Clebsch potentials and variational principles in the

Clebsch Potentials and Variational Principles in the Theory of Dynamical Systems

HANNO RUND

Communicated by J.C.C. NITSCHE

1. Introduction

In the course of the past few decades the application of Clebsch representations to the general theory of two- and three-dimensional flows in fluid mechanics has given rise to significant new insights, particularly as far as variational principles are concerned. This is evident, for instance, in the work of ECKART [-4] and of SELIGER 8,: WHITHAM [10], whose techniques would appear to have been motivated by the use of Clebsch potentials by BATEMAN [1], [2] in his treatment of some fairly special cases. Since the original formulation of Clebsch representations is concerned solely with the decomposition of vector fields in 3-dimensional Euclidean spaces, one might be inclined to suspect at first sight that this particular feature would severely restrict the scope of further applications of this concept. However, it was recently found that the latter can, in fact, be extended in several ways to yield decompositions of tensor fields on manifolds of arbitrary dimension, independently of the admissibility or otherwise of a Euclidean metric. Since this generalization assumes a relatively simple form for totally skew-symmetric tensor fields, it admits direct applications of such Clebsch representations to the theory of electromagnetic fields ([-7], [9]).

Having overcome the original restrictions with regard to the dimensionality of the configuration space, it is natural to inquire whether the method of generalized Clebsch representations can also serve a useful purpose in the theory of finite- dimensional Hamiltonian systems, and it is to this question that the present note will address itself. We shall consider a classical holonomic dynamical system of n degrees of freedom, defined by a Hamiltonian function H(t, x ~, p j), in which x j, p~ denote the generalized coordinates and momenta respectively. The covariant vector fields pj are represented in terms of generalized Clebsch potentials: it is found in section 2 that the classical canonical equations then guarantee the existence of a so-called superpotential ~b, which is solely a function of t and of the Clebsch potentials; moreover, the latter satisfy a system of differential equations whose formal structure is identical with that of the given canonical equations, the role of the Hamiltonian being played by q'. This in turn leads to a generalized Hamilton-Jacobi equation for the Clebsch potentials and hence to an extension of the notion of the complete figure associated with a single integral variational

306 H. RUND

problem in the sense of CARATHI~ODORY ([3], p. 249), it being recalled that this figure is based upon the somewhat restrictive representation pj = OS(t, xh)/~x ~. In the present context, however, the curl ofpj is not thus restricted; indeed, it is found that the evolution in time of this curl is always described by the vanishing of a Lie derivative, which allows for a direct integration of these evolution equations.

A Lagrangian A, depending solely on t and on some of the Clebsch potentials and the first derivatives thereof, is associated in section 3 in the usual manner with the superpotential 4~, the latter now being regarded as a Hamiltonian. It is found that the relationship between A and the Lagrangian L associated with the given Hamiltonian H is such that the Clebsch representation ofpj may be interpreted as a B~icklund transformation between the respective sets of Euler-Lagrange equations. Since the Clebsch potentials ofpj are by no means uniquely determined by p j, there exists a non-trivial class of Clebsch gauge transformations, whose properties are examined in some detail in section 4. This analysis is quite similar to the theory of canonical transformations, and again involves B~icklund transformations, giving rise to a fundamental invariant on which the subsequent analysis is crucially dependent.

By virtue of the peculiar structure of the Clebsch representation of pj it is possible to reinterpret the canonical equations satisfied by the Clebsch potentials as the Euler-Lagrange equations of a multiple integral problem in the calculus of variations in the n + 1 independent variables t, x ~. In this manner a multiple integral variational principle is associated in accordance with an essentially unique procedure with the original Hamiltonian system (Section 5). More precisely: the Euler-Lagrange equations of this principle imply the validity of the canonical equations satisfied by the Clebsch potentials, and this in turn implies the validity of the original canonical equations, and conversely. If, moreover, the Hamiltonian H depends on potential fields whose behavior is governed by field equations which are the Euler-Lagrange equations associated with some Lagrange density, the latter may be adjoined to the Lagrangian of the aforementioned multiple integral problem, giving rise to a single action principle which simultaneously determines the field equations and the equations of motion (that is, the original canonical equations). The underlying Lagrangian depends on the Clebsch potentials only via the gauge-invariant quantity referred to above and is therefore itself invariant under Clebsch gauge transformations.

In section 6 the Hamiltonian complex of our multiple integral variational principle is derived; again, simple considerations concerning the gauge lead directly to a gauge-invariant energy-momentum tensor. The special case when H is identified with the non-relativistic Hamiltonian associated with a charged particle in an electromagnetic field, subject possibly also to additional forces, is examined in the final section 7, and, in the absence of pressure terms, the Lagrangian thus derived by a systematic application of the procedure outlined above reduces to a Lagrangian proposed by SELIGER • WHITHAM [10] for the study of plasma waves.

Before concluding this section, let us briefly glance at some basic facts concerning Clebsch representations. The first such representation was introduced in 1857 by Clebsch, who showed that any differentiable vector field on a 3- dimensional Euclidean manifold can always be decomposed locally into the sum of a gradient and a scalar multiple of another gradient. Instead of using the aforementioned extension to skew-symmetric tensor fields on manifolds, we shall

The Theory of Dynamical Systems 307

here be concerned with a generalization which results directly from the theory of differential forms.

Let Yj(x h) denote a differentiable type (0, 1) tensor field on an n-dimensional differentiable manifold, the latter being referred to local coordinates x h. [Here, and in the sequel, Latin indices j, h, k, ... range from i to n, the summation convention being operative throughout.] This field determines an invariant 1-form

(1.1) ~o = Yjdx a,

whose class p is defined to be the minimum number of variables required for its representation. Since p < n/2 or p < (n - 1)/2, according as n is even or odd, this gives rise to the so-called canonical forms of co, namely

(1.2)

or

(1.3)

respectively,

~o=Ql dP1 + ,.. +QPdP,,

~o=d~, + Ql dPl + ... + QP dPp,

in which ~,QI , . . . ,Qp, P~ . . . . ,Pp, denote systems of independent functions of x h ([-5], pp. 32, 42).

Thus, when n is even, it follows from (1.2) and the representation (1.1) ofo) that the components of an arbitrary differentiable type (0, 1) field can always be represented, at least locally, in the form

(1.4) Yj=Q ~Uxj,

where the implied summation over ~ extends from 1 to n/2. Similarly, when n is odd, the corresponding representation of the field is given by

(1.5) Y j = ~ + Q" 8P~ 8x j'

where the summation over e extends from 1 to ( n - 1)/2. Clearly (1.5) reduces to the original representation of Clebsch when n = 3. Accordingly we shall refer to (1.4) and (1.5) as generalized Clebsch representations, and the functions 0, Q', P~ which appear therein will be called Clebsch potentials. The latter are obviously not uniquely determined by the field Yj, and are therefore subject to certain gauge transformations whose properties will be examined presently.

2. Superpotentiais Associated with Canonical Vector Fields

Let us consider a classical holonomic dynamical system with n degrees of freedom, this system being specified by a Hamiltonian function H(t, x J, p j), in which x j,pj respectively represent the generalized coordinates and momenta, while t denotes the time. It is assumed that H is of class C z, and that

{ 82H (2.1) det \ ~ ! +0.

