A NNALES DE L ’I. H. P., SECTION A
L UIS B EL
Hamiltonian Poincaré invariant systems
Annales de l’I. H. P., section A, tome 18, n
o1 (1973), p. 57-75
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57
Hamiltonian Poincaré Invariant Systems
Luis BEL
Universidad Autonoma de Madrid, Departamento de Fisica, Canto Blanco, Madrid-34
Vol. XVIII, no 1, 1973, Physique théorique.
ABSTRACT. - In a
preceding
paper weproposed
a new definitionof Hamiltonian Poincare Invariant
Systems (H.
P. I.S.).
The mainidea of this definition is to
keep
Poincare transformations canonical and torequire
that the Poisson Brackets of relativephysical positions
vanish. In this paper we
generalize
to any number ofparticles
oneof our results.
Namely,
that for each H. P. I. S. the InducedSystem
at the Center of Mass is
lagrangian.
We obtain also for N = 2 thegeneral equations
whose solutions wouldgive
all H. P. I. S.INTRODUCTION
The Predictive
(1)
Relativistic Mechanics(P.
R.M.)
of Npoint-like
structureless
particles
has beendeveloped recently.
In P. R.M.,
a
system
of Npoint-like particles
is describedby
a Poincare InvariantSystem (P.
I.S.),
i. e., anordinary
second ordersystem
of differentialequations
with definite invarianceproperties.
Twoequivalent
forma-lisms have been used to write down these
equations.
In the Time-Symmetric
formalism a P. I. S. is describedby
a newtonian-likesystem
(1) The word Predictive is here used in contradistinction to the word deterministic to emphasize that the physical systems being considered have a finite number of
degrees of freedom.
ANNALES DE L’INSTITUT HENRI POINCARE
58 L. BEL
of differential
equations
where the are restricted to
satisfy
anappropriate system
of non linearpartial
differentialequations (D.
G. Currie[1],
R. N. Hill[2], [3],
L. Bel
[4], [5])..
In theManifestly
Covariant formalism the samephysical system
is describedby
asystem
of differentialequations
inMinkowski
Space-Time :
where
again
the are restricted tosatisfy appropriate equations (Ph.
Droz-Vincent[6],
L. Bel[5]).
Both formalisms haveadvantages
and inconveniences. In this paper we shall use the
Time-Symmetric
formalism
only.
One of the unsolved
problems
of P. R. M. is togive
a convenientdefinition of Hamiltonian P. I. S.
(H.
P. I.S.).
The first and natural definition that was considered led to trivialsystems only,
i. e., to noninteracting particles (D.
G. Currie-T. F. Jordan-E. C. G. Sudarshan[7],
J. T. Cannon-T. F. Jordan
[8],
H.Leutwyler [9],
R. N. Hill[3]).
The rootof the
difficulty
was first identifiedby
E. H. Kerner[10],
andclearly
exhibited
by
R. N. Hill[3],
aslying
in theassumption
that thei. e., the
physical positions,
were assumed to havevanishing
Poisson-Brackets
(P. B.)
among themselves. Several new definitions whichpermit
to circumvent this « No Interaction » theorem havealready
been considered. We may mention those of R. N. Hill-E. H. Kerner
[I I],
L. Bel
([5], [12])
and C. Frondsdal[13].
But we believe it is safe to say that the status of theproblem
is stillconfusing
and thatgeneral agreement
on a convenient definition of H. P. I. S. has not
yet
been reached.The main idea in the definition
proposed
in reference[12]
was tokeep
Poincare transformations canonical whiledropping
the commuta-tion of the
xa’s
andrequiring only
that the P. B.’s of the relativepositions
among themselves vanish :
This paper deals
again
with theimplications
of this definition. We gene- ralize to any N andsimplify
theproof
of the theorem of reference[12].
Namely,
that for each H. P. I. S. the InducedSystem
at the Centerof Mass in
lagrangian.
We establish for N =2,
thegeneral equations
whose solutions would
give
all H. P. I. S. in the sense of ourdefinition,
and we prove
that,
at the Center of MassSystem,
the Center of Mass lies on theline joining
thepositions
of the twoparticles
and that the motion of thesystem
isplane
motion.VOLUME A-XVIII - 1973 - NO 1
1.
