Singular lagrangians: some geometric structures along the Legendre map

TL;DR

This paper introduces new geometric structures along the Legendre map that clarify the relationship between Lagrangian and Hamiltonian formalisms for singular Lagrangians, providing tools for analyzing vector fields, symmetries, and presymplectic forms.

Contribution

It presents novel geometric structures and vector fields that enhance understanding of the Legendre map in singular Lagrangian systems, addressing projectability, kernels, and symmetries.

Findings

Constructed vector fields in velocity space for singular Lagrangians.

Provided methods to compute the kernel of the presymplectic form.

Characterized dynamical symmetries in the Lagrangian formalism.

Abstract

New geometric structures that relate the lagrangian and hamiltonian formalisms defined upon a singular lagrangian are presented. Several vector fields are constructed in velocity space that give new and precise answers to several topics like the projectability of a vector field to a hamiltonian vector field, the computation of the kernel of the presymplectic form of lagrangian formalism, the construction of the lagrangian dynamical vector fields, and the characterisation of dynamical symmetries.