A K-contact Lagrangian formulation for nonconservative field theories

Abstract

Dynamical systems with dissipative behaviour can be described in terms of contact manifolds and a modified version of Hamilton's equations. Dissipation terms can also be added to field equations, as showed in a recent paper where we introduced the notion of k-contact structure, and obtained a modified version of the De Donder–Weyl equations of covariant Hamiltonian field theory. In this paper we continue this study by presenting a k-contact Lagrangian formulation for nonconservative field theories. The Lagrangian density is defined on the product of the space of k-velocities times a k-dimensional Euclidean space with coordinates sa, which are responsible for the dissipation. We analyze the regularity of such Lagrangians; only in the regular case we obtain a k-contact Hamiltonian system. We study several types of symmetries for k-contact Lagrangian systems, and relate them with dissipation laws, which are analogous to conservation laws of conservative systems. Several examples are discussed: we find contact Lagrangians for some kinds of second-order linear partial differential equations, with the damped membrane as a particular example, and we also study a vibrating string with a magnetic-like term.