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 sα, 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.
- Publication:
-
Reports on Mathematical Physics
- Pub Date:
- June 2021
- DOI:
- 10.1016/S0034-4877(21)00041-0
- arXiv:
- arXiv:2002.10458
- Bibcode:
- 2021RpMP...87..347G
- Keywords:
-
- contact structure;
- field theory;
- Lagrangian system;
- dissipation;
- k-symplectic structure;
- k-contact structure;
- Mathematical Physics;
- High Energy Physics - Theory
- E-Print:
- arXiv admin note: text overlap with arXiv:1905.07354