308 H. RUND

We shall regard the variables x J as the local coordinates of a differentiable manifold X,, and we shall be concerned with time-dependent class C 2 vector fields p~ = p~(t, x h) on X,. The canonical equations associated with the Hamiltonian H are represented as usual as (2.2) dpj = OH dx J ~H

dt Ox j' dt Opj

Let us write the first of these in the form

op, lop, Opq h=_ H Oph., Ot ~- \Ox ~ O x # Ox ' -U~x ~ x '

where it is to be understood that the derivatives ~h= dxh/dt are given by the second member of (2.2), so that

(2.3) Opj + (Opj Oph] OH OH OH Op h gt \t~X h OXJ] t~ph OX J Oph OX j"

This suggests the introduction of the function

(2.4) h(t, x j) = H( t, x J, pj(t, xh)),

by means of which (2,3) can be expressed as

(Op, gp,] OH Oh (2.5) ~P~ ~- - - =

Ot \OX h OX j / Oph ~X j"

A field pi(t, x h) for which the expression on the left-hand side of (2.5) is a gradient will henceforth be called a canonical vector field.

If the given Hamiltonian H is a scalar relative to arbitrary coordinate transformations on X. - a s will be assumed h e r e - the canonical momenta pj are type (0, 1) tensors admitting Clebsch representations of the form

(2.6) PJ = Q= 0P= Ox-~, c~ = 1 . . . . . n/2,

when n is even, while, when n is odd,

(2.7) pj =~-~i+ Q~ 0x-~, c~ = 1 . . . . . (n - 1)/2.

[Here, and in the sequel, summation over repeated Greek indices ~, fl . . . . is implied with ranges as specified.] Since the field pj is time-dependent, the Clebsch potentials qJ, Q', P= will, in general, be dependent on t also, and accordingly it will be supposed that the potentials are class C z functions of the n + 1 variables (t, xh).

Let us now rewrite the left-hand side of (2.5) in terms of the Clebsch representation, assuming, in the first instance, that n is odd, in which case (2.7) is to be applied. Clearly

0pj 02~/ 0Q ~ 0P~ 02P~

(2.8) & - = ~ T x ~-~ gt OxJ+Q~otOxJ'


and

(2.9) 9Pi - Ox n

so that

9~0 OQ = oP, 9~P~ OxhgxJ + 9x ~ 9x j kQ" 9xhgxJ,

(2.10) Opt Op._ 9~ 9Q �9 oQ = o/,, 9x h Ox ~ Ox J Ox h Ox ~ Ox h"

On introducing the generalized 'convected' derivatives of Q', P~, namely

dQ = 9Q ~' 9Q = OH dP~,_gP= OP~ OH (2.11) dt - - Ot Jf O X h Oph' dt 9t + 6~X h 9ph'

it is seen that (2.10) yields

(op, 9pq 9. io~ de = 90 = de I i0P, 0e ~ 0e" 0e 1 (2.12) \9x h 9xQ ~ph = \O~X ~ dt Ox J dt ] - \Ox J 9t 9x i & !"

With the aid of (2.8) and (2.12) we may now express the left-hand side of (2.5) as

(2.13) ~ + (Opj Oph] OH (OP~ dQ = OQ ~ dP~]+ O~ /0~9+ =OPt\ ~ox h oxJ: ~= ~ dt OxJ dt : 0~ ~ Q -~)

This relation is merely an identity which is a direct result of the Clebsch representation (2.7). However, when (2.13) is applied to the condition (2.5) for a

canonical field p j, this requirement assumes the form

OP~ dQ = OQ ~dP~ 0~ (2.14) Ox j dt 9x ~ dt 9x j=O ( j = l , . . . , n ) ,

where the function q3=~(t, x h) is given by

(2.15) ~ = - h - O ~ - O ~gP~ 9t ~ ~-"

The left-hand side of each of the n relations (2.14) consists of the sum of 211(n - 1)] + 1 -- n derivatives with respect to x ~, so that

0(Q =, P,, ~) (2.16) 0(x 1 . . . . . x") 0.

" P But this implies the existence of a function rb(t, Q , ,) such that

(2.17) ,~(t, xh)= rb(t, Q~, P~),

which we shall regard as a superpotential for the Clebsch potentials inasmuch as (2.14) can now be expressed as

OPt, dQ ~' 9Q" dP~ 9rb (2.18) Ox ~ dt Ox ~ dt =~x j'

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yielding dQ ~ ~4 dP~ OcI)

(2.19) dt =~-~' dt ~Q~"

Conversely, let us now suppose that the Clebsch representation (2.7) possesses a superpotential in the sense of (2.18). Substitution of the latter in the identity (2.13) then gives

) (2.20) ~P~ + - + Q~' + 4 ~t \~X h OX j] t~ph ~X j

which is identical with the condition (2.5) for a canonical field with

(2.21) h (;~Ot-+ ~ P ~ + 4 ) = - Q ~ - .

If, in the second instance, it is assumed that n is even, the Clebsch representation (2.6) must be used. The resulting analysis proceeds along lines similar to that of the argument presented above: in fact, one merely suppresses the potential q/, at the same time noting that the index e now ranges from 1 to n/2. Thus a direct analogue of (2.14) is obtained in which ~3(t, x h) is given by

(2.22) ~ = - h - Q ~- ,

while, instead of (2.16), we now have

O(Q1 . . . . , Q,/2, p~, .. . , e , / 9 4 0 . (2.23) ~3(x 1 . . . . . x")

Thus the system Q" = Q~(t, xh), P~ = P~(t, x h) may be inverted to yield x ~ = xJ(t, Q~, P~), which gives rise once more to a superpotential (2.17) by means of (2.22). Again, the existence of this superpotential, in the sense of(2.18), implies the condition (2.5) with

The results obtained above may be summarized as follows:

Theorem. In order that the field p~(t, x h) be canonical it is necessary and sufficient that there exist a superpotential 4 = 4(t, Q~, P~) of the potentials of the Clebsch representations (2.6) and (2.7) in the sense of (2.18), or equivalently, such that

dQ ~ 04 dP~ 04 (2.25) dt =~--~' dt aQ ~"

When n is odd, the resulting relation between 4 and h is given by

oO ~P~+ O-T-+ O~ ~ - 4 + h = 0 , (2.26)

while, when n is even,

(2.27) ~ - ~+h=O.


The following observations may serve to clarify the significance of the relations (2.26) and (2.27). Again, for the case when n is odd, let us substitute for Pi from (2.7) in the function h as defined by (2.4), so that (2.26) assumes the explicit form

~ =0

This relation may be regarded as a first order partial differential equation for the Clebsch potentials. Differentiation with respect to x ~ yields, after a little rearrangement,

0Q ~ (0P~+0~ ~/~ ~P,~ /02P~ 0u ~P~

aP~a,/, ~2r c~2~, aH 0H

or, if we apply the definition (2.11) of the generalized convected derivative,

dt

Substitution from (2.7) then gives

(2.29) OQ~ /dP~ ~ k OP~ ~ dpj ~H 0 : + - 2 T + ~ = .

This is the relation which we have been seeking; it is immediately evident that it is valid also when n is even, in which case it results directly from (2.27) with the aid of (2.4) and (2.6). Now, the relation (2.29) is satisfied for any set of solutions if, Q', P~ of the partial differential equation (2.28), in which the function q~(t, Q~, P~) may be chosen arbitrarily: in particular, /f this set of solutions is such that the equations (2.25) hold, it follows that the original canonical equations

(2.30) dpj OH 0 dt t-O-~xJ=

are satisfied, while conversely the latter imply the validity of (2.25) whenever the solutions Q~, P~ of (2.28) are independent.

