Vistas in Astronomy, Vol. 37, pp. 265--268, 1993 Printed in Great Britain. All rights resolved.

0083-6656/93 $24.0O @ 1993 Pergmnon Press Ltd

DYNAMICAL SYSTEMS AND ANHOLONOMIC CONSTRAINTS

David Hartley,t Robin W. Tucker , and Philip A. Tuckey§

School of Physics and Materials, University of Lancaster, Lancaster LA1 4YB, U.K.

A b s t r a c t

The Dirac analysis of constrained Hamiltonian mechanics is one of the conventional precursors to the quantisation of classical systems. We reformulate this analysis in the language of exterior differential systems, starting from the Lagrangian, moving through the generation of primary and secondary constraints and leading to the con- struction of symmetry generators for gauge symmetries. This reformulation extends the procedure to systems involving anholonomic elements.

We have found it useful to reformulate Dirac's procedure for analysing a constrained dy- namical system in the language of exterior differential systems. The basic idea is to recognise that the solutions of a (constrained) dynamical problem can be put into correspondence with integral manifolds of a closed involutive exterior differential system. Such a reformulation offers one a number of advantages. These include the analysis of higher order problems where the La- grangian depends on higher order accelerations, and coordinate independent problems in which the constraints arise naturally from the underlying geometry of the problem. This generality has been exploited in examples from the theory of relativistic elastica (Hartley, Tucker and Tuckey 1991). Furthermore, for systems that exhibit a symplectic structure, it is straightforward to generate equations for the local symmetries corresponding to first-class constraints using the notion of an i sovec tor .

t e-mail: D.HartleyOuk.ac.lancaster

e-mail: R.W.Tucker@uk.ac.lancaster

§ e-mail: P.A.Tuckey@uk.ae.lancaster

266 D. Hartley et aL

1 E u l e r - L a g r a n g e S y s t e m

The basic setting used here for variational problems in mechanics is as follows. Let X be an n-dimensional manifold, {0 ~} ( i - - 1 ,2 , . . ,r) be a set of 1-forms on X, and let w be a further 1-form on X, with w A 01 A. . . A 0 ~ ~ 0 at all points of X. The {8 i} will be called the constraint system and w the independence form. Together, they specify a set {f} of integral manifolds: 1-dimensional immersions f : C --, X satisfying

f ' 8 i = 0 (1)

and

at all points of C.

# 0 (2)

Within this basic setting, a variational problem is given by specifying a 1-form ~0 on X and requiring that the action

be stationary under variations of the immersion f which preserve the constraint system and independence condition. That is, the problem is to find those f amongst the set determined by {0i;.w} for which the action is stationary. The condition that f must satisfy is

f*ivdA = 0 for all V e T(X) (4)

where (5)

and the p, are functions on X to b e determined.

It is useful to extend the manifold X to Y = X x R r, introducing the {pi} as r new coordinates rather than functions to be determinedduring solution. The immersion f lifts to an immersion into Y, which will continue to be denoted by f . The exterior differential system

S = { i v d A l V e T(Y)}

with independence form w will be called the Euler-Lagrange syatem.

(s)

There is no difficulty in accommodating Lagrangians which contain higher-order derivatives of the configuration variables. For a p-th order problem, X is the jet bundle JP(R --* Q), with constraint system {dq (c ') /- q(~+l)idT}, where {q(¢)i} are jet bundle fibre coordinates, T E R and a = 0, 1 , . . . , p - 1. The Euler-Lagrange system S yields the usual Euler-Lagrange equations for higher order problems. For details of this and other examples, see Griffiths(1982), Hartley and

Tucker(1990).

Dynamical Systems 267

2 Constraint Analysis

The key concept is that of involution of an exterior differential system on Y with respect to the

independence form w. For the class of problems considered here (where w is a 1-form), this is the condition that

o~^~#o (7)

for all 1-forms ct in the exterior system at all points on Y where all O-forms in the system vanish. This condition singles out those solutions of the exterior differential system for which w corresponds to an independent evolution variable.

