I

DYNAMICS OF CONTINUA AND PARTICLES FROM GENERAL CQVARIANCE OF NEWTONIAN GRAVITATION THEORY

Ch. DUVAL * and H.P. KUNZLE

Centre de Physique Théorique, C.N.P.,S. Marseille

ABSTRACT : The principle of general «variance, which states that the total action functional in General Relativity is independent of coordinate transformations, is shown to be also applicable to the four-di:nensional geometric theory of Newtonian gravitation. It leads to the cor­ rect conservation (or balance) equations of continuum mecnanics as well as the equations of motion of test particles in c gravitational field. The degeneracy of the "metric" of Newtonian space-time forces us to intro­ duce a "gauge field" which fixes the connection and leads to a conserved current, the mass flow. The particle equati'oreare also derived from an invariant Hamiltonian structure on the extended Galilei group and a minimal interaction principle. One not only finds the same equations of notion but even the same gauge fields.

JULY 1976 76/P.849 x U.E.R. Scientifique de Marseille-Luminy

** Supported in part by the National Research Council of Canada

+ On leave from Department of Mathematics, • University of Alberta, EDMONTON (Canada T6G 2G1) POSTAL ADD5ESS : Centre de Physique Théorique, CNRS 31, chemin Joseph Aiguier F-13274 MARSEILLE CEDEX 2 France -2-

1. INTRODUCTION

The covan'ant space-time formulation of Newtonian gravi­ tation theory goes back to Frank [f| and Cartan P2j and has since been developed by various authors £3-8 J • The Newtonian gravitational field equations and the balance equations of continuum mechanics are" here formulated in terms of a connection and its curvature on a 4- dimensional space-time exactly as in General Relativity theory, except t.iat the Lorentz metric is replaced by a Galilei structure [7l, i.e. essentially a degenerate "metric" that singles out a class of local reference frames related to each other by homogeneous Galilei trans­ formations. Neither the equations for particles, nor for fields were obtained from a covariant variational principle, but rather they were found by analogy with the relativistic theory.

In fact, it seems that such a Lagrangian formulation can­ not exist, at least not in terms of the given variables. For example, on the tangent bundle of flat space-time taken as evolution space E of an isolated 1-particle system, there does not exist a Galilei invariant Lagrangian, while a Lorentz invariant one does [91. In par­ ticle mechanics the problem can be solved in two ways. Either one keeps the original evolution space but gives up the Lagrangian and requires only an invariant Hamiltonlan structure to exist (i.e. a presyrcplectic 2-form C on TlR. which defines- both equations of motion and Poisson brackets), or one enlarges the evolution space E to tome E such that a projection mapTTrE —•* E exists and attempts to find an invariant

• 1-form 9 such that el & B TV*V . Such a Cartan 1-form 6 is normally obtained from a Lagrangian (e.g. &= |^=^ dc\ L9>10J)>

but it may exist in cases where no Lagrangian is^defined (like on certai evolution spaces for particles with spin [_11 » 12J ) and replaces tne la ter for almost all purposes. In particular, the minimal interaction principle can be directly applied to Q (though not to G~ ), In

6/P.849 the last section we will illustrate this procedure by constructing a model of particle with spin in a Newtonian gravitational field starting from group theoretical considerations and the minimal interaction prin­ ciple. The same procedure starting with the Poincaré group would lead tc the corresponding relativistic model in a curved space-time.

For a classical field theory, it is not yet too well under­ stood what local differential geometric object corresponds to the Hamil- tonian structure in mechanics (for attempts to identify this-object see [l3| ). On the other hand it is always possible to enlarge the number of variables on which the Lagrar.gian density may depend (i.e. to intro­ duce some sort of "g^uge fields"). It might give new insights into the structure of field theories to be able to derive the covariant form of tr.ô Newtonian equations of gravitation this way. But this appears to be a non trivial open problem.

What will be solved in this paper is the related problem of how to derive the correct conservation (balance) equations of non relativistic continuum theory from the principle of general covariance. That means we require some action functional to be invariant under arbi­ trary space-time coordinate transformations as well as gauge transforma­ tions for the introduced gauge fields. This approach is well known in General Relativity (see, e.g. [}{]). We wish to show that the same principle applies just as well to the classical Newtonian theories.