POINCAR~
INVARIANT SYSTEMSLet be the Lie
algebra
of the Poincare group and letH, Pi, Ji,
K be a basis of P such that(2) :
Let
E3
be the euclidean 3-dimensional space andE6
the fibertangent
bundle of all
vectors v tangent
toE3
and with norm less than 1(#2 1).
We shall note
VcN,
and call theco-phase
space, the 6N-dimensional
manifold :V6N
=(Ê6r’i.
For each cartesiansystem
of coordinates ofE3, trivially
extended toV,x,
eachpoint
A ofV6N
will have coordinates(-4. vv) 0.
thexa’s being
the coordinates of theprojection
of A intothe basis of
V6N,
i. e.,(E3)N.
Let
(V6 N)
be the infinite dimensional Liealgebra
of all vector fieldsof
V,N.
If21
and2 belong
theirLie-Bracket (L. B.)
is defined
by
where yA (A, B,
... =1, 2,
..., 6N)
is anarbitrary
coordinatesystem
ofV6N (5).
DEFINITION 1. - A Poincare Invariant
System (P.
I.S.),
with 6 Ndegrees
offreedom,
is anhomomorphism 03C8
of into the infinite dimen- sional Liealgebra r (V6 N)
such that(2) i, j, k = 1, 2, 3. The summation convention will be used throughout. The latin indices will be raised or lowered without change of sign. Tiijk is the Levi-Civitta symbol. The speed of light in vacuum is taken equal to 1.
(3) a, b, c, ... = 1, 2, ..., N.
(4) The summation convention is also used for the indices A, B, C, ....
(5) A possible choice of
yA
is yi- x1, , , . ~ y~-3+f ~ xN~
ANNALES DE L’INSTITUT HENRI POINCARE
60 L. BEL
-+- + ~ -+
H, P;, J;, K;, being
theimages by 03C8
ofH, P;, Ji, Ki,
and .ebeing
the-+
Lie derivative
operator acting
onfunctions,
i. e., if E E Jfi(V 6 N)
and ()is a function
When necessary we shall refer to
(1)
and(2)
to remind the Lie Bracket- -~ ~ -~
relations of the vector fields
H, Pi, J i, Ki.
Let us define
~.a
asFrom
(1), (4), (5), (6),
and fromwe obtain
Therefore,
the vector fieldsH, Pi, Ji, Ki
will be known as soon as weknow the functions
~.a.
These functions are restricted
by equations (2)
to be solutions of thefollowing system
of differentialequations (6) :
where ~a = 1.
For each solution of these
equations,
thesystem
of second orderordinary
differentialequations
can be
interpreted
as theequations
of motion of Npoint-like
structurelessparticles
in P. R. M.([1], [2], [3], [4], [5]).
(6) The summation convention is used also for the indices a, b, c.
VOLUME A-XVIII - 1973 - NO 1
2. HAMILTONIAN P. I. S.
a. Let o- be a
sympletic
form on i. e., a 2-form(second
rankcovariant skew
symmetric
tensor (jAB = -(jAB)
withdet 03C3 ~
0 and suchthat
(’) :
The
following
definitions and results ofsymplectic geometry
willbe used below
([14], [15], [16]).
DARBOUX THEOREM. - For any
given point of V6N
therealways
existsa local coordinate
system,
whose domain contains thispoint,
such that ~ takes theform (8) :
where a =
1,
..., N.Any system of
coordinatesfor
which a takes thisform
is called a canonicalsystem of
coordinates.rda
and are saidto be canonical mates.
The P. B. of two functions + and E with
respect
to a isby
definitionthe function
a-1
being
definedby
ocB =~. Using
canonical coordinates theexplicit expression
for(15)
is the usual one,namely :
The P. B. is anticommutative :
and as a direct consequence of
(13)
we have for any three functionsThese
equations being
known as the Jacobi identities.(7) In general, for any p-form g (covariant completely skew symmetric tensor
of rank p), the exterior differential operator d is defined by
where 6 is the kronecker tensor. The operator d has the property d2 = 0.
C) In general if S is a p-form and Q a q-form, their exterior product is the q + p- form
ANNALES DE L’INSTITUT HENRI POINCARE
62 L. BEL
If 2. is a function of several other functions
qr v,
thenLEMMA. - Given N
independent functions ya
such thatthen there
always
exists N canonical mates i. e., N newfunctions
such that is a coordinate
system o f V6 N
and such that a takes fheform (14).
