POincare-Cartan integral invariant and canonical transformations for singular lagrangians

In this work we develop the canonical formalism for constrained systems with a finite number of degrees offreedom by making use of the Poincare-Cartan integral invariant method. A set of variables suitable for the reduction to the physical ones can be obtained by means of a canonical transformation. From the invariance of the Poincare-Cartan integral under canonical transformations we get the form of the equations of motion for the physical variables of the system.


INTRODUCTION
It is known that many interesting physical systems are described by singular Lagrangian. Some examples are provided by the electromagnetic, the gravitational, the Yang-Mills fields, and some relativistic models. 1 Features of all these theories are the in variance under certain transformations and the presence of relations (constraints) among the canonical variables, which restrict the motion to a hypersurface of the phase space.
A method for developing the canonical formalism and the quantization of constrained systems was proposed by Dirac. 2 The constraints are classified into two groups (firstclass and second-class), depending on their algebraic properties with respect to Poisson brackets. The dynamics of the system is generated by an extended Hamiltonian, obtained by adding a linear combination of first-class constraints to the canonical one. One must take into account the presence of second-class constraints by working with generalized Poisson brackets (Dirac brackets). The problem of the quantization is complicated by the search for a set of variables independent and canonical with respect to Dirac brackets. Instead of following the Dirac technique, these variables can be directly obtained, as suggested by Shanmugadhasan, 3 as a subset of the variables of a canonical transformation, whose existence is based on some theorems on involutory systems. 4 ,5 and function groupS. 6 We want to stress that this method, as well as the Dirac brackets technique, is a local one; in fact the existence of the canonical transformation is only locally guaranteed. 4 In this work we pursue the study of the extension of the formalism of the Poincare-Cartan integral invariant to constrained systems with a finite number of degrees of freedom, which one of us began in Ref. 7, and making use of the fnvariance of the Poincare-Cartan integral under canonical transformations, the equations of motion for a set of variables free with respect to second-class constraints are easily obtained. Furthermore, working in this reduced space of the variables independent with respect to second-class constraints, a canonical transformation which isolates the gauge-indepen-dent variables from the gauge-dependent ones is performed. This is the great advantage of this technique with respect to the Dirac one. An interesting result is that, for Lagrangians homogeneous of first-degree in the velocities, this procedure corresponds to the Hamilton-Jacobi method.
In Sec. 2 we review and extend the Poincare-Cartan integral formalism for constrained systems. Section 3 is devoted to the introduction of the concept of canonical transformation and to the proof ofthe invariance of the Poincare-Cartan integral under canonical transformations. In Sec. 4 we perform the canonical transformation extended to the second-class constraints and the Hamilton equations for the new variables are obtained. In Sec. 5 the Hamilton equations for the set of variables free with respect to first-and secondclass constraints are obtained.

POINCARE-CARTAN INTEGRAL INVARIANT FOR CONSTRAINED SYSTEMS
The Poincare-Cartan integral invariant plays a fundamental role in standard classical mechanics since, from its invariance, it follows that the equations of motion of the dynamical system are Hamilton canonical equations. 8 In Ref. 7 this result was generalized to systems described by singular Lagrangians.
Let us now review the essential points ofthis generalization. Let us consider a dynamical system described by a singular Lagrangian L = L (q"q,,(), (s = t, ... ,n).
(2.1) Due to the singularity of the Lagrangian, the motion of the system is restricted to a hypersurface of the phase space, determined by a set of constraints. Let fla(q"p,) = 0, (a = 1, ... ,T -W), be first-class constraints and w= dtL, I" it is possible to show that the integral 1= £ (PsDqs -HcDt), Then, the necessary and sufficient for Eqs. (2.6) be Hamilton equations is that the Poincare-Cartan integral (2.5) be invariant.
Proof Firstly, see that the in variance of the Poincare-Cartan integral is a sufficient condition.
The parametric equations for the dynamical paths that form the tube are qs = q,(p,a), Ps = ps(p,a), t = t(p,a) (O<a</). If we agree that d means differentiation with respect to I' and D with respect to a, by invariance we have Integrating by parts, dividing by dl' = dt Irr and using Eqs.
(2.6) we get Since rr is an arbitrary factor we obtain ( aHc ~ The Dqs and the Dps are not independent, since C must belong to S. So they must satisfy (2.14) anf3 anf3 -Dqs + -Dps =0.