The procedure is, however, somewhat less straightforward. It cannot be based directly on the Galilei structure and the connection as primary variables. Vie find it necessary to introduce two "potentials"

A rt and LA that together with the Galilei structure determine the connection but not in a one-to-one way. Thus there appears a "gauge group", invariance with respect to which implies that the corresponding current is conserved. In this way the concept of mass current appears in the Newtonian theory separately from the stress-energy tensor, in

76/?.349

f -<;-

contradistinction to the raiativistic case, similarly as the mass appears in a different role in Galilei and Poincaré invariant mechanics, res­ pectively.

The principle of general covariance has been cast into a particularly elegant and effective form by Souriau [l5j which permits it to be directly applied also in the limiting case of matter distribu­ tion with support on a single worldline. Then it yields the equations of motion for test particles in a given external field. He obtain this way the equations of a particle with mass, spin and quadrupole moment in a Newtonian gravitational field, and find that they agree with those deri­ ved by means of the minimal interaction principle.

Section 2 reviews Souriau's approach to the principle of general covariance and test particle motion in General Relativity; section 3 the basic definitions for Newtonian structure. In the next two sections we derive the balance equations for continua and the equa­ tions of motion for particles, respectively, in the Newtonian theory. The last section deals with the Hamiltonian model of spinning particles.

2. PRINCIPLE OF GENERAL COVARIANCE

The Palatini method of considering the stress-energy tensor I ' of rn.it.tcran d non gravitational fields as' the variational derivative of a Lagrangian density with respect to the space-time metric ^.v has been restated by Souriau |_15j in the following form : Consider the set S of all Lorentz metrics â«/i on a given 4-dimonsional manifold V and a 1-form J" en the manifold

' ' S is an open subset of the set of sections of a vector bundle and as such has the structure of an infinite dimensional manifold. Ke will r.ot make this more precise than is usually done in the context of variational calculus used in physical applications (see, e.g., ricH).

76/P.849 -5-

*/% ^ (2.1) 'V ' where the variation °*J is interpreted as a tangent vectorto 5 at g and the integral required to exist.

The principle of general cqvariance now demands that J be a 1-form not only on S but rather on the quotient "manifold"

H =$/$*& (v ) (where "*>'$0(V ) denotes the group of all C °° diffeomorphisms of V with compact support) since any two'metrics related by a diffeomorphism are physically equivalent. This means thatf J~' must vanish on all "vectors" H. tangent to the oroit of g under $;# [v) in S, in other words that

for all vector fields fc on V with compact support (where «fç denotes the Lie derivative v/i th respect to J" ). This is well known to be equivalent to the conservation equation of the stress-energy ten­ sor T71 ,

v^-r-y» = „ (2.2)

Here the connection has been taken to be the unique sym­ metric Lorentz connection. One could replace S by the set ^ftV'^/O lV*J/*r = ° } without requiring that F^ be symmetric and (2.1) by '

TcrCv^ffjrrjj^ e'C IV^ }^3 (2.3) then ?

M;i^v©T^w, V

75/P.849 -6-

(21 leads again to equation (2.2) if one defines* '

T*.f*+ w^©•CM -r? -.*{•<-© -\ &' : y <2-4> n 1 r\~A and now requires that 1 «A and Vyi are symmetric. If, however, torsion is admitted as, for example, in the Einstein-Cartan theory flT], \^„ can be decomposed into the Levi-Cività connection

ant a cont W/s î * orsion tensor. Part of ©y is again ab­ sorbed in T i and the rest interpreted as a spin density which will appear as a source term in (2.2). (The ©-' has been called hyper- momentum by Hehl et al. [l8]). Since in this principle only the (global) functional is considered as a physical quantity it is clear tnat in (2.3) ~T"fl and ©»' could only be individually well defined local fields if there were no constraint at all between %*e/± anc* ^«/i • ^ l~v/i is to be a symmetric lorentz connection, then either of T'y1 or G^J/1 can be dropped and the physical stress-energy tensor defined by (2-4) (with either "T"*/1 or ©*/" equal to zero). Then equa­ tion (2.2) will still be a consequence of the general principle- of co- variance.

The advantage of Souriau's approach is that the same prin­ ciple also leads very easily to the equations of motion of test particles in the gravitational field (and other external fields). The method is summarized as follows : Suppose the functional J has its support on a worldline A in V and is a distribution of, say, no higher than second order. Then it can be assumed to be of the fora

7Lhl,\i.T\.©firmer'» }* "••' A '

(2) ( ) and f "1 denote symmetrization and antisymraetri2ation respectively.

