Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems

The current interest in constrained systems was spawned by Dirac· and Bergmann in their study of the canonical formalism of gravitational fields. Since that time several people contributed to the building of a mechanics for such systems. In particular the Lagrangian,4 Hamiltonian,4,S Hamilton-Jacobi,6 and geometrical formalisms have been studied. For a time this field of research had little more than mathematical interest, but now with the increasing interest in gauge theories (any theory with gauge transformations is a theory of constrained systems), more people are beginning to use this formalism at the classical and quantum level. On the other hand, constrained systems with a finite number of degrees offreedom have been used to construct an N-body relativistic mechanics of direct interactions whose corresponding quantum mechanics, which is multitemporal, is related to the Bethe-Salpeter equation. Despite increasing interest, the mechanics of these systems is not as elaborate as the corresponding mechanics for unconstrained systems. For example, the equivalence between the Lagrangian and Hamiltonian formalism has not been definitely established. 10--12 In this paper we give an explicit and complete proof of this equivalence. We construct an implicit inverse relation between velocities and momenta, i.e., the inverse Legendre transformation. Using that we deduce the Hamilton-Dirac equations from Euler-Lagrange equations. Neither is a set of normal differential equations, therefore the uniqueness and existence theorem cannot be applied. This means that, at most, we will only have solutions in a submanifold of the respective spaces and in general these solutions will not be unique. A careful analysis shows that given a solution of the Euler-Lagrange equations we can construct a solution of the Hamilton-Dirac equations and vice versa. Next we look for the appropriate submanifold of the tangent bundle (TQ) and a submanifold of the cotangent bundle (T*Q) where the solutions exist. These submanifolds are constructed through an iterative procedure. In a given local chart they are characterized by a set of functions that are called constraints. The Hamiltonian formalism as developed in this paper differs from the usual development. .,2 The first class primary constraints playa privileged role. Other constraints are either first or second class with respect to them. These constraints that are first class with respect to the primary first class constraints can be associated with Lagrangian constraints that are FL-projectable (or weakly FL-projectable). Those that are second class in the Hamiltonian formalism have associated non-FL-projectable Lagrangian constraints, It is also shown that all constraints other that the primary constraints have either a symmetric or antisymmetric Poisson bracket (PB) structure with the first class primary constraints. The paper is organized as follows. In Sec. II we show that if we have a solution of the Euler-Lagrange equations we can construct from it a solution of the Hamilton-Dirac equations and vice versa. In Sec. III we develop an algorithm for the determination of the Hamiltonian constraints. In Sec. IV we develop an analogous algorithm for the Lagrangian constraints and we relate the Lagrangian and Hamiltonian constraints.


I. INTRODUCTION
The current interest in constrained systems was spawned by Dirac• and Bergmann 2 in their study of the canonical formalism of gravitational fields.Since that time several people contributed to the building of a mechanics for such systems. 3In particular the Lagrangian, 4 Hamiltonian,4,S Hamilton-Jacobi,6 and geometrical formalisms 7 have been studied.For a time this field of research had little more than mathematical interest, but now with the increasing interest in gauge theories (any theory with gauge transformations is a theory of constrained systems), more people are beginning to use this formalism at the classical and quantum level.
On the other hand, constrained systems with a finite number of degrees offreedom have been used to construct an N-body relativistic mechanics of direct interactions 8 whose corresponding quantum mechanics, 9 which is multitemporal, is related to the Bethe-Salpeter equation.
Despite increasing interest, the mechanics of these systems is not as elaborate as the corresponding mechanics for unconstrained systems.For example, the equivalence between the Lagrangian and Hamiltonian formalism has not been definitely established.10--12 In this paper we give an explicit and complete proof of this equivalence.We construct an implicit inverse relation between velocities and momenta, i.e., the inverse Legendre transformation.Using that we deduce the Hamilton-Dirac equations from Euler-Lagrange equations.Neither is a set of normal differential equations, therefore the uniqueness and existence theorem cannot be applied.This means that, at most, we will only have solutions in a submanifold of the respective spaces and in general these solutions will not be unique.
A careful analysis shows that given a solution of the Euler-Lagrange equations we can construct a solution of the Hamilton-Dirac equations and vice versa.Next we look for the appropriate submanifold of the tangent bundle (TQ) and a submanifold of the cotangent bundle (T*Q) where the solutions exist.These submanifolds are constructed through an iterative procedure.In a given local chart they are characterized by a set of functions that are called constraints.
The Hamiltonian formalism as developed in this paper differs from the usual development..,2The first class primary constraints playa privileged role.Other constraints are either first or second class with respect to them.These constraints that are first class with respect to the primary first class constraints can be associated with Lagrangian constraints that are FL-projectable (or weakly FL-projectable).Those that are second class in the Hamiltonian formalism have associated non-FL-projectable Lagrangian constraints, It is also shown that all constraints other that the primary constraints have either a symmetric or antisymmetric Poisson bracket (PB) structure with the first class primary constraints.
The paper is organized as follows.In Sec.II we show that if we have a solution of the Euler-Lagrange equations we can construct from it a solution of the Hamilton-Dirac equations and vice versa.In Sec.III we develop an algorithm for the determination of the Hamiltonian constraints.In Sec.IV we develop an analogous algorithm for the Lagrangian constraints and we relate the Lagrangian and Hamiltonian constraints.

