Nonholonomic constraints in $k$-symplectic Classical Field Theories

TL;DR

This paper develops a $k$-symplectic framework for classical field theories with nonholonomic constraints, enabling the use of projection operators and symmetry analysis to study constrained systems.

Contribution

It introduces a novel $k$-symplectic formalism for nonholonomic field theories, including methods for handling constraints and symmetries.

Findings

Projection operators relate constrained and free solutions.

Symmetries lead to nonholonomic momentum equations.

Framework simplifies the application of nonholonomic mechanics tools.

Abstract

A $k$-symplectic framework for classical field theories subject to nonholonomic constraints is presented. If the constrained problem is regular one can construct a projection operator such that the solutions of the constrained problem are obtained by projecting the solutions of the free problem. Symmetries for the nonholonomic system are introduced and we show that for every such symmetry, there exist a nonholonomic momentum equation. The proposed formalism permits to introduce in a simple way many tools of nonholonomic mechanics to nonholonomic field theories.