76/P.849 -7-

where t is any curve parameter of -A and \ ' s o < ^-^z - » and Q} /** has the same symmetries as the curvature of a symme­ tric Lorentz connection. By the principle of general covariance we have H«^V^r?-0 VW*

for all vector fields v on V with compact support.

^ In particular, for J =• /\^i, £ where A and B are t* -functions vanishing on j\ and £ any C vector field with compact support equation (2.6) leads to o = f ©7120* £* d* A £ wnich is equivalent to \ÙJ beinç, of the form for some M . -^ , where V is the tangent vector to A with respect to t. Using this and now putting Ç^r A Ç * with /{|j\ = O we have from (2.6)

A where • :- V Vj, and integration by parts has been used. Equa­ tion (2.7) is equivalent to

76/P.849 for some In defined along -A. . Substituting again into (2.6) gives

X Vr Q \?0 (2.9) - M + i< - < T"f Define pi* 6*: r .2 then (2.9) becomes

x î --jR% v 5

Z. * '^ Equations (2.10) and (2.11) are now fairly accepted as equations of motion of a test particle if O ' is interpreted as its spin, x as its 4-momentum and Q / as its (mass) quadrupole moment (e.g. F19» 15, 201 arid references given therein).

It is well known that they do not form a complete set of equations for all dynamical quantities necessary to describe the state of motion of a test particle, unless additional conditions are imposed, the most natural being

(when çy* r 0 ) [l9. 12, 20] . Note that again not all of the coefficients T^1 , £jV • Q7i'r' ln equation (2.5) can

76/P.049 -9-

be interpreted physically in an unambiguous way. To obtain the equa­ tion of motion of a monopole particle ©_/^ and Q r can be assu- med zero. For the dipole or multipole case the term T" "/ is not necessary.

uanoxwi STRUCTURES

A Galilei structure on a four-dimensional C -nr-nfold V can be given by a pair of tensor fields (fA & ) "here Y^ is symmetric and % spans the kernel of "/"A at every point of V. (See L?J for details and proofs for all the material of this section.) A vector U is called unit timelike if U H\, = ' > it is spacelike if U ^ = o . The three-dimensional distribution

1 of hyperplanes Sx fom.ed by a ! spacelike tangent vectors at each K. G. V* is completely integrate, in particular, if the 1-form !/< c p^ cix.'1 is closed : dy - o • The leaves of this folia­ tion then carry a natural Riemannian metric induced by J7 .

Linear symmetric Galilei connections (i.e. satisfying

V* ^ =o and 7* ^ a 0 ) (3.1) exist if and only if Tp is closed and are determined up to an arbi­

trary 2-form F = '/_; r«„ lx "A eU P • In fact, the general solution of (3.3) for syxnetric p"f can be written in the form c;*<. £yr.j,^,, ,r, r *v^ (3.2) where U is an arbitrary given unit tinelike vector field and

76/?.849 -10-

n ' •* I *n is the umq-.c symmetric Galilei connection such that U . is geodesic and «rotational, i.e. such that ([3, p. 197])

U V U = o and V U1 = o (3.3)

(where indices are as always raised by means =f yr,.

The curvature tensor of any symmetric Galilei connection has the properties

We will only consider ? subclass of sytrmetric Galilei connections, called Newtonian connections which are characterized by satisfying in addition

or. equivalently, by

dF : 0 (3.5)

-/» any Galilei connection that is obtained as a limit of symmetric Lorentz ' connections [21^. It means that we confine ourselves to conservative forces. We will call the triple f"Y /, ip , V\,„ ) on V a Newtonian structure if (3.4) holds.

A unit timelike vector field U (which can be con­ sidered as an observer field) is also necessary to define a "generalised inverse of If 7* by means of

? y u r o «d \J^c S,f- c^uf* (3.6) "'J -J

76/P.049 -11-

Any orientable Galilei manifold carries a canonical volume element vol which can be written as /\ Vol» where Vol , is the Riemanni3n volume element of *S •

It has been shown in [7] that an asymptotically Euclidean Newtonian space-time satisfying the field equations

is completely equivalent to the classical Newtonian gravitation theory {with a potential *u satisfying the Poisson equation ^^ ?ttnkp in Euclidean S-sp^ce).