aqs aps
Introducing a set of Lagrangian multipliers la' If3 Is z -

CANONICAL TRNASFORMATIONS AND POINCARE:-CARTAN INTEGRAL INVARIANT
Let us now extend the concept of canonical transformation to constrained systems, by introducing, as in standard classical mechanics, the following Definition: Given a dynamical system, constrained by Following the usual procedure of standard classical mechanics we will prove the following theorem: Theorem 2: Let Eq. (2.18) be the equations of motion of a dynamical system; a transformation Qs = Qs(q,p,t), P s = p.(q,p,t), (3.4) for which two functions Kc and F exist so that where C is an arbitrary closed contour in the extended phase space, that we will take lying on S. Let C be the contour obtained from Cby means of the transformation (3.4). Then the Poincare-Cartan integral is invariant under the considered transformation. In fact, from Eq. (3.6) we get The left-hand side ofEq. (3.7) is invariant under displacement of the contour along the tube of the dynamical trajectories, solutions ofEq. (2.18) and lying on S. The righthand side will be invariant under displacement of the contour C along the tube obtained by means of the transformation (3.4) from the proceeding. On the other hand, the transformed trajectories obey a system of first order differential equations. Thus, by repeating the proof of Theorem I and by taking into account the explicit dependence of the constraints on the time, we get (3.8) with K given by Eq. (3.3), and therefore the transformation is canonical.

A SET OF CANONICAL VARIABLES INDEPENDENT WITH RESPECT TO THE SECOND-CLASS CONSTRAINTS
In Ref. 7, as we have reviewed in Sec. 2, the Hamilton equations for a constrained system have been obtained [Eq.
(2.18)]. The variables qs andps are not independent, since they must satisfy Eqs. (2.2) and (2.3). A suitable method for isolating the true independent variables has been developed by Shanmugadhasan. 3 His theory is based on two theorems on function groups6.9 and involutory systems 4 . 5 that we recan without giving proofs.

Pf=d 1(Q) ,P;,t )Ii'p'
and in terms of the old variables

4.20)
On the other hand, if we take the total derivative with respect to t ofEq. (4.19) and use the stationarity ofthenp's we have   {np',Kc}}' and (j = 1,... ,n2) ' (4.25) (4.26) where the last eqUalities of Eq. (4.25) define K, Ifwe denote the set of variables which are independent with respect to second-class constraints by R   Summing up, we have shown that it is possible, by making use of a canonical transformation, to write the equations of motion for a set of variables which are independent with respect to second-class constraints. As shown in Ref. 3, we have the following relation between Dirac brackets and Poisson brackets defined in the reduced space R: ! , }* = I , JR .
Therefore the variables which are canonical with respect to Dirac brackets are directly obtained by means of this canonical transformation.

UNCONSTRAINED VARIABLES
A further step can be done by extending the transformation to include first-class constraints too.
In fact, Theorem 4 guarantees that it is always possible, at least locally, to replace the lia by an equivalent set P e = 0, (e = n. + l, ... ,n z ), (e = n l + l, ... ,n z ).

(5.3)
We observe, from Eq. (5.3), the local character of this technique. Thus, in general, we will have to repeat the procedure we wiIl develop in thefollowing, for the different sheets of the hypersurface (4.30).
. Let us notice that from Eqs. (4.9) and (5.3) we have (5.4) or, following the terminology of the function groups, Q; (that from now on we wiIl call Q.) and P e form a noncommutative function group of dimension 2(n znl)' By applying again Theorem 3 we can construct a canonical transformation Qj,Pj,l-Qk,Pk>Qe 'P, (k = l, ... ,n.),(e = n l + I, ... ,n z ),

CONCLUSIONS
Making use of the Poincare-Cartan integral for constrained systems, we have shown that the invariance of this integral enables us to write the equations of motion for a dynamical system as Hamilton equations. We want to observe that with this procedure, all the first-class constraints appear in the Hamiltonian, because we cannot introduce any distinction between them. Recent papers, by Cawley l3 and Frenkel, 14 have shown, with some examples, that not all the first-class secondary cqnstraints generate gauge transformations and therefore not all the first-class constraints appear in the Hamiltonian. 15 Thus we are investigating an algebraic procedure in order to take into account this result. Furthermore, we have introduced a definition of canonical transformation, which is the trivial generalization of the usual one, and shown that the Poincare-Cartan integral is invariant under this transformation. Then, following Shanmugadhasan,3 we have performed a canonical transformation such that a subset of the new variables is equivalent to the second-class constraints. The reduced set of variables, independent with respect to second-class constraints, is nothing but the set of variables which are canonical with respect to Dirac brackets.
A further step is done by performing a new canonical transformation in the reduced phase space which isolates the variables corresponding to first-class constraints. This transformation is very useful because it isolates also the gauge independent variables from the gauge dependent ones. The evolution of these gauge dependent variables, contrary to the result of Shanmugadhasan, consistently depends on arbitrary functions.
When" He = 0 this technique becomes equivalent to the Hamilton-Jacobi method. Explicit examples (the free relativistic point and a model of two interacting relativistic particles) have been already studied 16; presently we are investigating the possibility of extending this technique to continuous systems, studying the relativistic string model (see Nambu and Scherk in Ref. 1).