Constraint algorithm for singular field theories in the <i>k</i>-cosymplectic framework

Abstract

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of k-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of k-precosymplectic structure, which is a generalization of the k-cosymplectic structure. Next k-precosymplectic Hamiltonian systems are introduced in order to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a submanifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.

Erratum: Constraint algorithm for singular field theories in the $ k $-cosymplectic framework

“…-a linear function T 1 k Q → R on the fibers of the bundle T 1 k Q → Q; -an arbitrary function R k × Q → R. In this way, the difference in the treatment of the Lagrangian (2) with respect to the Lagrangian (1) is that ∂f µ j ∂x α = 0, both in the Lagrangian and the Hamiltonian formalisms. With these changes, most of the equations of section 6.1 of [1] are correct, as well as its conclusions. Notice in particular that, for Lagrangians of the form (2), Reeb vector fields always exist and can be taken to be R α = ∂ ∂x α .…”

“…A simple affine Lagrangian model. Here we analyze a simple Lagrangian of type (2), which should replace example 6.2 in [1]. Lagrangian formalism: The configuration manifold is R 2 ×Q = R 2 ×R 2 , with coordinates (x 1 , x 2 ; q 1 , q 2 ).…”

“…Thus, the final constraint submanifold is S 1 . Hamiltonian formalism: The Hamiltonian formalism takes place in the bundle R 2 ×⊕ 2 T * Q, which has coordinates (x 1 , x 2 , y 1 , y 2 , p 1 1 , p 2 1 , p 1 2 , p 2 2 ). The Legendre map…”

“…Its image is the submanifold P of R 2 × ⊕ 2 T * Q given by the primary constraints p 1 1 = q 2 , p 2 1 = 0 , p 1 2 = 0 , p 2 2 = −q 1 ; so, we can describe P with coordinates (x 1 , x 2 , q 1 , q 2 ). In P we have the forms η 1 = dx 1 , η 2 = dx 2 ; ω 1 = dq 1 ∧ dq 2 , ω 2 = dq 1 ∧ dq 2 , and the Reeb vector fields R…”

“…Consistency of field equations can be analyzed by means of a constraint algorithm. In our recent paper [1] we extended this analysis to the non-autonomous case. In this case the geometric setting is provided by the notion of k-precosymplectic structure.…”