He end this section by deriving some formulae that will later oe needed. Covariant derivatives will always refer to P^» given by {3.2) except when specified otherwise. From {3.2) w. have ' f F ^"H--VVf1 P X > -i î thus, in view of (3.3),

Wr V l/ -- - F* U? (3.7)

VC- « ^ 1 F*/1 s 2. Using (3.6) we can derive from thi>se equation..

X F, = .A Vxr„ V, U (3-9)

The general variation of "^ is obtained from (3.6) directly as /

76/P.349 -12-

(3.10 M Y(< p.) A a particular case of which is

v i (3.11) ^v^ V^

4. BALA.-;C£ EQUATIONS FOR CONTINUA IN THE NEWTONIAN THEORY

Taking S now to be the set of all Galilei structures (-%T tyg } on V , one might think of replacing (2.1) by

But as shown in [211 the balance equations following from the general covariance of this functional lack the acceleration term of the classical

equations. This is not surprising because (if ^ • PK ) do not fully determine the connection. Even if we require the connection to be sym­ metric, we must therefore take this functional to be f. ^u*, m-J^y^ *si* ®?K} } "* (4-i) where ^ r In the relati- vistic theory the o Pj« can be expressed in terms of o'2~s, cor a Newtonian connection p ^ compatible with the Galilei struc­ ture WÛ have by (3.2,5) ' v1 - rT ip V

76/P.E19 -13-

where 1 ^A is characterized by (3.3) in terms ^f derivatives of [V r, 4"a~ 1 . and a unit vector field u . A ^ is an arbit. ary 1-form. The variation ^Hx/i can thus be expressed in terns of ile W could be considered fixed. However, there is no canonical choice for U. and it seems therefore best to regard the connection fixed (in a some­

what redundant way) by ( *$ J / if/^ ) and the pair (A^, U )• Thus (^.1) can be replaced by

%{M^)y-[\-i^u'^^st* J"f/I«+ K.^J.W (4.2) We now require that be a 1-fonn on the manifold of equivalence classes of Newtonian structures. Thus J must in prrticular be left invariant by any "gauge -transformation", i.e. a change ("f^ £ A'U"J"* C 7"^£/^*, ù*J that does not change the Newtonian structure (TA^, Pj%) • To find the infinitesimal gauge transformation suppose therefore that »?/* $"<&. *• S^JL * o . Then IU Su. : o and varying equation (3.9) gives {in view of (3.10, u)) V U - 27 P AI A[* pip a H f - 2 Tf VSue *** v V ft r Pi - ur *\ (w )• But 2.1 è a r therefore we find "y F- \i - Ï* Vi

In other words variât ons such that S "Y ' T o , atfj ° ° anc!

4 **.. •r-/

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for any W and

Therefore «J must vanish for all such variations and we have from (4.2)

for arbitrary spacelike gU and ^ with compact support. This implies that VJ*--. (4.1)

where K. is an arbitrary scalar.

Using (4.5) we now state that sj must vanish on varia­ tions that are infinitesimal diffeomorphisms with compact support, that is

by Stokes' theorem and (4.4), where

76/P.819 -15-

Since (4.6) must hold for all vector fields Ç with compact support, we have 7p^--^pVft V Jfi- (4-8! Equation (4.4) states that O is a conserved current which is most naturally interpreted as the mass flow. Assume therefore * that

-X*r « V* } V* tfa = j (4.9) and confine considerations to a region where the mass density y° is positive. Now, with the help of (3.9) expression (4.3) can be written as

Vp^ .. . j,£A ^ «4.10» with ' : a V V . But we wish to replace all references to * it *i the arbitrary field U by one to the physical matter flow V . It is easily seen that V " V * * ^ v/»> ~v*f- ^ where v^y/" n 5.-^V . Then fro.-n (4.10)

V

76/P.849 -16-

where (3.11) has been used as well as

But by (4.4) and (4.9) this can be written as v-- - x V^ (412) provided one defines

Equation (4.12) is the closest analogue to the relativistic conservation law (2.2) one can find in the Nev/tonian theory, since it is not possible to absorb the right hand side in the divergence. The relativistic stress energy tensor therefore decouples in the Newtonian theory in a tensor describing stresses and energy flow, and the matter flow vector J" - a V . where TVK has the property

which follows at once from (4.7) and (4.13) since Xryy^-i - c> This description seems quite satisfactory since it is rather analogous to the situation in particle mechanics. Contravariant syrrr.etric tensors "T y have also been used in Newtonian theory (e.g. [8])» but they do not describe stresses and energy (flow).