II. THE EQUIVALENCE THEOREMS
We consider an N-dimensional configuration space Q and a function L, the Lagrangian, defined in its tangent bundle TQ.If the Hessian matrix is singular, neither the existence nor uniqueness theorems for SODE holds.This means that the possible solutions of (2.2) lie in a submanifold of TQ and given a point of that submanifold we can have more than one solution passing through that point.We shall assume in the following that the rank of the Hessian matrix W is constant in all TQ and is n -m I' If this is not the case, our considerations will only hold in an open region of TQ where this condition is satisfied.

A. The map FL
The fiber derivative of the Lagrangian is the application (FL) of the tangent bundle on the cotangent bundle T *Q given by FL(q,q) = (q,p), where We shall also assume that FL(TQ) =MoCT*Q is a submanifold of T *Q, locally defined by the constraints <I>~O) (q,p) = 0, p, = 1,2, ... ,m " which are the primary constraints.
We also assume rank --p,-= mi' This condition excludes ineffective constraints at this level.
In the following we will disregard Lagrangians that have ineffective constraints at any level.The primary Hamiltonian constraints (2.3) are identified at the Lagrangian level, i.e., <I>~O) (q,flJ (q,q) )=0, (2.5a) or equivalently where FL* is the pullback application.From (2.5) we deduce a<I>(O) oflJ .~q,flJ (q,q)J-;:-!-= 0, cJPI dqj and since oflJ Joqj is the Hessian matrix element Wij' we have a basis for the null vectors of W: A basis for the kernel of differential application FL * can be written in terms of (2.7) : . .a rp, = r'p, (q,q)-;-:-.
Due to the condition (2.4) given a point ofthis leaf we can determine the parameters if in terms of the coordinates of this point.This means that for a given point (q,p)eMoand all its possible anti-images we have the relation (2.17) Ifwe now consider Eq. (2.17) as a system with (q,p) as data and q as unknowns, we show in Appendix A that there are no solutions if the data are out of M o ; whereas if the (qp)eM o , the solutions of (2.17) are obviously given by (2.16).Therefore we conclude that the relations (2.17) and (2.2) are equivalent.Therefore Eq. (2.17) is the inverse Legendre transformation; note that Eq. (2.17) is an implicit equation for q.
Let us observe that the application of r to both sides of (2.17) gives rp,vy=l)p,y' (2.18)This means that all the functions vp, (qq) are not FL-projectable.However, we shall see in Sec.III that some of these functions admit a canonical form when restricted to a suitable submanifold of TQ.Now if we take the derivative of (2.11) with respect to qjJ we have where (qp) = FL(qq).If we use Eq. ( 2.17) we have (2.21 ) Therefore, Eq. ( 2.20) can be written as aL aB a4>( 0) Let us now consider the equations of motion.
Theorem: If q(t) is a solution of Euler-Lagrange equations (2.1 ) in configuration space, the lifting to T *Q given by (q(t),p(t») with p(t) defined by (2.2), is a solution of the Hamilton-Dirac equatic;ms (2.25) and (2.27).
If we consider the canonical symplectic structure of T*Q we can write Eqs.(2.25) and (2.27) in terms ofPB as ~~ = {q,Hc} + VI'( q, ~~ ){q, ct>~0)}, (2.28) (2.29)These equations are not written in the normal form, in the same sense as the Euler-Lagrange equations of motion, (2.1 ), therefore the possible solutions of those equations lie in a submanifold of T *Q and the solution passing through a point of that submanifold is not necessarily unique.
Equations (2.28) and (2.39) can be written in a normal form if one introduces m 1 arbitrary functions of the evolution parameter AI' (t), and also imposes from the outset the primary constraints

