Constrained dynamics: generalized Lie symmetries, singular Lagrangians, and the passage to Hamiltonian mechanics

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-called second-order, Euler-Lagrange vector fields (SOELVFs)-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.

Generalized Lie symmetries and almost regular Lagrangians: a link between symmetry and dynamics

“…The symmetries of the Euler-Lagrange equations of motion were recently used to study the constrained dynamics of singular Lagrangians [1]. The focus was on almost regular Lagrangians [2][3][4][5], and it was found that for these Lagrangians the Euler-Lagrange equations of motion admit a generalized Lie symmetry (also known as a local gauge symmetry).…”

“…is not unique for almost regular Lagrangians, it was shown in [1] that the action of ym  on a general solution to this equation-and in particular, on the second-order, Lagrangian vector field (SOLVF)-will result in a vector field that is no longer a solution of equation (1). Thus, not all solutions of the energy equation have Gr ym  as a symmetry group.…”

“…Thus, not all solutions of the energy equation have Gr ym  as a symmetry group. It is, however, possible to construct solutions to equation (1) for whom ym  does generate a group of symmetry transformations [1]. These vector fields are called second-order, Euler-Lagrange vector fields (SOELVFs).…”

“…These vector fields are called second-order, Euler-Lagrange vector fields (SOELVFs). As the evolution of the dynamical system for singular Lagrangians must lie on Lagrangian constraint surfaces [5], a Lagrangian constraint algorithm for SOELVFs was also introduced in [1] to construct such solutions to the energy equation. It was then shown that these SOELVFs, along with the dynamical structures in the Lagrangian phase space needed to describe and determine the motion of the dynamical system, are projectable to the Hamiltonian phase space.…”

“…While [1] focused on the generalized Lie symmetries of the Euler-Lagrange equations of motion and whether the dynamical structures constructed in the Lagrangian phase space are projectable to the Hamiltonian phase space, in this paper the focus is on the symmetries of the action itself and the impact these symmetries have on the evolution of dynamical systems. This impact is found to be quite broad, surprisingly restrictive, and unexpectedly subtle.…”