Just as in the relativistic theory the tensor can be decomposed in different reference frames and then interpreted physi­ cally. The most canonical one is the "matter frame". Therein view of (4.14)

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/ -17-

where °| Yu = o and ^^3 = ° = J^ V , ana one can interpret £ as the specific (non mechanical) internal energy den­ se . sity, <* US the energy lor heat) flow and b- . as the stress tensor. This decomposition corresponds to the well known one in Relativity where a nutter flow vector p V is introduced ad hoc, namely to

Kith CJ Vj r o , f- I V» « O

It was shown in [2lT that equations (4.12) for the decora- position (4.15) are equivalent in a flat Galilei space-time to the clas­ sical, balance equations of continuum mechanics, namely to

A where V = ( 1, V ) , /I = I,?, 3 and k -- - PDC, - 2 ?0^ V ' is a body force density expressed in terms of the only non-vanishing components of I~I

EQUATIONS OF MOTION OF TEST PARTICLES

We now show that the principle of general «variance applied to the 1-form 3" on the manifold of .Newtonian structures on V also implies the correct equations of motion of a test particle just as in the relativistic theory. This cou'i I again be done for a monopole or any multipole particle, but we will perform the calculation only for a parti­ cle having mass, spin and a quadrupole moment.

76/P.849 -18-

The starting point is thus the Newtonian equivalent of equation (2.5), that is, in view of (4.2), ***

whore .A. is again a worldline and t any curve parameter. In contradistinction to the relativistic case it is not possible to drop

all the "potentials" $ f*/* » $ «fot , £A u and Ju" without loosing the mass term in the equatioreof motion. This is rela­ ted to the different role the mass plays in nonrelativistic and relati­ vistic theories and will be discussed from another point of view in the next section.

First we exploit again the gauge invariance (4.3) of T, namely

O* J -flJ1 «.„ «• '-;*"• •+ d «i, * 1 *t (5.2) AA J for arbitrary spacelike and tX with compact support. For

W £M : o and JC = £_£ , A |A = o wo have

o = [ I* £ A JC otb for all _X A and therefore J = MV if V is the tangent vector to -A with respect to t, which we will now choose such that V <£\ «I •an d where u is a scalar.

"^ Now »= Uv )t

for all X with compact support where '.'sV^is the derivative with respect to worldtime alongVV • Thusni^oand n is a constant of the motion,obviously to be interpreted as the mass of the particle. ChoosingJU» and Su"arbitrary spacelike we find from(5.2),similarly as in (4.5)

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for an arbitrary scalar \e. (where (1.11) has been used).

The invariance under diffeomeorphisnis now gi/es

• A***

for all r with conpact support, where

As in section 2 we successively choose Ç r A% Ç " • f r A Ç * , ^ arbitrary with A)j^; 111A - ° and find ©? = v' ' for some M?p then

A (5.3) T Q -4-Q -r" P V for some ^ , and finally

M^R1: v (5.4) \r + 0^ if also C 3-D) is u> ed.

76/P.849 -20-

Equations (5.3) and (5.4) correspond to (2.R) and (2.9}, respectively, but they still contain the gauge field U . To elimi­ nate U- we proceed also by raising indices (by means of Y ' }. which çives

where (3.4) was used and the symmetries

were assumed for the quadrupole tensor. This suggests that we define

J V* U 4- t i-f (5.5) then we find

which is formally identical with (2.11). But from (5.5) together with (5.4) we find

wMch also agrees formally with the relativistic equaticn (2.10).

Note that all references to the gauge fields fl * and U. has disappeared in (5.6) and (5.7). It is thus reasonable to regard 2. and £ r as physical quantities describing 1-momentum and spin of the particle.

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To get a deterministic systerr of equations for the spin- ; ^ - o } we need again an auxiliary con­ dition. Ejt there is less a;:.i>ig!jity than in the relativistic case and tr.e simplest one,

5°^ f = 0 (5.S)

that ccr.es to mind is already satisfactory. Indeed, contracting (5.6) with "tyn and using (5.8) and (5.5) gives

Thus (5.7) and (5.6) reduce to

respectively, for a particle with mass and spin in a Kewtonian gravita­ tional field. They agree with those for a rigid test body model in Newtonian space-time obtained by Baxter [ 22l who also showed that they reduce to the classical three-dimensional equations in a flat Galilei space-time.

