Constraints in Hamiltonian time-dependent mechanics

TL;DR

This paper explores the peculiarities of constraints in Hamiltonian time-dependent mechanics, highlighting differences from conservative mechanics and providing explicit constructions for degenerate systems.

Contribution

It develops a framework for understanding constraints in time-dependent Hamiltonian mechanics, including the Koszul-Tate resolution for degenerate Lagrangian systems.

Findings

Poisson bracket does not define dynamics in time-dependent Hamiltonian mechanics.

Relations between Lagrangian and Hamiltonian solutions are established.

Explicit Koszul-Tate resolution for degenerate systems is constructed.

Abstract

In Hamiltonian time-dependent mechanics, the Poisson bracket does not define dynamic equations, that implies the corresponding peculiarities of describing time-dependent holonomic constraints. As in conservative mechanics, one can consider the Poisson bracket of constraints, separate them in first and second class constraints, construct the Koszul-Tate resolution and a BRST complex. However, the Poisson bracket of constraints and a Hamiltonian makes no sense. Hamiltonian vector fields for first class constraints are not generators of gauge transformations. In the case of Lagrangian constraints, we state the comprehensive relations between solutions of the Lagrange equations for an almost regular Lagrangian and solutions of the Hamilton equations for associated Hamiltonian forms, which live in the Lagrangian constraint space. Degenerate quadratic Lagrangian systems are studied in details. We construct the Koszul-Tate resolution for Lagrangian constraints of these systems in an explicit form.