Thesefunctions
areonly de fined
up to thetrans formation
Let us consider a transf ormation of
V~ N,
This transformation is
by
definition a canonical transformation if it leaves invariant a. Or moreexplicitly
ifwith A’ =
A,
B’ = B.For S to be the
generator
of a oneparameter
group of canonical transformations the necessary and sufficient condition is that(9) :
If
S
satisfiesequation (23),
then there exists a function E such thatObviously,
this function E is definedonly
up to an additive constant.Formula
(24)
can also be read fromright
toleft,
i. e., for each functionE,
the vector E whose
components
are solutions of(24)
satisfies(23).
(9) Given a p-form S2 and a vector
field 8
a useful formula to know iswhere
t(s)
is the interior product operator defined byFrom this compact formula for the Lie derivarive of a p-form, and from d2 = 0
it is trivial to see that e
(s)
dQ = de(s)
Q.VOLUME A-XVIII - 1973 2014 ? 1
If E satisfies
(23)
and 03A6 is any function then+ +
If
Ei
andE2 satisfy (23)
thenb. For obvious reasons, which include
preparation
towards a quan- tizedtheory
or relativistic statisticalmechanics,
P. R. M. will have to bedeveloped by giving
anappropriate
definition of Hamiltonian P. I. S.(H.
P. I.S.).
A natural definition would be thefollowing :
An H. P. I. S. is a P. I. S. such that there exists a
symplectic
form asatisfying
thefollowing
conditionsConditions
(27)
express the Poincaré invariance of u. In otherwords,
-+ -+
they
express that the Poincaré transformationsgenerated by H, P;,
+- -+-
J;
andK;
are canonical transformations. Conditions(28)
areequivalent
to
saying
that the P. B. of the among themselves vanish:Unfortunately,
as has been provenby
manypeople ([7], [8], [9], [3]),
the
only
P. I. S.’s for which asymplectic
form o- can be found satis-fying (27)
and(28)
are thosewith i03B1
=0,
i. e., thefree, non-interacting, particle systems.
c. It becomes then necessary to
find
a new definition of H. P. I. S.The first new one to be
proposed by
T. N. Hill-E. H. Kerner[ 11] is just,
in our
opinion,
aboundary
condition to berequired,
ifpossible,
forinfinite
inter-particle separation;
the second oneproposed by
L. Bel[5]
does not convey
enough
information to be useful either. We still believe that the definition weproposed
in[12]
does conveyenough
information and this paper is
essentially
devoted todevelop
the conse-quences of this definition.
DEFINITION. - An H. P. I. S. is a P. I. S. for which there exists a
symplectic
formsatisfying
thefollowing
two sets of conditionsANNALES DE L’INSTITUT HENRI POINCARE 5
64 , L.BEL ;, .
The choice of the numerical value of a, in the second
writing, being,
of course,
arbitrary.
’
The idea
is, thus,
tokeep
the Poincare transformations as canonical transformations and tomaintain only part
of the restrictions contained in(29).
Remark 1. - We may notice that in the I-dimensional space case, for two
particles,
theonly
P. B.(31)
to be considered is ’which vanishes
identically. Therefore,
in this case, the second set ofconditions
(31)
are not restrictions at all. This case which is a verydegenerate
one, has beenextensively
consideredby
R. N. Hill([3], [17]).
d.
Among
the first consequences which can be derived from the Defini- tionabove,
we may mention thefollowing : using (24)
we can associateto each of the vector fields
H, Pi, Ji, Ki,
acorresponding
functionH,
Pi, Ji, Ki
such that the P. B. of these functions areexactly
those of(1)
and
(2),
i. e.,(1)
and(2)
without the neutral elements which would be ingeneral present
because of the arbitrariness in the additive constants mentioned above.Thus,
we shall referagain
to(1)
and(2)
to remind the P. B. relations of the functions
H, Pi, Ji, Ki.
From
(4), (5), (6), (25)
and(30)
it follows thatH, Pi, J~
willbe,
of course,interpreted
as the TotalEnergy,
the TotalMomentum,
and the totalAngular
Momentum. Andwill be
interpreted
as the coordinates of the Center of Mass.We shall assume that
together
withPj
and Ri is an admissible coordinatesystem
ofV~N.
It
might
bepossible
to prove thisassumption.