6. KAHILTOKIAH STRUCTURE FCR THE SPINNING PARTICLE^3*

In this final section we derive the equations of motion of a particle with mass and intrinsic spin only in a Newtonian gravitational field, no'; as a liait case of the conservation laws of continuous matter but strictly in the framework of Haniltonian particle mechanics. We will

'"'' To keep this section reasonably short, details of calculations that are similar to those in ["ill or flol are left out.

76/P.E49 -22-

obtain equations (5.9) (together with a syraplectic structure on the phase space) by applying a minimal intention principle to the equations in flat space . This cannot be done on the level of the equations of motion or the synplectic structure, rather it must be applied to a

Lagrangian or the Cartan 1-form, But in the space-time picture such a Galilei invariant 1-fomi does not exist which neans we have to introduce the- extended ^ililei group. In this way we will understand, at least from one point uf view, why the gauge fields A and U had to be introduced.

Let G be the proper Galilei group and G its non trivial 11-dimensional extension, parametrized as follows (e.g. [23])

fi. i^rCr^.a^K.X) j R£SO(3);«.,U em.* j c, X e (R I with the composition law

, l , , (r1U,9,g,X)ft l(i>', R',v).-(r+r )kT a + Rl»>+f^ ,RR', X") where l i v :QT »\+(rRi.') + uv .

Then li : 6 -* Q :(T, !»,«, R, X),..,^,!»,*,!*) is a surjective horcomorphism, the extension map.

Define also f :Ç-*V": (t,k, ft, R ) *_» (y , \> ) £nd Tf = p <» " • They are surjective maps onto the space V of translations.

He will now identify V with the (flat) space-time and consider G and G as fibre bundles over V , They are in fact prin­ cipal (frame) bundles with, as standard fibre the homogeneous groups Gi^ and C, » respectively, acting on the right |_24] . To make

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f -23-

this more explicit we will write' ' H r (,*• ; ^o je 1 J*/ ^ when considered as bundle space (where Cx-) = C*-D/ *r ) • («£) ifeja,) correspond to (f, t> ) , a. and R , respectively).

= s hen The right action of * Gi c J^ f*R^)V ' ^

which on G is exactly the correct action if fe ) is interpreted as a Galilei frame M .

Define

9.:= B* iJUo (6.1) where &« e_b = ?£ . Then (6 ) is the fundamental IR - valued 1-fonn on the bundle of Galilei frames G . We can pull it back to Q by means of h . To introduce also a connection form we first let t fo i^ i ^ i ^jiiv ^« V 0^ *0 De a basis of the Lie algebra Ê of G such that the non vanishing brackets are

(WO-

If we now write a *£• -valued 1-form w on G as

' ' Capital Greek and Latin indices run from 1 to 3, the former denote space components, the latter frame compon'.its and are raised and lowered by means of 5 and £ An,

76/P.S49

i -24-

and ir.pose the conditions that it be a connection form, i.e. satisfy [24]

* l

«.>(£ ) r A

- c1» for every A *i » where A is the corresponding generator of the right action on Ç , then

«/. ^ _ AA

r c A , . T ° rr„A S -\ •A ti > z r/s. •* / which is recognized as the standard action of the Galilei group on flat space-tirr.e and its natural lift to the frarce bundle. We will now inter-

76/P.849 -25-

pret the extended fran-..- bundle as en evolution space for the Galiïean spinning particle mot'"vatc-d by the relativistic model [ill. For the free particle v/e thus wibh to choose an <~ -invariant Cartan 1- for;- t whose exterior derivative will define tile Maniltonian struc-

It is easily verified that o and "•> , <-J defined by (6.1) and (6.2) are tp" -invariant 1-forms on G and it takes only a little more work to see that the same is true for

7 := to _ : à ee 9 _ J e e, 9 (6.3) j, V/x ° ° VA ° c The nest general -C ~ -invariant 1-form on G is thus G

where X , G , , M H ,_, and m are eleven arbitrary constants. "*. J rt I ils= r Li S j