The Euler-Lagrange system S does not generally satisfy the involution condition.

In terms of exterior systems, the constraint procedure for a first order problem may be described as follows. The 1-form A on Y with 0 i = dq i - qidT and ~ = L(q ,q)dr can be

rewritten using the Hamiltonian H(q, (I, P) = Pi(1 i - L as

A = - H d r + pidq i.

T h e Euler-Lagrange system becomes

{ dq ~ - Hp~ d'r

S = dpi + Hqi d'r

H~ dT.

Solving as many of the equations H~ = 0 solutions leaves a system

{

(8)

(9)

as possible for the {qi} and pulling back using those

dq ~ - Hpi dr

dpi + Hq~ dT

¢=dr

(10)

in which some of the q' have been eliminated in Hp~ and Hq~. This system is clearly not in involution with the independence form dr. Rather than solve the Ca to overcome the involution problem, they can s imply be appended to the exterior system as O-forms, giving

S' = dq i - Hp, dT (11)

dpi + Hq~ dT.

Since f*~b~ = 0 implies that f*d¢~ = 0 also, we should close the system under exterior differen- tiation. Because f is one-dimensional we need only include the 1-forms d~b~ and then re-examine the involution condition. Since the Ca depend solely on the {qi,pl} , dea can be rewritten as

d¢,~ = x , d r (mod S') (12)

(This means that the expressions in S ~ have been used to simplify d¢~). If any of the X~ are independent of the ~b~ then S ~ does not give an involutive system. If this is the case, then some

268 D. Hartley et aL

of the X~ may yield solutions for some of the remaining qa, but the rest are new constraints. The q~ solutions may be used to pull back to a smaller spare and the new constraints appended as further 0-forms to the system. By repeating this procedure until involution is achieved, the Dirac constraint analysis is reproduced. The constraint analysis for a p-th order problem is

immediate.

For more general problems, e.g where the Euler-Lagrange system involves anholonomic ele- ments, it may not be possible to make a distinction between 1-forms corresponding to configura- tion variables {qi} and those corresponding to velocities {qi}. To extend the constraint analysis to these problems it is necessary to overcome this difficulty. To do this, it may be noted that it is no more necessary to solve for the qi at any stage of the procedure than it is to solve for the qi or pi. It is possible simply to append all of the 0-form involution conditions to the system at each step without distinguishing between constraints and velocity solutions. This time, including the exterior derivatives of the 0-forms in the system will lead to some 1-forms containing dq i terms

as well as new involution conditions. These new 1-forms are just the exterior derivatives of the previous velocity solutions.

Thus the constraint analysis procedure for the 9eneral problem is as follows. Start with the Euler-Lagrange system S and check if it is in involution with the independence form w. I f it is not, then append the involution conditions as O.forms to the system and include their ezterior derivatives. Check this eztended system for involution and repeat until an involutive system is

achieved.

As an example of this approach, we have solved the problem of finding a time-like curve in Minkowski space-time which optimises the action for a Lagrangian L(~;1,~2) where ~1 and ~2 are the acceleration and torsion of the curve. Dereli, Hartley, Onder and Tucker (1990), (Hartley, Tucker and Tuckey 1991).

This work has been carried out with support from a BP Venture Research Fellowship.

R e f e r e n c e s

Dereli T, Hartley D H, Onder M and Tucker R W 1990 Phys Left B 252 601

Dirac P A M 1950 Canad J Math 2 129

Griffiths P A 1983 Ezterior differential systems and the calculus of variations (Birkh~user)

Hartley D H and Tucker R W 1990 in Geometry of low-dimensional manifolds, LMS Lecture Note Series 150 (Cambridge University Press)

Hartley D H , Tucker R W and Tuckey P A 1991 J Phys A (Math Gen) 24 5253-5265