It should be observed that the above construction entails a substantial generalization of the concept of the complete figure of CARATHISODORY for single integral variational problems ([3], [6]). Let us recall that the complete figure depends upon the existence of a principal function S = S(t, xn), in terms of which the canonical momenta admit a representation of the form

#S (2.31) pj - Ox J,

while, at the same time, the function S is required to be a solution of the classical Hamilton-Jacobi equation

312 H. RUND

(2.32) aS ( a S ) ~-~ + H t, xh, ~X~ =0.

If a representation of p~ such as (2.31) is indeed admissible, we may choose the potentials in the Clebsch representation (2.6) and (2.7) as follows:

(2.33) O=O, Qe=af, P~ =a~S, 4 = 0 .

Under these circumstances, the partial differential equation (2.28) reduces to the Hamilton-Jacobi equation (2.32), and, moreover, the relation (2.29) automatically implies the canonical equations (2.30). Thus the specialization of our theory as implied by the substitutions (2.33) leads directly to the essential ingredients (2.31), (2.32), and (2.30) of the complete figure. Accordingly we may regard (2.28) as a generalized Hamilton-Jacobi equation.

The distinction between the two theories is obviously reflected in the properties of what we shall call the generalized vorticity tensor, namely

ap~ aph (2.34) r h axj,

whose Clebsch representation is already displayed in (2.10). In order to obtain a compact expression for the convected derivative of (2.34), let us introduce the notation

(2.35) vi(t ' Xh ) _ 8H( t, x h, ph(t, X')) apj

in terms of which it follows directly from the definition (2.11) that

(2.36) ~ ~,~x~l=~s \dt ! ax' ax ~' d--t \ax' l -Ux~ \ dt I

Accordingly differentiation of (2.10) yields

(2.37)

aQ ~ av I ax i ax,r

dt axS \-dtl ax h a# -o~ +~x h \ dt I Ox s Ox h \-d-i!

av, VP, ao. aO:l_av' VP, ao " ao.71 +-g~ Lax" & ' & ' ax" J ax h t & ~ ax' ax' ax' 1"

If it is now assumed that the canonical equations (2.25) are satisfied, the first four terms on the right-hand side may be expressed as

a (atb aQ = a~ aP=] a~ a2Q = a~ a2P~

= -&--7 ~ c~x ~ ~-~ ~x~! + aQ~ ~xJ,~x ~ + ae~ ~xJax ~ a (a~ aP~ a~ aQT'l_ a~ azP~ a~ a2Q ~

+~x h ~-~ ~x j ~ aQ ~ axJ ! aP~ axJ ax h eQ~ ~xJ & ~ =0"


Moreover, if we substitute from (2.10) in the remaining terms in (2.37), we obtain

dfDhj __ OV l ~V t

dt co h j ~x j - ~o 0 Ox h'

o r

(2.38) ~v COhj = O,

where the left-hand side denotes the L i e der iva t i ve o f ~ohj wi th respec t to the f i e l d (2.35).

In order to interpret this result, let us consider an n-parameter family of integral curves of (2.2), represented in the form

(2.39) # = xJ(t, uh),

it being supposed that x j = u s at some initial time t = t o. Our considerations will be restricted to a region in which the Jacobian

(Ox' (2.40) J = det ~Ouh ]

is non-vanishing. We may thus define the quantities

Ox h Ox j 2~k(t, u v) = %j ( t , x p) ~u ~ &k, (2.41)

so that Ou t ~u k

(2.42) a~hj(t , x p) = 2tk(t , U p) OX h Oxj,

where u h = uh(t, X j) is the inverse of(2.39). But, according to a general lemma (see the Appendix), the condition (2.38) implies that 02 lk(t, uh) /o t = O. Since, by construction, Ouh/Ox j = 6~ at t = t 0, it then follows from (2.41) that 2 lk(t, U p) = 2 zk(U p) = ~0 lk(tO, UP), which we shall denote by co~ for the sake of brevity. Thus (4.42) assumes the form

(2.43) OR l OU k

ohj(t, xV)=a~~ ax ~ & j ,

which clearly describes the evolution of the vorticity tensor, whose time dependence is contained entirely in the coefficients Ou~/Ox h on the right-hand side. In par t icular , / f ~ohj = 0 at t = to, this p r o p e r t y will pers is t f o r subsequen t values o f t.

3. The Associated Single Integral Variational Problems

The condition (2.1) on the given Hamiltonian H(t , x h, Ph) e n s u r e s the existence of an appropriate Lagrangian L(t , x h, 53). Indeed, by virtue of (2.1) we may solve the second member of (2.2) for p j = p j ( t , x h, Ych), and put

(3.1) L( t, x h, 2n) = _ H ( t, x h, Pn( t, x k, ~*) ) + 2 J p j( t, x h, 2h).

It then follows as usual that

314 H. RUND

OL OH OL OH OL (3.2) & & ' Ox j = ~x j, ~5=P~

identically, from which it is immediately evident that the canonical equations (2.30) are identical with the Euler-Lagrange equations

(3.3) Ej(L)=0,

where

(3.4) Ej(L)-=~ ~ 0x j.

Thus the equations (2.2) represent the extremals of the single integral variational problem as defined by L on the manifold X,.

The peculiar canonical (or Hamiltonian) structure of the equations (2.25) suggests, on the other hand, that the function ~(t, Q~, P~) should be regarded as the Hamiltonian of a single integral variational problem on a q-dimensional differentiable manifold Xq, where q = n/2 or (n - 1)/2 according as n is even or odd, the local coordinates of Xq being represented by Q'. Thus, by analogy with (2.1) it will now be assumed that

t02 t (3.5) det \0P, 6P~] 40,

so that the first member of (2.25), namely

0~ (3.6) - -

may be solved for P~ = P~(t, Q~, 0~), these being regarded as the canonical momenta associated with the tangent vector 0 ~ = dQ~/dt of any trajectory Q~ = Q~(t) on xq. The Lagrangian associated with �9 is given by

(3.7) A(t, Q~, Q~) = - ~(t, Qo, P~(t, Q', Q')) + Q~ P~(t, Q~, ()~),

for which the relations

OA 04~ ~A 0~ OA (3.8) & &, 0Q ~ - 0Q,, 8Q =P~

are satisfied identically by virtue of (3.6). Thus the second member of (2.25) is identical with the Euler-Lagrange equation

(3.9) E,(A) =0,

where d ( O A ) c3A

(3.10) E~(A)=~ ~ g 0Q ,.

These simple conclusions may serve to shed some light on the construction of the previous section. The canonical equations (2.2) and (2.25) respectively represent


extremals of variational problems on X, and Xq, the respective Lagrangians being defined by (3.1) and (3.7). If, in particular, the Clebsch potentials satisfy the generalized Hamilton-Jacobi equation (2.28), the Euler-Lagrange equations (3.3) imply the Euler-Lagrange equations (3.9), and conversely, whenever the solutions Q,, W of(2.28) are independent. In other words, the Clebsch representations (2.6) or (2.7), subject to the partial differential equation (2.28), represent a Bi~cklund transformation which relates the extremals of the Lagrangian L on the manifold X, to the extremals of A on X q.

From (2.6) or (2.7) we have, using (2.11),

p~yd=d~dt_+ Q ~ dP~ OP~] Z - ot / '

(3.11)

in which we put ~ = 0 or ~0 + 0 according as n is even or odd. Alternatively, we may write

�9 d (3.12) P, Q~= - (~t + Q" - ~

which is merely an identity resulting from the form of the Clebsch representation. If we now substitute from (3.1) and (3.7) in (3.12), we obtain the following identical relation between the Lagrangians and Hamiltonians of our variational problems:

(3.13) A = _ L _ ( O ~ + Q ~P~ ) d

If, in particular, the potentials satisfy the generalized Hamilton-Jacobi equation (2.28), this reduces to

d (3.14) A = - L + ~ (0 + Q" P~).