Two such invariant 1-forms 0 and 6 can be called equivalent (in the sense of the Hamiltonian structure they define) if there exists a diffeon;orphisni £ : Ç —y G, such that t1) j>~ B c 0 , and (ii) o fa ; /j c for all ge

T L 1„. 20+i m«- c . «KR?ui ,

A 2*; R^^ . M* ,

Afl M s P.".. R", M

™ = ^ (6.4)

76/?. S-Î9 -26-

The constant n is trus an invariant under such changes,

filJ and if vv] =L o the only other invariants turn out to be s*= - S S.R

,ri tA 3 v.-here S' := M'" <.=t;T'Ç £ -' which can be interpreted as the

K (intrinsic) spin of the particle and £0-(-ti) j£ 2 K • the "internal energy" T23J . One recovers here the classification of Galilei invariant syrplectic structures obtained by Souriau [ïf] using the orbits of the co-adjoint representation.

We will assume that m > D and use the transformations (6.4) to bring 9 into the form

6 5 9= î09°+_l ^^ «-"* < - > which is now only preserved under space rotations. The exterior deri­ vative of © then becomes (see equation (6.9) below)

It is left-invariant, of course, and thus of constant rank, whence pre- sympclctic. (Koto that

By computing the kernel of a" {e.g. by the nethod of

[ll]and [l6] ) it is found that a curve rv->(y*(r),e ^) f X(t) ) is tangent to kn, o~ if and only if

p dlx-Vd-r = v" ; axf/dr =v°e0 , A£ /AT = * , (6.6)

(6.7) i^/Ars \e\S\ , Ajc/ir ,4 where v , ^ and C are three arbitrary parameters. Therefore War

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flat Galilei frame e' = ¥° ) that the space-time components et c>" of the spin S /;= e." e^ >SA* are constants of the motion. A * Remark ': *\, Right translations on G , while they map a Cartan 1-form into an equi­ valent one as far as the induced (pre)-symplectic structure is concerned, do not leave the fibre structure of GJ over space-time V invariant. • Thus if other choices of the constants are made then in (6.5) the space- time motion will be different, it can in fact be unrecognizable, In a sense, we have here incorporated an auxiliary condition corresponding to (2.12)or (5.8). We propose now to apply a "minimal interaction principle" to the Cartan 1-form 0 given by (6.5). This means that we now consider V as a space-time equipped with a possibly curved Newtonian structure and replace G by an 11-dimensional extension P of the bundle of Galilei frames over V. He interpret tv> and u) , now given by (6.2), as components of the connection form corresponding to the Newtonian connection on V . But the generalization of ~j to curved space is somewhat less evident. In (6.3) eo * can be considered given by (6.2) for some f\ (" ) but the only way to write the other two terms "covariantly" is at the expense of introducing a unit timelike vector field U and replacing "JT by

Tf -- J- i "Ï ee S *• . X„ < e£ & » («-a) Z *n, o o ° A To compute the exterior derivative of t? we first note tnat

from (6.2) we obtain, using the fact that 5 e a [• is a Galilei frame, that

j* K A p-<^ri. rr* « x *Afr and then

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r "• £>* ^.-lFr^ AG\(«A/o+%W/,)9A

A/R. Ali

and where U ^ s = © 2 <* * , F,^ = •*?,.*.* \« * pi wa/U - &' 6.^ Vu1* . (To derive this the first Cartan equations I

d%0« » , i«A. ©"AW'1. U>\ A.©* have been used). Then using also the second Cartan equations

t"i 2 tA " • c

2. • °r A ' ^ k t^^^W^V^,wefind

ABS Al& A i. if, *+ ^ Afto£ • • ryi )

+ " VB AW .1 3 W A w, . (6.9)

e For a curve ^ V-» £•*."(>:); ^_(~) • -Xfc) ) to be tangent to Wir (T we find again by the method of ("ill

f a «". = ?"c^ + irf ^^V-wll/^;^ «A-^

76/P.849 where " :=(io?/s).Z)V« , 6Ô°V\ ' :;r e^e^ S and V ' , X and Ç are arbitrary. The phase space has thus dimension 8. Choosing

Z to be the worldtirr.e we have V a I and hence V = ^0 . The space-time components 5 ' of tne SP^" satisfy (5.8) and (C.il) implies

S*? = o

Now we note that we get exactly the same equation; of motion as (5.9) provided the first term on the right hand side of (CIO) vanishes. But this is the case by (3.9) if the so far undetermined "gauge fields"

n c< and U. satisfy (3.2), i.e. define the same connection VJ^n that was already given in terms of 6-> and UJ ** .

