracting 3 X i=1

vαi (Pα )

∂Pαi ∂x

from both sides. The subtrahend is the gradient of the following function (if we take into account the expression for the relativistic velocity vα in terms of the momentum Pα ): s P2α (mα c)2 . 1+ mα (mα c)2 Hence, after reduction to Gromeka form, this term is a gradient even in the relativistic case. But the Lorentz force is non-gradient and, therefore, we transform it by combining it with the similar term vα ×rot Pα and moving vα ×(rot Pα −(eα /c)H) to the left-hand side. Taking the curl of both sides, we then obtain the following equation for the vorticity in the stationary case: eα = 0. rot vα × rot Pα − H c But we also have a continuity equation, the first equation in (49): div(nα vα (Pα )) = 0. Hence it follows by the Arnold–Kozlov lemma [17]–[19] (see also the appendix below) that the vector fields vα ,

rot Pα − (eα /c)H nα

commute, that is, rot Pα − (eα /c)H vα (Pα ), = 0. nα


535

Notice that H = rot A, where A is the vector potential, and the second vector field is of the form eα rot Pα − A , c where Pα − (eα /c)A is the generalized momentum of the electromagnetic field. We again obtain the same consequences, but for other vector fields: the velocity vα and the vorticity rot Pα −(eα /c)A of the generalized momentum commute. Therefore, by Arnold’s argument [17], [18], each surface of motion is topologically the surface of a torus or cylinder, or a plane. It is well known that the Vlasov equation admits hydrodynamical substitution in the N -layer (and even in the continuum-layer) variant; see [13], [14]. In the N -layer case, this is expected to generalize to the Vlasov–Maxwell and Vlasov–Poisson equations as well as to other equations of this type considered in this paper (see also [61]–[63]). We now show what happens in the case of the Vlasov–Maxwell equation (the N -layer case): fα (x, p, t) =

N X

nαµ δ(p − Pµα (x, t)).

µ=1

In the continuum-layer case we have Z fα (x, p, t) = nα (µ; x, t)δ(p − Pα (µ; x, t)) dµ. The resulting equations are as follows: ∂nα (µ; x, t) + div(nα vα (µ; x, t)) = 0, ∂t 3 ∂Pα (µ) X i ∂Pα (µ) 1 + vα = eα E + [vα (Pα ) × H] , ∂t ∂xi c i=1 Z X ∂H div E = 4π eα nα (µ; x, t) dµ, rot E − = 0, ∂t α Z 1 ∂E 4π X rot H − =− eα nα vα (Pα ) dµ, div H = 0. c ∂t c α For every µ we have its own equation of the form (49) and its own continuity equation. The conclusion is the same for each layer µ. Thus, in all cases of the Vlasov–Poisson–Poisson and Vlasov–Maxwell equations we obtain that the stationary solutions of their hydrodynamical consequences (that is, MHD-equations and EMHD-equations) have the following topological structure. The three-dimensional space splits into two-dimensional tori, products of a circle and a line, or planes. We have shown that the Arnold–Kozlov approach is very general, and it seems that in concrete problems this approach can be deepened to bring the results into an analytic form.

536


§ 7. Conclusion In this paper we have considered microscopic, energetic and hydrodynamical consequences of the Liouville and Vlasov equations using the analogy between them. Liouville’s equation was successfully applied by the classical scientists Maxwell, Boltzmann, Gibbs and Poincaré to describe ensembles of particles. In particular, Poincaré obtained a new form of the H-theorem, which was later developed in [64], [65]. It would be interesting to understand to what extent can this form be transferred to Vlasov’s equation. The latter, whose fundamental and universal nature has been confirmed in the last ten years, contains many other questions for physicists as well as mathematicians. It applies whenever there is an ensemble of particles with long-range action. In particular, again using the analogy with Liouville’s equation, we wonder whether it satisfies von Neumann’s ergodic theorem and its time averages coincide with Boltzmann extremals [65]–[67], as well as whether one can construct a theory of Vlasov’s equation for Calogero–Moser potentials. Another significant problem, which follows from the material considered here, is the possibility of developing the analogy between Gibbs potentials and Godunov’s double divergence form of the evolution equations. Developing our approach to generalizing the Hamilton–Jacobi method in the non-Hamiltonian case is also very promising. Appendix. The Arnold–Kozlov lemma (on commuting fields) Lemma. Let R and P be vector fields in three-dimensional space with div R = 0, div(nP) = 0 and rot(R × P) = 0, that is, the curl of their vector product R × P vanishes. Then the vector fields R/n and P commute. Under the hypotheses of the lemma, we shall find all functions ϕ(n) and ψ(n) such that the commutator of the fields ϕ(n)R and ψ(n)P vanishes: [ϕ(n)R, ψ(n)P] = 0. Using the identity rot [v × w] = [v, w] + v div w − w div v, we have rot(R × P) = [R, P] + R div P − P div R. To calculate [ϕR, ψP], we use the Newton–Leibniz formula and the definition of [v, w]: ∂vj ∂wj [v, w]j = wi − vi . ∂xi ∂xi Therefore, [ϕv, w] = ϕ[v, w] + v(∇ϕ, w),

