Constrained Dynamics: Generalized Lie Symmetries, Singular Lagrangians, and the Passage to Hamiltonian Mechanics

TL;DR

This paper explores the symmetries of singular Lagrangian systems, demonstrating how they relate to Hamiltonian mechanics and providing a unified framework for constrained dynamics through generalized Lie symmetries.

Contribution

It introduces the concept of generalized Lie symmetries in singular Lagrangians and shows their role in linking Lagrangian and Hamiltonian formulations of constrained dynamics.

Findings

SOELVFs can be constructed from second-order Lagrangian vector fields.

All SOELVFs are projectable to the Hamiltonian phase space.

Lagrangian constraint algorithm is equivalent to Hamiltonian stability analysis.

Abstract

Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation\textemdash called second-order, Euler-Lagrange vector fields (SOELVFs)\textemdash with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.