The equivalence between the Hamiltonian and the Lagrangian formulations for the parametrization invariant theories

The link between the tratment of singular Lagrangians as field systems and the canonical Hamiltonian approach is studied. It is shown that the singular Lagrangians as field systems are always in exact agreement with the canonical approach for the parametrization invariant theories.

The equations of motion are obtained as total differential equations in many variables as follows: α, β = 0, n − r + 1, ..., n, a = 1, ..., n − r where z = S(t α ; q a ). The set of equations (4,5) is integrable [3,4] if If condition (6,7) are not satisfied identically, one considers them as new constraints and again testes the consistency conditions. Hence, the canonical formulation leads to obtain the set of canonical phase space coordinates q a and p a as functions of t α , besides the canonical action integral is obtained in terms of the canonical coordinates.The Hamiltonians H ′ α are considered as the infinitesimal generators of canonical transformations given by parameters t α respectively.
In ref. [5] the singular Lagrangians are treated as field systems. The Euler-Lagrange equations of singular systems are proposed in the form with constraints where In order to have a consistant theory, one should consider the variations of the constraints (9), (10).
In this paper we would like to study the link between the treatment of singular Lagrangians as field systems and the canonical formalism for the parametrization invariant theories.

Prametrization invariant theories as singular systems
In ref. [3] the canonical method treatment of the parametrization-invariant theories is studied and will be briefly reviewed here. Let us consider a system with th action integral as where L is a regular Lagrangian with Hessian n. Parameterize the time t → τ (t), withτ = dτ dt > 0. The velocitiesq i may be expressed aṡ where q ′ i are defined as q Denote t = q 0 and q µ = (q 0 , q i ), µ = 0, 1, ..., n, then the action integral (13) may be written as which is parameterization invariant since L is homogeneous of first degree in the velocities q ′ µ with L given as The Lagrangian L is now singular since its Hessian is n.
The canonical method [1][2][3][4] leads us to obtain the set of Hamilton-Jacobi partial differential equations as follows: where H t is defined as Here, p τ i and p t are the generalized momenta conjugated to the generalized coordinates q i and t respectively.
The equations of motion are obtained as total differential equations in many variables as follows: Since vanishes identically, this system is integrable and the canonical phase space coordinates q i and p i are obtained in terms of the time (q 0 = t). Now, let us look at the Lagrangian (17) as a field system. Since the rank of the Hessian martix is n, this Lagrangian can be be treated as a field system in the form thus, the expression can be replaced in eqn. (17) to obtain the modified Lagrangian L ′ : Making use of eqn (8), we have Calculations show that eqn. (28) leads to well-known Lagrangian equation Using eqn. (20), we have In order to have a consistent theory, one should consider the total variation of H t . In fact Making use of eq. (10), one finds Besides, the quantity H 0 is identically satisfied and does not lead to constriants.

Classical fields as constrained systems
In the following sections we would like to study the Hamiltonian and the Lagrangian formulations for classical field systems and demonstrating the equivalence between these two formulations for the reparametrization invariant fields.
A classical relativistic field φ i = φ i ( x, t) in four space-time dimensions may be described by the action functional which leads to the Euler-Lagrange equations of motion as One can go over from the Lagrangian description to the Hamiltonian description by using the definition then canonical Hamiltonian is defined as The equations of motion are obtained aṡ

Reparametrization invariant fields
In analogy with the finite dimensional systems, we introduce the reparametrization invariant action for the field system as where Following the canonical method [1][2][3][4], we obtain the set of [HJPDE] as where H t is defined as and π (τ ) i , π t are the generalized momenta conjugated to the generalized coordinates φ i and t respectively.
The equations of motion are obtained as Now the Euler-Lagrangian equation for the field system reads as Again as for the finite dimensional systems, equations (44,45) are equivalent to equations (47) for field systems.

Conclusion
As it was mentioned in the introduction, if the rank of the Hessian matrix for discrete systems is (n−r); 0 < r < n, then the systems can be treated as field systems [5]. The treatment of Lagrangians as field systems is always in exact agreement with the Hamilton-Jacobi treatment for reparametrization invariant theories. The equations of motion (21, 22) are equivalent to the equations of motion (28, 29). Besides the the variations of constraints (31) and (32) are identically satisfied and no further constraints arise.
In analogy with the finite dimensional systems, it is observed that the Lagrangian and the Hamilton-Jacobi treatments for the reparametrization invariant fields are in exact agreement.