Thus, since under these circumstances the difference between A and - L is a total derivative, the aforementioned BiJcklund transformation is a variational B~cklund transformation, which is not surprising in view of the general theory of the latter [8].

4. Clebsch Gauge Transformations

Needless to say, the potentials which occur in the Clebsch representations (2.6) and (2.7) are by no means uniquely determined by the vector field pj. Consequently these representations admit gauge transformations of the potentials:

(4.1) 0P, Q~, P~)~(0, Q', P~),

which are subject solely to the requirement that

0." xJ- O# Q" OxJ"

[Again we shall take tp to be zero or non-zero according as n is even or odd.]

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This condition is equivalent to

(4.2) Q~ dP~ - Q~dP~ - (ft, +1 - P, + 1) dt = d(t~ - ~),

where, for the sake of brevity, we have written

(4.3) p , + l = ~ +Q" /3,+, + ( ~ &,

An equivalent, but more convenient formulation of (4.2) is given by

P~dQ=- P. dQ" +(P.+ x - P.+ 1) d t= - d z , (4.4)

where

(4.5) - ~p_ ~p_ ~=(O-O+O ~ - g J .

The condition (4.4) is very similar to the defining relation of a classical canonical transformation. Indeed, from (4.4) it is evident that Z = Z( t, Q#, 0~), with

~X (4.6) /~= - ( ~ ,

which gives a representations of the type

(4.7) P~ = P~(t, Qa, Oa),

Let us suppose that

~Z

= P , ( t ,

det \~-0-~] ~0,

in which case we can solve the second member of (4.7) for (~ as a function of (t, Q~, P~). This in turn is substituted in the first member of (4.7),which yields/~ as a function of these variables also. It is therefore inferred that the gauge transformation (4.1) gives rise to a transformation of the type

(4.8) Q~ = O~(t, Q~, P~), P~ = P~(t, Qa, P~).

Moreover, when the first member of (4.8) is substituted in Z(t, Q~, 0~), we obtain the function

(4.9) O(t, O ~, P~)= Z(t, g=, O~(t, O~, P~)).

The condition (4.4) may now be expressed in the following form:

(4.10) ~0 dt ~_~g ~ ~0 = - & _ _ dQ - ~ - ~ dP~,

which naturally entails certain integrability conditions. The latter are expressed most readily in terms of Lagrange brackets, these being defined as usual by


�9 \ ~ u Ov ~v a u / '

for any pair of variables (u, v). A routine calculation shows that (4.10) is equivalent to the conditions

(4.12) [Q~, Q#] =0, [Q', Pp] =6~, [P~, PJ =0,

(4.13) [Q#, t] = ~ ( p , § I - p , + j , [P~, t ] = (ig,+ 1 -p, ,+ j .

With the aid of (4.12) one can then follow the standard procedure ([6], p. 90) to show that the functional determinant of (4.8), namely

a(Q ~, P~) J

a(o< P~)

is such that j z = 1, which implies the existence of the inverse

(4.14) Qp=Qp(t,Q%P~), P~ = Pp(t, Q% P~).

The derivatives of (4.8) and (4.14) satisfy certain conditions which will be found to be useful in the sequel. In order to obtain these relations, we note that, by virtue of (4.8),

(4.15) OQ" [O/] d/~ ~ e/] dp 0 OP~ \~t d t + ~ d Q + ~

: [t, P j dt + [Q~, PJ dQ ~ + [P. PJ dP.

which is merely an identity resulting from the definition (4.11). However, if we now substitute from (4.12), (4.13), and (4.14) on the right-hand side, we obtain

- 0 ~ (fi"+l -P"+l)dt+bfdQ~= [-~-f OP~ (fi"+ + ~ oQ dP~,

and a comparison with the left-hand side of (4.15) then yields

OQ ~ a/] OQ p aO" 0Q ~ a (4.16) 0Q~/--0P~ ' e/] - 0P~' #t -eP~ (/T"+I-p"+I)"

A similar procedure, applied to the expression (#fi~/OQ ~) dQ ~ - (OQ'/OQ ~) dP~, leads to

(4.17) ~P~ ~/] 0P~ eQ" ~P~_ 0 a0" aQP' aP~ aQ,' aZ-- aQ~ (P"+ ' -P"+ ' )

The relations (4.16) and (4.17) represent the conditions which we have been seeking: they are both necessary and sufficient in order that the transformation (4.8) be such that (4.4) is satisfied.

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These conditions will now be used to determine the behavior of the super- potentials ~b(t, Q', P~) under the gauge transformation (4.1). To this end we first note that, by virtue of (4.16),

dQ ~ aQ ~ aQ~ dO. ~ aQ~ dP~ - - ~

dt at ~Q~ dt a~ dt

aQ~ a~ d09 aOE dP~ - a t ~ aPe dt aP~ d-~"

If we denote by ~(t, Q~, P~) the transform of ~(t, Qa, P~) under (4.1), we may express

this as dQ p a~ a~ [dQ ~ a~\ aQ ~ [d~ a~ \

a~a~ a~aO_ ~ a# aQ~

or, if we apply the third member of (4.16),

dO ~ a~ a~ - a4 aO =/d~ a4\

(4.18) a + ~ [~-~+(~.+~-p.+,)].

A similar calculation based on (4.17) yields

(4.19) a aQP [~-'P+(tL+ 1-p.+,)].

Now, from the general theory of section 2 it is evident that the canonical equations (2.25) must be gauge-invariant. However, the relations (4.18) and (4.19) clearly indicate that this implies that

and

8 ae~ - - [ ~ - ~ +(~.+ 1 - p . + 1)3 =0 ,

8 aQ~ E'b- # +(P"+ l-P"§ 1)3 = ~

Thus the quantity in the square brackets above is at most a function of t. The latter may be ignored, since it can be absorbed in the Lagrangian A (and hence in q~) without affecting the Euler-Lagrange and the canonical equations of the underlying variational problem. Reverting to our original notation as displayed in (4.3), we therefore conclude that the quantity

(4.20) ~(t, Oa, Pe)+8~-~+Q " - a/5~ cb t T i = ('Q~'P~)+~-~ +0~ at 8P~

o i

represents an invariant of the gauge transformation (4.1).


This result entails some interesting consequences. First, we observe that by analogy with (3.7) the transform /1 of the Lagrangian A under the gauge transformation is given by

dt ' or, if we use (4.4) and (4.3),

a e q _ - az

from which ,~ may be eliminated by means of (4.20) to yield

dQ = dx dx (4.21) A = - 4 ~ + P ~ dt dt A dt

Thus the application of the gauge transformation gives rise to a new Lagrangian which differs from the original merely by a total derivative. Because of the invariance of the canonical equations under gauge transformations, the latter are therefore variational B~icklund transformations of the canonical equations into themselves.

Second, it should be noted that (4.4) and (4.3) also imply that

(4.22) ( ~ + O~'~-] ( ~ t + Q~' = at ! at"

With the aid of the second member of(4.6) the relation (4.20) can thus be expressed in the form

aX (4.23) *(t, Q~,P~)=* (t, Q~, ~ ) + ~ .

It therefore follows that , /f the function g as given by (4.5) can be chosen such that it is a solution of the Hamilton-Jacobi type equation

(4.24)

then, in the resulting gauge,

(4.25) ,b(t, 8 ) = 0.