But were itnot,
it wouldhave to be introduced if we
want,
as it is the case, thedynamical systems
we are
considering
to have a newtonian limit.VOLUME A-XVIII - 1973 20142014 ? 1
3. THE INDUCED LAGRANGIAN SYSTEM
a. Let us consider an H. P. I. S. From
(31)
and(32)
we havewhere
According
to the Lemma ofparagraph
2 a, there exists canonical mates of these 2 Ncoordinates, Pj,
that we shall note andQi.
Interms of these 6 N
coordinates, a
will take the formFrom
we obtain
On the other
hand,
from the thirdequation (30)
and from the fact thatand
Qi
are definedonly
up to a transformation"
it can be seen that no
generality
is lostby assuming
thatThe first set of
equations (40)
express that arefunctions
ofxfx) and Ua only.
b. From
(12)
we havewhere
~)
as the are, as a consequence of the first set ofequations (11),
functions ofand via
and may therefore be consideredas functions of
xtX)’
andPy.
But since from(1)
we know that theP J ’s
ANNALES DE L’INSTITUT HENRI POINCARE
66 L. BEL
are constants of motion it follows that the solutions of
(12)
with initial conditions such thatPi
= 0 willsatisfy
thesystem
differentialequations
where the are the functions of
xtX)
and obtained from theby putting
in thesePi
= 0. We shall call thesystem (44).
The InducedSystem
at the Center of Mass.d. We shall prove now the
following
theorem :THEOREM. - For each H. P. I. S. the Induced
System
at Centerof
Mass is
Lagrangian.
To prove that
system (44)
isLagrangian
we have to prove that there exists asymplectic
form r* :where the are functions of and
~,
and such thatwhere
-+-
H is the vector field of thecophase
space of thesystem,
defined
by
Let us prove that
equations (46)
are satisfied if we define as the functions obtained from the of(38),
which can be considered asfunctions of
xtX)’ vtxy
andPj, by putting
in thesePj = 0,
i. e.,by defining
(7* as thesymplectic
form inducedby 03C3
on any of thehyper-
surfaces
~o
withequations
Let us
consider,
ingeneral,
thefamily
ofhyper-surfaces
withequations
The vector field H can be written as
where
HL
istangent,
at eachpoint,
to one of thehyper-surface ~
andHT
is transversal. Let uswrite,
on the otherhand,
thesymplectic
form(38)
as
VOLUME A-XVIII - 1973 - N° 1
with
From the first set of
equations (30),
we haveBut from
it follows that on
~o :
On the other hand
Therefore, considering equation (53)
on we obtainAnd since the first term does not contain any term with a
dPi product,
both terms must be null. Formula
(46),
and thus ourtheorem,
havebeen
proved.
Let us consider now more
precisely
thehypersurface
~* withequations
From
it
follows, using
similar notations asbefore,
that on ~* :where +-
Jj
is the vector field ofV6(N-1),
identified toI*, given by
therelations
By
similararguments
to those usedbefore,
it is easy to prove thatLet H* and
Jj
be theEnergy
and theAngular
Momentum of the InducedSystem (44).
SinceH,
the TotalEnergy
of theoriginal system,
ANNALES DE L’INSTITUT HENRI POINCARÉ
68 L. BEL and H* are
given by
using
thedecompositions (51)
and(52), together
with(55)
and(56),
we obtain
where
Therefore,
up to an additive constant :And
similarly,
we could prove that4. GENERAL
EQUATIONS
OF H. P. I. S. FOR N = 2a. From now on we restrict ourselves to the
simplest
case whereN,
the number of
particles
involved in thesystem,
is 2. Theproblem
ofconstructing
H. P. 1. S. isequivalent
in this case to theproblem
ofobtaining
ten functionsH, P,, Ji, Ki
and six functionsxa (a
=1, 2)
of the 12 variables
xi, Qi,
7ri andPi
such that : 1.r’i -xz
= xi.2. The Poisson brackets
(16)
of these functions among themselvessatisfy
the commutation relations(1)
and(2).
3. The Poisson brackets of
x~~
withPy, J j
andKy satisfy
the commutation relations(32)
and(34),
where in the latter thev’ s
are definedby (33).
In
fact,
let us assume that we have found such functionsxa
andH, Pi, Ji, K~ satisfying
the conditions1,
2 and 3 above. To thelatter,
we can
associate, using
formula(24)
and thesymplectic
formten vector fields
H, Ji, Pi,
K; whoby
construction willsatisfy
equa- tions(1), (2), (4), (5), (6),
and(30).