This is perhaps not so surprising if compared with the relativistic model in [if]. There the Cartan 1-form $ can be formu­ lated covariantly in terms of §wj and • V^n and it is also under­ stood that the V«ri represent the Levi-Cività connection.

ACKNOWLEDGMENTS

We thank J.M. Souriau for several discussions. One of the authors (H.P. K.) acknowledges the hospitality of the Centre de Physique Théorique, C.N.R.S. Marseille.

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REFERENCES

[l] P. FRANK Sitzungsber. Kgl, Akad. Wiss. Wien, Math.-naturw. Kl. 118, Abt lia, 373 (1909).

[2] E. CARTAN Ann. Scient. Ec. Norm. Sup. ip_, 325 (1923), and 11, 1 (1921). fi] R.A. TOUPIN L-J Arch. Rat. Mech. Anal. }_, 181 (1958).

A. TRAUTMAN M C.R.A.S. Paris, 257_, 617 (1963), and in B. Hoffmann (ed.) : "Perspectives in geometry and relativity", Indiana University Press, Bloogminton (1966).

[6] P. HA VAS Rev. ted. Phys. ,2i, 93B (1961).

[6] H.D. DOHBROWSKI and K. HORNEFFER, Nachr. Akad. Wiss. Gb'ttingen, Math.-phys. Kl. 233 (1961) and Math. Z., 86_, 291 (1961).

K. HORNEFFER Arch. Rat. Mech. Anal., 23_. 239 (1966).

[7] H.P. KUN2LE Ann. Inst. H. Poincaré 17 A, 337 (1972). ré] W.G. DIXON J Commun, math. Phys. 15, 167 (1975). .

[9] J.M. SOURIAU "Structure des systèmes dynamiques", Dunod, Paris (1970).

[lC] R. HERMANN "Differential geometry and the calculus of variations", Academic Press, New York (1963).

fill H-p- KUNZLE L J J. Math. Phys. 13_. 739 (1972).

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[Ï2l C. BbVAL, H.H. FLICHE, J.M. SOURIAU L C.R A.S. Paris, 274, 1032 (1972)

C. CUVAL Thèse t!c 3c:ne Cycle, C.P.T., C.N.H.S. Marseille (1972). [ni R. HERKANK "Lie algebras and quantum mechanics", f;cr,jamin, New 'Cork (1970). H. C-OLOSC:-::::DT, S. STERNBERG ,T,n. Inst. Fourier 23_, 1., 203 (1973).

J. KIJOUSKI Commun, math, Phys. 30, 99 (1973).

J. KIJOI.'SKI, W. SZCZVRBA Commun, math. Phys. 46, 183 (1976).

[l4j C.W. HISNER, K.S. THORNE, J.A. WHEELER "Gravitation", Freeman, San Francisco (1973).

[l5] J.M. SOURIAU Ann. Inst. H. Poincarê 20_A, 315 (1974).

[l6l H.P. KUNZLE J Commun, math. Phys. 27., 23 (1972). n.T\ A. TRAUTPAN Bull. Acad. Polon. Sci., sér. sci., math. astr. phys. 20, 185, 503, 895 (1972), and 2T, 345 (1973).

F.W. HEKL GRG 4, 333 (1973), and 1, 491 (1974).

[is] F.W. HEKL, G.D. KERLICK, P. VON DER HEYCE Preprint Koln (1976).

[19] W.G. DIXON Muovo Cimento 34, 317 (1964), and Proc. Roy. Soc. London A 314, 499 (1970).

[20l I. BAILEY, W. ISRAEL Cœsnun. nath. Phys. 42, 65 (1975).

76/P.849 -32-

[21J H.P. KUiCLE ERG I, (1976). [22] J.M. BAXTER M. Se. Thesis, Univ. of Alberu, Ur.ont.zn (1976). [23] J.H. LEVV-LEELOND in E.H. Lcebl (ed.) : "Group theory and its applications". Vol. 2, Academic Press, New York (1971). [24] s. KOBAYASH:, K. NOHIZU "Foundations of differential geometry" , Vol. I, Interscience, New York, (1963).

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