5. The Associated Multiple Integral Variational Problem

It will now be shown that the peculiar structure of the Clebsch representations (2.6) or (2.7) leads directly to a reformulation of the canonical equations (2.25) as the Euler-Lagrange equations of a suitably defined (n + 1)-fold integral problem in the calculus of variations. In order to exhibit this equivalence, let us recall that, for a

0 A 0 a 0 a] of such a problem, in which 0 A denotes m Lagrange density ~ = ~ ( t , x h, , ,,, ,hi dependent functions, 0,at = ooa/at, O~ : ~On/ax h, A = 1 . . . . . m (m arbitrary), the Euler- Lagrange expressions are given by

(5.1) . - a 0 A

320 H. RUND

Now, again assuming that the Clebsch potentials independent, we observe, firstly, that when n is odd,

OPh ~Ph OP~ Oph (5.2) CO j = 6 {, - , atY Ox h oP~,j =(y~{' so that

(5.3)

in (2.6) and (2.7) are

OH OH OH OH 0P~ 0H 0H QZ. 0~,j 0pi 0Q ~ 0pj 0x j' 0P~, i 0pj

Secondly, if v = v(t, x h) denotes some differentiable scalar field, we have, using (2.11),

tdt ~ { W + ~ oxJ1 ok (5.4)

t 0pj I [77+~\ op/j vow' which is merely an identity resulting from the definition of the generalized convected derivative. However, when we substitute from the third member of (5.3), the identity (5.4) becomes

whose formal structure begins to resemble that of (5.1). Indeed, if we now provisionally define a Lagrange density &P~ by putting

r (5.6) Z I : v LQ ~ H(4 x ~, p~) + ,~(t, Q~,

it is evident that (5.5) may be expressed as

which would be a true Euler-Lagrange expression if the scalar field v were to satisfy an equation of continuity, in which case the coefficient of Q~ on the right-hand side would vanish. But, because of the first member of (5.3), this coefficient may be written as

~i+~x e ( vaH\-~-_ l=~-+~xTx j a y O {v 014 ~ ~,j, t cO,~

(5.8) o o + 2 (v oH

where we have defined a second provisional gagrange density by

(5.9) ~LP2 = V [~t + H(t, xh, ph) ] .

Thus the term in question is indeed an Euler-Lagrange expression. Finally, we also have, using (2.11) and the second member of (5.3),


ag, X eu v

or, since s as defined by (5.6) does not depend on any derivatives of Q=,

/dP~ aq~ \ ( 5 . 1 0 ) v [dV+yd)= -

Clearly the Lagrangians (5.6) and (5.9) may be combined into the Lagrange density

(5.11 v, e, e ; <,, < , , < > - v xh, kVt at

where it is to be understood that the momenta Ph which appear as the arguments in the term VI-I(t, xh, ph) of ~q~ o are to be expressed in terms of the Clebsch representations (2.6) or (2.'7). With ~~ 0 thus defined, the relations (5.?), (5.8), and (5.10) now yield the system

(5.12) v = Ep= (~o) - Q~Eo(~gYo),

(5.13) v \d t + = -Eel(2#~

(5.14) c?t I-ff~xS \ @j l =E*(s176

We have therefore established the following result: When n is odd, the canonical equations (2.25) are implied directly by the Euler-Lagrange equations

(5.15) G ( ~ o ) : 0 , E~(~eo) = 0, G, (~G) =0

associated with the (n + 1)-fold multiple integral variational problem defined by the Lagrang,ian (5.11). Conversely, the canonical equations (2.25) imply the Euler- Lag, range equations (5.15) of that problem whenever the scalar field v satisfies the equation of continuity

ev (5.16) t?t ~-~x j \ Opjl =0.

It is remarkable that tfiis equivalence depends crucially upon the introduction of the field v(t, xh); however, from a physical point of view this is quite plausible in the light of condition (5.16).

For the case when n is even, the representation (2.6) does not contain the term OO/Sx j, and accordingly the first member of (5.3) must be replaced by ~?H/O~,j = O. Thus the corresponding substitution in (5.8) is not valid in this case, and hence the left-hand side of(5.8) is not necessarily an Euler-Lagrange expression. Therefore, in the formulation of the above result for even values of n, the first member of (5.15) must be replaced by the condition (5.16), no further amendments being required.

In passing it should be observed that, by virtue of(4.20), the Lag,rang,Jan (5.11) is invariant under all Clebsch gauge transformations.

322 H. RUND

Because of the peculiar structure of the Lagrangian (5.11) the left-hand side of the generalized Hamilton-Jacobi equation (2.28) is identical with -Ev(~o) , and thus (2.28) is equivalent to the Euler-Lagrange equation

(5.17) Ev(5('o) =0.

Following the procedure which leads from (2.28) to (2.29) it is now found with the aid of (5.12) and (5.13) that, when n is odd,

(5.18)

8Q ~ 8P EQ~(~o) + ~ [EP~ (~o) - Q~E~,(s ] =

Ox J

Therefore the Euler-Lagrange equations (5.15), (5.17) imply the canonical equations (2.30). This is true also when n is even, again subject to the proviso that the first member of (5.15) be replaced by (5.16).

Let us suppose now that a potential function V(t, x h) occurs in the Hamiltonian function H, it being supposed that the behavior of V is determined by field equations which are the Euler-Lagrange equations associated with some Lag- rangian ~'f( t , X h, ~ l/,,t, V,,h). The variational principle defined by the Lagrange density

(5.19) 5~ =LP0 +5r

yields not only the equations of motion (2.30) but also the field equations to be satisfied by the potential V. This phenomenon reflects one of the principal advantages which result from the construction of our (n + 1)-fold integral problem in the calculus of variations.

The following simple example may serve to illustrate this point. Let us assume that n = 3, and that

1 (5.20) H =~mm pjpJ + V(t, xh),

where m is a constant (mass of a test particle), while V is supposed to be a solution of Poisson's equation. The Lagrangian (5.19) is now given by

(5.21) [8~ ~ 8P 1 )

~f =v t ~ + ~ 2 ~ + q ) ( t , Q,P)+~--~mpjp~ + V + I V ~V.i,

in which Pi is represented by (2.7). According to our theory, the Euler-Lagrange equations (5.15) and (5.17) give rise to (2.30), which in this case is simply

(5.22) dpj _ 8V dt 8x s'

while the additional Euler-Lagrange equation Poisson's equation

82V (5.23) t3x~ 8x j = v.

E v ( Y ) = 0 is identical with

[dp; 8H\ = v


This particular example is, perhaps, too naive to illustrate the crux of the method. This is due to the fact that the potential V, by virtue of its dependence solely on (t, xh), cannot affect the Clebsch representation (2.7) of the momentum pj = c~L/~?2 j. However, a somewhat less trivial special case, which does not avoid this issue, will be discussed in detail in section 7.

Remark. In the general theory above it was tacitly assumed that the Hamiltonian H is independent of the field v. However, in some cases of physical interest it may well happen that H contains terms which depend on v. Although the formal structure of the Lagrangian (5.11) remains unchanged under these circumstances, it is evident that o_9o o is no longer linear in v. In fact, one now has

0H (5.24) _ E v ( ~ O ) = ~) - 1 ~ o nt- l,' ~ - .