[v, ϕw] = ϕ[v, w] − w(∇ϕ, v).


537

This yields that [ϕR, ψP] = ϕ[R, ψP] + ϕR(∇ϕ, ψP) = ϕψ[R, P] − ϕψP(∇ψ, R) + ϕψR(∇ϕ, P) = ϕψ rot(R × P) − R div P + P div R − ϕψP(∇ψ, R) + ϕψR(∇ϕ, P) = ϕψR (∇ϕ, P) − div P + ϕψP div R − (∇ψ, R) . Here we have used the identity rot(R × P) = 0. Since we want the last expression to vanish identically, all brackets must be equal to zero. By hypothesis, div R = 0 and, therefore, (∇ψ, R) = 0. Since R is an arbitrary field, we have ∇ψ = 0 ⇒ ψ = const. For the other bracket we obtain 1 (∇ϕ, P) − div P = 0, div(nP) = 0, (∇ϕ, P) − ϕ div P = ϕ2 div P . ϕ It follows that ϕ = λ/n (λ = const1 ) because subtraction of the function ϕ2 div(nP) = 0 yields the equality 1 ϕ2 div − λn P = 0 ϕ and we conclude from the arbitrariness of P that ϕ = λ/n, λ ∈ R1 . Thus we obtain more: the Arnold–Kozlov solutions ψ = const and ϕ = const1 /n have been shown to be unique in a certain class. Bibliography [1] A. A. Vlasov, “On vibration properties of electron gas”, J. Exper. Theor. Phys. 8:3 (1938), 291–318. (Russian) [2] A. A. Vlasov, Non-local statistical mechanics, Nauka, Moscow 1978. (Russian) [3] N. N. Bogolyubov, Problems of a dynamical theory in statistical physics, Gostekhizdat, Moscow 1946; English transl., Studies in Statistical Mechanics, vol. 1, North-Holland, Amsterdam; Interscience, New York 1962. [4] V. P. Maslov and P. P. Mosolov, “The asymptotic behavior as N → ∞ of the trajectories of N point masses interacting in accordance with Newton’s law of gravitation”, Izv. Akad. Nauk SSSR Ser. Mat. 42:5 (1978), 1063–1100; English transl., Math. USSR-Izv. 13:2 (1979), 349–386. [5] Yu. A. Volkov, “Solutions of the Vlasov equation in Lagrange coordinates”, Teoret. Mat. Fiz. 151:1 (2007), 138–148; English transl., Theoret. and Math. Phys. 151:1 (2007), 556–565. [6] L. D. Landau and E. M. Lifschitz, Theoretical physics, vol. 2: Field theory, 5th ed., Nauka, Moscow 1967 (Russian); German transl. of 4th ed., L. D. Landau und E. M. Lifschitz, Lehrbuch der theoretischen Physik, vol. 2: Klassische Feldtheorie, Akademie-Verlag, Berlin 1967. [7] I. P. Pavlotskii, Introduction to weakly relativistic statistical mechanics, Keldysh Inst. Appl. Math. Academy of Sciences of the USSR, Moscow 1987. (Russian) [8] W. Pauli, “Relativit¨ atstheorie”, Encyklop¨ adie der mathematischen Wissenschaften, vol. 19, Teubner, Leipzig 1921, pp. 538–775; Russian transl, 2nd ed., corr. and augm., Nauka, Moscow 1983.

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Received 17/SEP/15 Translated by A. V. DOMRIN