III. HAMILTONIAN FORMALISM
In the preceding section we assumed the existence of solutions of the equations of motion and we have shown the equivalence between the Lagrangian and Hamiltonian formalism.Now we study the submanifold where those solutions exist, we will use an iterative procedure.Let us begin with the Hamiltonian formalism, the Hamilton-Dirac equations of motion are Eqs.(2.28) and (2.29): where ct>~0) are the primary Hamiltonian constraints and vI' (q,q) are known function of q and q.We know, from Appendix A, that (3.1) have only solutions if the initial conditions belong to the submanifold Moe T*Q.In that case, a curve passing through a point of Mo will be a solution of (3.1) if that is, the solution (q (t) ,p (t) ) must belong entirely to Mo.In general, Eqs.(3.2) will be restrictions for the initial conditions.We write (3.2) as To discuss the content of (3.2) it is necessary to know the rank of the PB matrix between the primary constraints, i.e., If all these conditions are satisfied on Mo the analysis is finished, if this is not the case Eqs.(3.13) are new restrictions on the initial conditions, which we call secondary constraints.Note that some of these constraints can be automatically satisfied on M o , but in order to use a more compact notation we will continue to use the SUbscript /-lo for all secondary constraints.
Let M I , be the new submanifold defined by A curve passing through a point of MI will be a solution of These stability conditions can be written explicitly 2956 J. Math.Phys., Vol. 27, No. 12, December 1986 o = {<I>~!),H~I)} + vvo(q,q) {<I>:.!), <I>~~)}. (3.15b) M, In Eq. (3.15) a PB matrix appears between the primary firstclass constraints on Mo and the secondary constraints.As is shown in Appendix B this matrix is symmetric: Let m 2 -m3 be the rank of this matrix.Due to this symmetry property we can introduce a new set of constraints with the following properties: (3.17 m 2 -m3 of the <I>:.!).The<l>~~) are a linear combination of the <I>~) and the <I>~:) are the same linear combination of the <I>~!).This means that we also have This means that the labeling of new constraints is compatible with their stability, therefore we have a sort of hereditary property.It should be noted that <I>~~) are first-class constraints on MI' At this point, we consider the stability condition for the constraints <I>(!).From this we obtain a canonical expression v, for the functions v , (qq): ", The evolution on Ml is given by where H( 2)=H( 1)+{H( 1) with the properties {<I>(!), H~2)} = O. (3.25) Now consider the stability of the remaining secondary constraints !£<I>(1) = {<I> (1) H(2)}:s<l>(2) the relations <I>~~) = 0 can be satisfied on Ml in which case the analysis is finished.Otherwise <I>~~) = 0, /-ll = 1, ... ,m 3 , (3.27) are tertiary constraints.At this level the evolution is restricted to the submanifold M 2 : In order to stQdy the stability of tertiary constraints ct>(2), we need to consider the PB matrix of the primary first- which define the same surface as the set ct>~~), ct>~~), with the properties The stability conditions for the tertiary constraints ct>(~), as in the previous case, enables us to obtain a canonical expression for the functions v •• Using that expression the evolu-

V2
tion on M2 is given by where (3.36) with the property (3.40) If the relations (3.39) are verified on M 2 , the analysis is finished.Otherwise we have more constraints and therefore we need further to require the stability of those constraints and the procedure continues as before.Let assume that our Lagrangian has a final submanifold M f where we have solution of the equations of motion (3.1).We write the constraints defining M f as 2957 J. Math.Phys., Vol. 27, No. 12, December 1986 ct>(0) The matrix C (j) is symmetric or antisymmetric depending on whether/is odd or even.
We have and since the analysis is finished we have Note that in the equations of motion there appear functions v" (qq) that are not determined canonically and are asso- ciated with the primary first class constraints on the submanifold M f .
Let us now study the relation between the procedure of Dirac brackets (DB) for second class constraints, 1 and the procedure developed here.Let us begin with the case with no tertiary constraints.The DB with respect to second-class constraints ct>(~) ct>(~) ct>(!) can be constructed in two steps.First we construct the DB for the constraints ct>~), i.e., (3.46)where (a (1) -1 is the inverse matrix of D (I) defined by where C( 2) is the matrix defined in(3.18) and K is a matrix constructed with the ct>( ~) constraints.