Thus, while the analysis preceeding (5.17) retains its validity, it can no longer be asserted that the Euler-Lagrange equation (5.17) is equivalent to the generalized Hamilton-Jacobi equation (2.28). Instead, it follows from (5.24) and (5.11) that

~9 ~ c~P~ h gH (5.25) -E~(L, a o ) = ~ - + Q ~- f+H( t , x ,ph, v)+cP(t, QtJ, P~)+V-c3-~,

and a straight-forward repetition of the calculation which leads from (2.28) to (2.29) indicates that (5.18) must now be replaced by

(5.26)

0Q . . . . 0~ ~xilSQ,(fol+ ~x) [Ee,(~o)-Q'Eo(~('o)] - v [E~(~o) ]

rdpj 0 . g =v / + S 5 ] "

Under these conditions the Euler-Lagrange equations (5.15) and (5.17) will obviously not yield the canonical equations (2.30) as such; however, it will be seen in section 7 that the additional terms involving the derivatives of H with respect to v may well be significant, if not indispensable from a physical point of view.

6. The Energy-Momentum Complex of the Multiple Integral Problem

It is well known that one may associate with any Lagrangian ~( t , x h, 0 A, 0~, 0,~) of a multiple integral variational problem a so-called Hamiltonian complex which gives rise to certain conservation laws whenever 2/' does not depend explicitly on some or all of the independent variables (t, xh). We shall now construct this complex for the Lagrangian (5.11).

When n is even, the dependent functions 0 A are Q', P~, v (e = 1 . . . . , n/2), that is, a total of n + 1 such functions. When n is odd, the dependent functions are Q', P~, ~, v (c~ = 1 . . . . , (n-1)/2), which again represents a total of n + 1 functions. We shall therefore adopt the convention that A, B, C, ... range from 1 to n + 1, summation over repeated indices being implied. Also, we shall set t = x "+ 1, so that 0~ = ~?OA/gt = 8 0 A / ( ~ X n + l = 0 A The Hamiltonian complex associated with ~ o is defined as , n + l "

324 H. RUND

usual ([6], p. 247) by

A 0 ~ 0 (6.1) /~B a = - 5eo6, + ~ 0cn, ,A

and the aforementioned conservation laws result from the identities

(6.2) dI2I] 3~t~~ ~- On.AEou(~o), dx B OX A ,

in which the operator d/dx B is defined to be

d O A O A O (6.3) = dx B OxB + O,B ~ + O,cB OOAc"

From (5.11) it follows that

0 ~ 0 OH OSo (OH+O+~ (6.4) Ox j = v -~x~' Ot = v \ ot Ot ] '

and thus, when the complete set of Euler-Lagrange equations

(6.5) EoB(~LPo) = O, B = 1 , . . . , n + 1,

is satisfied, the identities (6.2) yield

d /~ dH~ +' OH (6.6) dx j + d ~ = - V o x h,

together with d / ~ + , ~)'"+ ' (0H + 0q)' 1 ~ " n + 1 --V

(6.7) dx j § dt \Or Ot ]"

Clearly (6.6) and (6.7) represent conservation laws whenever OH/Ox j = O, OH/Ot = O, and 0~/0 t = 0, and accordingly we shall now explicitly evaluate the elements (6.1), that is,

~ . ~ o 0 A (6.8) HI,= -Lf03],+ ~ ,h,

(6.9) ~ J O~-~WO 0 A H,+ 1 - - 00-~-jj , t '

1 0 ~ o a (6.10) /4~.+ = ~ - 0j , ,t

(6.11) ~ = 0~~176 0 a H.+ - ~ C P o + ~ ,v

For the case when n is odd, we make the identification

(Q1 . . . . . Q(n-1) /2 191, . . . , P(n-1)/2, I~, V) (6.12)

= ( 0 1 . . . . . O(n-- 1)/2, o(n+ 1)/2 . . . , O n- 1 On, on+ 1),


while, when n is even,

(6.13) (Q1 .. . . . Q,/2, P1 .... . P,/2, v)=(01, ..., 0 "/2, 0 "/2+ 1 . . . . . 0", 0 "+ 1).

With these conventions the two cases can once more be treated simultaneously, it being understood that the derivatives of 4' are to be suppressed whenever n is even.

Since 5r o does not depend explicitly on derivatives of Q" and v, it follows from (5.11) and (5.3) that

`9-Wo 0 A `95fo ̀ gP~ `9~o `9~0 (`gH `gP~ `gH `94') `gH /`94' _~`gP~ - - - - = v - - ~ S - ~ + ~ `gx~ I , ,h `gp~,j `gxh + `94',j `gX h =Y - ~ , j `gxh ~ `94',j ~X h `gpj

or, in view of the Clebsch representation (2.6),

OLeo A OH O'h=VPh ̀ gpf

`9~o 0A=`95eo c~ `9o~j ,' `ge~,~. `gt

`gYo `9~o 0P, ~ , ~ OxJ

(6.14)

Similarly,

(6.15)

together with

(6.16)

and

(6.17)

`9 ~ o `94' OH/`94' + ~ ̀ g P~ \ ,

`9~o ,90 (`94' ~ ̀ gP~ \ `94',, `gx j = v ~x j + Q ~x~)=vp j ,

`9o a ,~-`9~,, ~t - - - - ~ ] )

`90. 9̀~ i~ -+u ~7)

On introducing the notation

(6.18) P. + ~ = ~ft + Q ~ `g P~ `gx i

once more, we can now represent (6.8)-(6.11) with the aid of (6.14)-(6.17) as a bordered (n + 1) x (n + 1) matrix:

/ j `gH `gH \ -~ [--~06h+VPh~--' vP"+I `9p~

(6.19) H n = ~ cpj

\ VP h, --~o+VPn+l/"

Evidently the terms /t~, H~,+ ~ are invariant under Clebsch gauge transformations, whereas this is not the case for the entries in the (n + 1)th column of(6.19). However, a gauge-invariant complex may be constructed with the aid of (6.19) in the following manner. In terms of the notation (6.3) we may write

d ,vo,+ ivo dt dxJ ~ `gp/=~ [~S+~ v ~

or, if we use (2.11),

326 H. RUND

~(v+)+J~ [v+~ O ( OI-It] dt \ ~pj] [ ~ - + ~ x j \ Opjl_l

(6.20) or dQ" [o,+ o, ee 1

+ v L & - OQ ~ dt ~ - ~ dt J"

This relation is merely an identity; but, when it is assumed that the Euler-Lagrange equations (5.15) are satisfied, it is found that

(6.21) ~ ( v ~ ) + T x s v~ =v 0t '

which represents a conservation law whenever Oq)/Ot=O. This state of affairs suggests the construction of the complex

(6.22) T~ = ( - S f ~ OH 0pj'

\ v p h,

which is related to (6.19) according to

(6.23) ~J =/4~,

(6.24)

v(P"+l+~)O-~----~jpj ] ,

-- ~Wo + v(p,+ l + ~ ) /

T.J+ 1 =/~i+ 1 + v~ OH 0p2'

~.+1=/~+1, fr.+ll_~.+l+v~.__.+l

From (6.6), (6.7), and (6.21) it then follows that

(6.25) dThJ dZ"+l h OH dx j dt =--V Oxh,

(6.26) dT~+x dT-"+ 1 + n+l OH

dx j dt - v Ot'

of which the second gives rise to a conservation law whenever OH~Or = O. Moreover, the Clebsch gauge-invariance of each element of (6.22) is a direct consequence of (4.20). We shall therefore regard the complex (6.22) as the energy-momentum complex associated with the Lagrange density (5.11). In terms of such an interpretation the element - 7","++~ would have to be regarded as the energy density associated with the system. It is remarkable that, after substitution from (5.11), this term assumes the simple form

(6.27) - T."]~ = vH(t, x h, Ph).

