On the Canonical Treatment of Lagrangian Constraints

A detailed program is proposed in the Lagrangian formalism to investigate the dynamical behavior of a theory with singular Lagrangian. This program goes on, at different levels, parallel to the Hamiltonian analysis. In particular, we introduce the notions of first class and second class Lagrangian constraints. We show each sequence of first class constraints leads to a Neother identity and consequently to a gauge transformation. We give a general formula for counting the dyna...

This is a review of the constrained dynamical structure of Poincare gauge theory which concentrates on the basic canonical and gauge properties of the theory, including the identification of constraints, gauge symmetries and conservation laws. As an interesting example of the general approach we discuss the teleparallel formulation of general relativity.

In this paper we examine an alternative formulation of the gauge principle in which the emphasis is shifted from the symmetry transformations to their generators. We show that the gauge principle can be entirely reformulated in terms of promoting constants of motion - which generate rigid symmetries - to constraints - which generate gauge symmetries. In our exposition we first explain the basic philosophy on mechanical systems, and then with the help of De Donder--Weyl formal...

There is a review of the physical theories needing Dirac-Bergmann theory of constraints at the Hamiltonian level due to the existence of gauge symmetries. It contains: i) the treatment of systems of point particles in special relativity both in inertial and non-inertial frames with a Wigner-covariant way of eliminating relative times in relativistic bound states; ii) the description of the electro-magnetic field in relativistic atomic physics and of Yang-Mills fields in a...

The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments.

The present article deals with general mechanics in an unconventional manner. At first, Newtonian mechanics for a point particle has been described in vectorial picture, considering Cartesian, polar and tangent-normal formulations in a unified manner. An extensive differential geometric notions have been used when motion on curved surfaces has been considered. Both the Lagrangian and the Hamiltonian formulations have been discussed with various examples. The relevant part of ...

The evolution equations of Einstein's theory and of Maxwell's theory---the latter used as a simple model to illustrate the former--- are written in gauge covariant first order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom [the spatial shift vector $\beta^{i}(t,x^{j})$ and the spatial scalar potential $\phi(t,x^{j})$, respectively] are not among the dynam...

A gauge-invariant formulation of constrained variational calculus, based on the introduction of the bundle of affine scalars over the configuration manifold, is presented. In the resulting setup, the Lagrangian is replaced by a section of a suitable principal fibre bundle over the velocity space. A geometric rephrasement of Pontryagin's maximum principle, showing the equivalence between a constrained variational problem in the state space and a canonically associated free one...

Classical mechanical systems with internal constraints will be studied using the extended symplectic formalism of Faddeev-Jackiw. Generalized brackets of the theory and equations of motion will be derived. A gauge system will be examined, leading to the associated gauge transformations. The results will be compared with the Dirac-Bergmann algorithm, which has already been reported in the literature. It will be shown that the symplectic approach is simpler and more economical ...

The application of the Legendre transformation to a hyperregular Lagrangian system results in a Hamiltonian vector field generated by a Hamiltonian defined on the phase space of the mechanical system. The Legendre transformation in its usual interpretation can not be applied to homogeneous Lagrangians found in relativistic mechanics. The dynamics of relativistic systems must be formulated in terms of implicit differential equations in the phase space and not in terms of Hamil...