On the other handequations (31)
will be satisfied as a consequence of condition 1
above,
and theparti-
cular form
(68)
of u.Therefore,
all the conditions ofparagraphs
1 and 2involved in the Definition of H. P. I. S. will be satisfied.
VOLUME A-XVIII - 1973 20142014 ? 1
Were it necessary to construct the functions of
(12) they
couldbe obtained from -
’
by writting
these P. B. in terms ofxa
and~,.
b. From
(68)
we haveFrom
(42)
and from similarequations,
satisfiedby xl
andPy,
we obtainthat the functions
J1
areFrom
[P~, H]
=0,
we obtainand from
[Ki, P j] = Oij
H weget
where the are
independent
ofQi :
From
[Ki, H]
=Pi
and(73)
we obtainand from
[Ki, Ky] = 2014
TJijk(73)
and(71)
weget
where
Equations [J;, H]
= 0 tell us that H is a scalar function and this canbe
satisfied, equivalently, by considering
H as a function of the scalars :Equations [Ji, Ki)
=K~, taking
into account(73), give
ANNALES DE L’INSTITUT HENRI POINCARE
70 L. BEL
and these
together
with(74)
tell usthat ff
are thecomponents
of a vectorindependent
ofQi.
Orequivalently, that
may be written aswhere are functions of the six scalars
(78).
c. From
(32)
weget
with
equations
that tell us thatp~
are thecomponents
ofvectors, indepen-
dent of
Q~.
To meet condition 1above,
we must haveFrom
(34),
where theva’s
are definedby (33),
weobtain, taking
intoaccount
(73)
and(81) :
These
equations
have to hold for a = 1 and 2.Using equation (76), equation (84)
with a = 2 can be writtenwhere
And
substracting equation (84)
with a = 2 from the sameequation
with a =
1,
we obtainWe may summarize the
preceding
results as follows :To construct H. P. I. S. with N = 2 we need to obtain functions
H,
fi
and ex i ofxi,
andPk,
solutions ofequations (75), (76), (85)
and(87).
fl
must have the form(80) ; ai
must havesimilarly,
as follows from(79), (82)
and(86),
the formand
H,
and must be functions of the six scalars(78).
Thus wehave,
alltogether,
seven unknown functions of six variables.VOLUME A-XVIII - 1973 - NO 1
5. THE INTEGRABILITY CONDITIONS
a. Obvious necessary conditions which could follow from
equations (75), (76), (85)
and(87)
would be the conditionsexpressing
theintegrability
conditions :
We have in hand all the elements to
effectively
calculate theseintegra- bility
conditions.Instead,
theintegrability
conditionscan not be calculated because none of the
general equations
sayanything
about the derivatives of aj or H with
respect
to7ri,
andtherefore, they
could not
give
rise to new necessary conditions.b. It would be
extremely
difficult to work outequations (89)
withoutusing
anappropriate technique.
Thistechnique
existsthough :
theexterior calculus for
p-forms
with values in a LieAlgebra.
Wepresent.
below the main results which will be
needed, using
a notationadapted.
to the future use of these results.
Let Q and ~ be
respectively
ap-form
and aq-form
where
~~1... ~q
are functions ofxi, Q~, Pj,
a coordinatesystem.
of a
symplectic
manifold.Therefore,
we know how to calculate the P. B..of any
pair
of these functions. We define the mixed exteriorproduct,
~]
as the p +q-form
withcomponents :
ANNALES DE L’INSTITUT HENRI POINCARÉ
72 L. BEL .
This mixed exterior
product
has thefollowing properties :
~i
1and Q2 being
bothp-forms.
where
/B
is the usual exteriorproduct symbol
and W is a r-formAnd
finally,
where d is the exterior differential
operator.
c.
Using
thisformalism, equations (75), (76)
and(85)
can be writtenwhere
’The conditions
expressing
that H is a scalar areand the conditions
expressing
thatfi
are thecomponents
of a vectorare
or
VOLUME A-XVIII - 1973 - N° 1
or
a and ai
satisfy
the sameequations.
The
Integrability
conditions(89)
areLet us work
out,
as anexample,
the first set ofintegrability
condi-tions
(107).