J',
Let us consider (3.47). in the case B = He.Using Eqs.(3.48), (3.12), and (3.24) we have Therefore our procedure on the MI surface is equivalent to the Dirac procedure.In Appendix C we explicitly prove this result for the case of no quartiary constraints.In order to give a proof in the general case, we need to consider a more geometrical formulation that takes into account the new structures we have found.Work in that direction is in progress.
Summing up, Eqs.(3.42) are equivalent on M f to the equations of motion generated by the total Dirac Hamiltonian: where H H(f + J) is the starred Hamiltonian 13 with respect to all second-class constraints.Therefore the OB is not the minimal structure to obtain the Hamiltonian equations of motion.

IV. LAGRANGIAN FORMALISM: RELATION BETWEEN THE LAGRANGIAN AND HAMILTONIAN CONSTRAINTS
In the previous section we have built a new scheme for the construction and classification of the submanifold of the Hamiltonian constraints.Now, we shall use these results to do the same with the Lagrangian constraints.
Using the Hessian matrix WIj' we can write the Lagrangian equations of motion (2.1) as WljqJ = a o where (4.1 ) If the rank of Wis n -m l , m J > 0, the Hessian will have m J null vectors YJ.t (q,q) such that The contraction of Eq. (4.1) with a null vector gives (1)_ ._XI' =air;.-O.

ApO
Let us now consider the stability of the FL-projectable constraints.We have If the relations XJ.!) = 0 are automatically verified in SI' the analysis is finished.Otherwise the X~) are the second generation of the Lagrangian constraints, which together with X~I) define the surface S2' Now it is necessary to study the stability of X~).It is possible to show that where K is the operator defined in (4.6).So the X~) are associated with the Hamiltonian constraints ~:.!).Remembering the splitting ( 3.17 ), it follows that we have the following splitting at Lagrangian level: This means that we have no more weakly FL-projectable constraints.However, we have the non-FL-projectable constraints x~t: 1).If we consider the stability of these constraints, we can obtain the undetermined fl ,acceleration in PI terms of the coordinate and momenta.At this point the analysis is finished.
Summing up, at every level a (weakly) FL-projectable constraint on a certain submanifold comes from the stability of a Hamiltonian constraint of the preceding level, which is first class with respect to the primary Hamiltonian constraint, while a non-FL-projectable constraint comes from the stability of a Hamiltonian constraint which converts a primary constraint to the second class.Also, if a certain number of velocities are canonically determined at a given level, the same number of accelerations are determined at the next level.

V. CONCLUSIONS
The equivalence between the Lagrangian and Hamiltonian formalism for constrained systems has been proved, in the sense that given a solution q( + ) of Euler-Lagrange equations of motion, the functions q(t) and pet) = 9(q(t) [dq(t)/dt]) are solutions of the Hamiltonian-Dirac equations of motion and vice versa.Note that neither of these equations is in normal form.This means that we can only have solutions in a submanifold of the respective space.These submanifolds are constructed through an interactive procedure.At the Hamiltonian level, our procedure differs from the standard one.All constraints are classified according to whether or not they are first class with respect to the primary constraints.We have seen that PB matrix of the primary first-class constraints on M o and the secondary, tertiary, ... constraints are either symmetric or antisymmetric.This implies that our final Hamiltonian H ~ f + 1) differs from the starred Hamiltonian of Komar and Bergman, but on the final submanifold M f they both yield the same evolution.
At the Lagrangian level, we have seen that the Lagrangian constraints can be obtained from the stability of the Hamiltonian constraints using the K operator (4.6).Furthermore, the Lagrangian constraints that are FL-projectable or weakly FL-projectable are the Lagrangian counterparts of the Hamiltonian constraints, which are first class or second class with respect to the primary Hamiltonian constraints.In fact at every level, a (weakly) FL-projectable constraint on a certain submanifold comes from the stability of a Hamiltonian constraint of the preceding level, which is first class with respect to the primary Hamiltonian constraint, while a non-FL-projectable constraint comes from the stability of a Hamiltonian constraint that converts a primary constraint to the second class.Also, if a certain number of velocities are canonically determined at a given level, the same number of accelerations are determined at the next level.