In conclusion it should be observed that this construction of the Hamiltonian and the energy-momentum complex is valid also when the function H depends on v (see the remark at the end of section 5). This is immediately evident from the fact only derivatives of v with respect to the independent variables t, x h (which are absent) can contribute to the formal structure of the right-hand side of (6.1).


7. Charged Particles Subject to Electromagnetic Fields

An important application of the theory outlined above is represented by the unified treatment of systems of charged particles subject to electromagnetic and other effects (such as pressure or gravitational forces). This case is of particular interest, since the presence of electromagnetic potentials is reflected in the expression for the generalized momenta p j, and therefore also in the Clebsch representation of the latter. This in turn requires that the complete Lagrange density (5.19) be constructed, where ~ I now refers to the electromagnetic field.

Let us begin, however, with the construction of Yo as defined by (5.11), which involves the explicit form of the Hamiltonian H(t, x h, Ph) of a particle of mass m and charge e, subject inter alia to an electromagnetic field. The latter is supposed to be represented by a 4-potential (A,~), giving rise as usual to the magnetic induction/3 and the electric field strength E:

(7.1) BJ=g jhk OAk ~q5 1 OAj ~xh, E j = - Ox j c t?t '

where the constant c denotes the velocity of light in vacuo. Additional forces acting on the particle are assumed to be represented by a potential V(t, x h) per unit mass. The (non-relativistic) Lagrangian associated with the particle possesses the well known form

(7.2) L = 1 m ~ j ~ j ..[_ ec- 1Ai2j_ e(a - mV,

where 2J = dxJ/dt (j = i, 2, 3). Thus

3L e (7.3) PJ = ~ 3 = m&J +-c A j,

and consequently the corresponding Hamiltonian is given by

(7.4)

Therefore

(7.5)

H(t, xh, ph)= ~-~ (P~-e Aj) ( P j - e A j ) +e(~+mV.

dp. c~H . e (c3A~ c~Ah' ~ e4) , c3V ~ t +~xSxJ= mSCJ + - ~n + e ~-) + m ~-LS, c \~x h ~x j l vx ~x

or, if we apply (7.1),

(7.6)

0H . e t~V dPJdt ~-~xj=mSCJ +ceJl~B~ych--eEj +m--~x j

where v j = ~J, the subscript j indicating the jth component of the vector in square brackets.

= m ~ - e E+ (,]x/3) + m V V , ~j

328 H. RUND

Since n= 3, the Clebsch representation (2.7) reduces to

~O aP (7.7) pj = ~ x j + Q Ox J,

and thus, because of (7.4), the Lagrangian (5.11) now assumes the form

50o=V [ ~ + Q Tt +rb(t, Q,P)+e~p+mv (7.8)

1 /Off -OP e Ak [dO OP e A]]

As a result of the appearance of the additional field functions At, q5 in 500 it is now imperative that the Lagrangian of the variational principle which governs the behavior of these fields be adjoined to 500. It is well known that this Lagrangian is given by

(7.9) 50f ~--- 1/3 0 ( B 2 _ E2),

where e o is the (constant) permittivity, it being understood that/~,/~ are represented in terms of the 4-potential in accordance with (7.1). The Euler-Lagrange expressions associated with (7.9) are

E~s(50s)=eo [-curlB+~ ~-t ] s, (7.10)

and

(7.11) E~(50y) =e o ?E~=e o div/~. ~x s

But from (7.8) we also have, with the aid of (7.3) and (7.7),

E ev (pj ~Aj)=~_vj, ,4j(50O)=mc - - ev . (7.12)

and

(7.13) E~(50o) = - ev.

Therefore, for the complete Lagrangian

(7.14) 5~ = 500 + 50y,

it follows from (7.10)--(7.13) that

(7.15)

and

(7.16)

Eaj(50)=eo [ _ c u r l / ~ + l 0/~ lev ] ,

C C j

E~(~) = eo [div/~ - e o i ev].


Since the Euler-Lagrange equations resulting from (7.15) and (7.16) are expected to be Maxwell's equations, it is evident that the scalar field v -wh ich had been introduced as a purely analytical device in section 5 - is to be interpreted as the particle number density; that is, if we denote the mass and charge densities respectively by p and Pe, we must put

(7.17) p=mv, p~=ev.

This interpretation of v is substantiated also by the fact that, under these circumstances, the Euler-Lagrange equation (5.16) guarantees the conservation of mass and charge. Moreover, if we denote the current density p~ 0 = e v 6 by J, we can now express (7.15) and (7.16) respectively as

[-curia+} lsl; c (7.18)

and

(7.19) E.~ (,,.~) =% [div/~ - % 1 pe],

while (7.6) is equivalent to

(7.20) dO + 1 +

The Lagrangian 2,e I depends solely on the 4-potential and therefore does not contribute to the Euler-Lagrange expressions of LP with respect to the functions Q, P, 0, v; accordingly these expressions are identical with those resulting from s being given by (5.12)-(5.14). Thus we may replace ~o by L,e in (5.18), which, because of (7.20), now becomes

~Q OP

(7.21) =[pdO (+ 1 B ) } + p V V ] . ~-pe~E+?(0•

Therefore, as expected, the four Euler-Lagrange equations

(7.22) Eq(s = O, Ee(s ) = O, Eo(~ ) = O, Ev(~ ) = 0

imply the dynamical equations of motion

dO (~ 1 I p ~=Pe~E +c(O X I2I )j + p V V,

together with the equation of continuity which results from (5.14):

(7.23) 0p 4- div (p0) = O. ?t

Moreover, the remaining four Euler-Lagrange equations

330 H. RUND

Eaj(~e)=0, E,(~e)=0 yield Maxwell' s equations

(7.24) div/~ =Co 1 pe, cu r l /~=1 t ? E + l e~ 1 j . c •t c

[Needless to say, the second set of equations usually associated with (7.24), namely

(7.25) div/~ = 0, curl/~ - c 0 t '

is a direct consequence of the representation (7.1) of/~ and/3. However, in passing it should be remarked that the representation (7.1), together with the Maxwell equations (7.24), can be derived from a single variational principle whose point of departure is a Clebsch representation in 4 dimensions of the electromagnetic field tensor

F OAc OAB (7.26) BC= ~x B ~?xC ,

where B, C = 1,..., 4, x* = ict, A 4 = iO ([7], section 6). Also, when V=0, the density (7.14) reduces to a Lagrangian proposed by SELIGER & WHITHAM ([10], p. 19) for plasma waves; this reduction is easily effected with the aid of (7.3) and (7.4).]

Now let us consider a more general situation, it being assumed that the Hamiltonian (7.4) depends also on v by virtue of an additive term U(v), that is,

(2.27) H(t, x h, Ph, V) = H(t, x h, Ph) + U(v).

Since v is regarded as a field function, this does not affect (7.5), (7.6), or (7.20). However, according to our remarks at the end of section 5, we must now use (5.26) instead of (5.18) in order to obtain the counterpart of (7.21). Because of (7.27), the additional terms which appear on the right-hand side of (5.26) are given by

[ ~ 0v ] U") Ov v ~xjxj(VU')+U'ff-~M = ( 2 v U ' + v 2 a#"

This, together with (7.20), is now substituted in (5.26) to yield

(7.28)

OQ OP o

= Vv . P)-~-P~ E + I ( g x B ) + o V V + ( 2 v U ' + v E U '') c j

In order to be able to interpret the additional terms on the right-hand side of (7.28) in terms of a pressure gradient, it is now supposed that an equation of state in the form

(7.29) p = f ( v )


is valid, so that

(7.30) Vp=f'(v) Vv.