From(98)
weget
and
using again (98), (99)
and(97),
we obtainUsing
now(92), (93)
and(94),
weget
But from
(93)
and(96)
it follows that the second line is zero.Therefore,
the
integrability
conditions we arelooking
for areequivalent
toBut these conditions are a consequence of
(98)
and(103). Thus,
assu-ming
that H is a function of the six scalars(78),
the first set ofintegra- bility
conditions(107)
are satisfied as a consequence ofequations (98)
.and
(99).
A similar
calculation,
but verylengthy
in the case of the first setof conditions
(108),
proves that theremaining integrability
conditions(107)
and
(108)
are consequences of(98)
to(105) (or
theequivalent equations
to this last
one).
Our conclusion
is, then,
that no more information is contained in the Definition of H. P. I. S. for N = 2 than that information contained inequations (98)
to(104) [together
with theanalogous equation
to(104)
for
x].
ANNALES DE L’INSTITUT HENRI POINCARE
74 L. BEL
6. THE CENTER OF MASS FORMULA
a. Out of the
equations (75), (76), (85)
and(87), only
the last onegives.
information about the functions
H, f;
and a j themselves taken atPi
= 0.Let us make more
explicit
this information.Using (80), (88)
and thescalar character of and we obtain
where we have written
and the P. B. are those that
correspond
to the inducedsymplectic:
form a* defined in
paragraph
2.b. From
(35)
and(73)
we haveand from
(81),
with a =2,
and from(86),
we obtainAnd
taking
this formula forPi
=0,
from(113),
weget
Equation (118)
tells usthat,
forP
=0,
the Center of Mass lies on the-straight
linejoining
thepositions
of bothparticles.
From
(67)
and(71),
it follows thatand therefore
Since
by
a space translation we canalways
makeRi
= 0 and thenx2
and ri arecolinear,
from(120)
we see that in this frame of referenceVOLUME A-XVIII - 1973 - N° 1
meaning
that both orbits lie on aplane.
This conclusion isobviously
invariant
by
spacetranslations,
and thereforegeneral
for frames of reference withPi
=0,
eventhough (121)
are not.c. It is clear that if the two
particles being
considered areidentical,
whichimplies,
inparticular,
thatthey
have the same mass, from(118)
itfollows that in this case
a 1 - 2 1 ~ Substitution of this particular
value
in
(114) gives
thenand
therefore,
is a function of xonly.
But since from(41)
and(73)
it follows that has been defined
only
up to a transformationwhere 03BB is an
arbitrary
function of x, we conclude that nogenerality
is lost
by assuming
= 0. H* isthen,
in this case, theonly
functionwhich remains
arbitrary
tospecify
aparticular
interaction. The choice ofH*,
which liesbeyond
the scope of this paper, will have to be madeon
physical grounds.
Knowing x~,
and H*equations (98), (99)
and(100)
wouldgive,
at
least, by
formal power seriesexpansions
withrespect
to the variablesPi,
the functions x~, and H.
[1] D. G. CURRIE, Phys. Rev., t. 142, 1966, p. 817.
[2] R. N. HILL, J. Math. Phys., t. 8, 1967, p. 201.
[3] R. N. HILL, J. Math. Phys., t. 8, 1967, p. 1756.
[4] L. BEL, Ann. Inst. H. Poincaré, t. 12, 1970, p. 307.
[5] L. BEL, Ann. Inst. H. Poincaré, t, 14, 1971, p. 189.
[6] Ph. DROZ-VINCENT, Phys. Scripta, t. 2, 1970, p. 129.
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[9] H. LEUTWYLER, Nuovo Cimento, t. 37, 1965, p. 556.
[10] E. H. KERNER, J. Math. Phys., t. 6, 1965, p. 1218.
[11] R. N. HILL et E. H. KERNER, Phys. Rev. Lett., t. 17, 1966, p. 1156.
[12] L. BEL, C. R. Acad. Sc., Paris, t. 273, série A, 1971, p. 405.
[13] Ch. FRONSDAL, Phys. Rev., D, t. 4, 1971, p. 1689.
[14] C. GODBILLON, Géométrie différentielle et Mécanique analytique, Hermann, Paris, 1969.
[15] J. M. SOURIAU, Structure des systèmes dynamiques, Dunod, Paris, 1970.
[16] R. ABRAHAM, Foundations of Mechanics, Benjamin, 1970.
[17] R. N. HILL, J. Math. Phys., t. 11, 1970, p. 1918.
(Manuscrit reçu Ie 30 novembre 1972.)
ANNALES DE L’INSTITUT HENRI POINCARE