Constraints in Hamiltonian time-dependent mechanics

Abstract

The key point of the study of constraints in Hamiltonian time-dependent mechanics lies in the fact that a Poisson structure does not provide dynamic equations and a Poisson bracket of constraints with a Hamiltonian is ill-defined. We describe Hamiltonian dynamics in terms of Hamiltonian forms and connections on the vertical cotangent bundle V*Q→R seen as a momentum phase space. A Poisson bracket {,}V on V*Q is induced by the canonical Poisson bracket {,}T on the cotangent bundle T*Q. With {,}V, an algebra of first and second class time-dependent constraints is described, but we use the pull-back of the evolution equation onto T*X and the bracket {,}T in order to extend the constraint algorithm to time-dependent constraints. The case of Lagrangian constraints of a degenerate almost regular Lagrangian is studied in detail. One can assign to this Lagrangian L a set of Hamiltonian forms (which are not necessarily degenerate) such that any solution of the corresponding Hamilton equations which lives in the Lagrangian constraint space is a solution of the Lagrange equations for L. In the case of an almost regular quadratic Lagrangian, the complete set of global nondegenerate Hamiltonian forms with the above-mentioned properties is described. We construct the Koszul–Tate resolution of the Lagrangian constraints for this Lagrangian in an explicit form.