Singular Lagrangian Systems on Jet Bundles

TL;DR

This paper revisits the jet bundle formulation of time-dependent mechanics, providing a detailed analysis of singular Lagrangian constraints, their classification, and the extension of Dirac brackets within a geometric framework.

Contribution

It offers a comprehensive constraint algorithm for singular Lagrangians on jet bundles, including classification and the use of auxiliary connections and cosymplectic geometry.

Findings

Detailed constraint functions for singular Lagrangians

Classification of constraints into first and second class

Extension of Dirac brackets using second class constraints

Abstract

The jet bundle description of time-dependent mechanics is revisited. The constraint algorithm for singular Lagrangians is discussed and an exhaustive description of the constraint functions is given. By means of auxiliary connections we give a basis of constraint functions in the Lagrangian and Hamiltonian sides. An additional description of constraints is also given considering at the same time compatibility, stability and second-order condition problems. Finally, a classification of the constraints in first and second class is obtained using a cosymplectic geometry setting. Using the second class constraints, a Dirac bracket is introduced, extending the well-known construction by Dirac.