Thus, if we choose the function U(v) in (7.27) such that

(7.31) v 2 U'(v)= f(v),

it follows that

(7.32) [2v U'(v) + v 2 U"(v)] Vv = Vp.

According to (7.28) the Euler-Lagrange equations (7.22) now imply the Eulerian equations of motion

d~ ~ c (7.33) p ~ =-Pe {E + l (g x B ) } - p V V - V p,

the other field equations (7.23) and (7.24) remaining unchanged. In conclusion we shall briefly consider the energy-momentum tensor of the

system described above. As is well known, the energy momentum tensor associated with the Lagrangian (7.9) is given by

f (7.34) Tff =~o(FAC FBc-�88 6A FC~ E ~ C D I ,

so that the total energy momentum complex of our system, namely

f (7.35) Tff = Tff + T if,

may be obtained directly from (6.12) and (7.34), where it is to be understood that the f

Hamiltonian (7.27) is to be substituted in (6.22). In partiuclar, since - 2 T 4 = eo(E 2 + Bz), it follows from (6.27) that the energy density is given by

- T 4 = �89 eo(E 2 + B 2) + v H(t, x h, Ph, v),

or, more explicitly, because of (7.4), (7.27) and (7.17),

(7.36)

2 V 81// 8 P e 8~9 t?P e -T2=�89176 )+~m (~xj+Q ~ x j - c Aj) (~xj+Q----8x j c Aj)

+peCk + p V+ v u(v).

The appearance of the last term on the right-hand side of (7.36) should not be surprising. For instance, in the case when the function f(v) in (7.29) is of the form My 7, where M and y are constants, it follows from (7.31) that U'(v)=My v-2, or

V(v) = ~ _ ~ v~- ~.

332 H. RUND

Under these conditions the term in question is simply

vU(v)= p y - l '

as expected. The expression (7.36) may, in turn, be regarded as the Lagrange density of a 3-

fold integral problem in the calculus of variations, the independent variables being the spatial coordinates (x x, x 2, x3), while the dependent functions are once more At, 4~, ~, Q, P, v. Since the corresponding action integral represents the total energy of the system, the variational problem thus constructed is of relevance to stability problems in plasmas, and has indeed been recently used as such in the presence of various constraints, the latter being prescribed by physical considerations [11]*.

Appendix

The analysis at the end of section 2 depends on the following simple lemma concerning Lie derivatives:

Let o9~ .... j~(t, x h) be a type (0, m) tensor field, whose transform under the time- dependent transformation (2.39) is given by

(A.1) ~xJ~ 63xJm

&, ...h,,(t, uZ)= ~j~ ...~(t, x ~) auh, auhm,

it being assumed that the Jacobian (2.40) is non-vanishing. Then the differential equations

(A.2) 2v ~oj, ... j,, = 0,

where

(A.3) v j = ~3xJ(t' uh! Ot

are equivalent to the conditions

(A.4) O2h .... hm =0. &

Proof. From (A.1) we have c~uh, ~uhm

(A.5) %~ ...j,,(t, x ~) =2< ...hm(t, u ~) ~xjl . ~ x j ,

SO that, by virtue of (A.3),

* In this connection the writer acknowledges with pleasure many valuable discussions with Professor D.R. WELLS of the Physics Department, University of Miami, which stimulated much of the analysis presented above.


a~ ~ ag~ ...j= ul a2hl...hm ~U hi aU hm

Ot ax t & axj~ ax~m

(A.6) au < d {aua.~ 1 au am + ~ "~h .... h,,, axa~ "'" dt \axJa! ax j"" //=1

But from (A.3) it follows that

d5 \ a x l l ~R m-{ a x l abl m '

so that

7i '--'\axJl ax' axJ

This is substituted in the sum on the right-hand side of (A.6) to give

~ au h' au ~e au h= av'

-- Ahl...h m ax j l aX l aX j= axJ#' #=1

which, because of (A.5) is equivalent to

av' -- O.)jl...jB_lljB+l...j m axj# "

/~=1

Thus (A.6) assumes the form

awj~ ...j~ ~, av l a~ ~ 1)l-~ (Djl . , . j#_ll j#+l. . . j m axJ#

at ax l ~= 1 (A.7)

_a2< . h= au h* auh-~ ~

& ax jl ax;~"

However, the left-hand side of (A.7) is simply the Lie derivative of coj1..4,, and accordingly it is inferred that (A.3) implies (A.4), and conversely.

Note added in proof. In the analysis above the assumption that the Clebsch potentials be independent was made repeatedly. It was pointed out by RICHARD BAUMEISTER that this implies unnecessary restrictions which can, in fact, be avoided by a modification of the construction of Clebsch representations as outlined in section 1. For a given field pj, let c denote the rank of the matrix (~o~j, pj), where o0hj is defined in (2.34). The integer p introduced in section ! is then given by c =2p, or c = 2 p + 1 , according as c is even or odd ([3], pp. 128 129). The Clebsch representations (2.6) and (2.7) can be replaced by

pJ=~+ __Z Q cT~J' where e = 0 or 1 according as c is even or odd. This device guarantees the independence of the Clebsch potentials. Nevertheless, an improvement of the argument given in section 2 establishes once more, for a given canonical field p j, the existence of a superpotential ~b(t, Q~, P~), for which the equations (2.25) are satisfied, irrespective of the value of c. For details of this and similar subsequent modifications, reference is made to Mr. BAUMEISTER'S forthcoming P h . D . dissertation, to be submitted to the University of Arizona in the fall of 1977.

Note. This research was sponsored in part by the National Science Foundat ion under Grant NSF G P 40370.

334 H. RUND

References

1. BATEMAN, H., Note on a differential equation which occurs in the two-dimensional motion of a compressible fluid, Proc. Roy. Soc. London Ser. A 125, 598 618 (1929).

2. BATEMAN, H., Partial Differential Equations of Mathematical Physics, Cambridge University Press, London and NewYork, 1959.

3. CARATr~EODORu C., Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Teubner Verlag, Leipzig und Berlin, 1935.

4. ECKART, C., Variation principles of hydrodynamics, Phys. Fluids 3, 421-427 (1960). 5. GOURSAT, E., Leqons sur le Probl6me de Pfaff, Hermann, Paris, 1922. 6. RUND, H., The Hamilton-Jacobi Theory in the Calculus of Variations, Van Nostrand, London and

NewYork, 1966; augmented and revised edition, Krieger Publishing Co., Huntington, N.Y., 1973.

7. RUND, H., Generalized Clebsch representations on manifolds, "Topics in Differential Geometry", Academic Press, New York, 111-133 (1976).

8. RUND, H., Variational problems and Bgcklund transformations, "B~icklund Transformations", Lecture Notes in Mathematics, Vol. 515 (ed. R.M. Miura, Springer-Verlag, Berlin-Heidelberg- New York, 199-226 (1976).

9. RUND, H., Clebsch potentials in the theory of electromagnetic fields admitting electric and magnetic charge distributions. J. Math. Phys. 18, 84~95 (1977).

10. SELXOER, R.L., & G.B. WHITHAM, Variational principles in continuum mechanics, Proc. Roy. Soc. London Ser. A 305, 1-25 (1968).

11. WELLS, D.R., Mass flow stabilization of plasmas confined by helical magnetic fields, Pulsed High Beta Plasmas, edited by D.E. Evans, Pergamon, Oxford and NewYork, 123 (1976).

Department of Mathematics University of Arizona

Tucson, Arizona

(Received January 20, 1977)

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