Nonautonomous k-contact field theories

Abstract

This paper provides a new geometric framework to describe non-conservative field theories with explicit dependence on the space-time coordinates by combining the k-cosymplectic and k-contact formulations. This geometric framework, the k-cocontact geometry, permits to develop a Hamiltonian and Lagrangian formalisms for these field theories. We also compare this new formulation in the autonomous case with the previous k-contact formalism. To illustrate the theory, we study the nonlinear damped wave equation with external timedependent forcing.

Multicontact formulation for non-conservative field theories

“…As in the Lagrangian case, these equations match those obtained in the k-contact and kcocontact formulations of non-conservative field theories [46,61,83].…”

“…Of course all these equations are the same as those obtained in the k-cocontact formulation of non-conservative field theories [83] and also match those of k-contact formalism when the Lagrangian function does not depend on the spacetime variables x µ [48,61]. Furthermore, equation (35) relates the canonical action form with the variational formulation through the Lagrangian density (see (43)).…”

“…This new definition must be a generalization of the so-called cocontact structure first introduced in [21] to describe time-dependent dissipative mechanical systems. It should also be consistent with the notions of k-contact and k-cocontact manifolds for non-conservative field theories [46,48,61,83] in the sense that these models must lead to the same equations when applied to the same situations. This new geometric structure should lead to a system of partial differential equations describing the field theory.…”

“…This is motivated by the following fact: If (P, Θ, ω) is a (pre)multicontact manifold, we can introduce a system of pdes associated with the (pre)multicontact structure. We want these equations, when expressed in coordinates, to coincide with those derived from the variational formulation [49], which are also those obtained in the k-(co)contact formulation of nonconservative field theories [46,48,61,83]. This variational principle is an extension of the Herglotz variational principle for contact mechanical systems.…”

“…So, for the case of non-conservative field theories, it was natural to consider a construction similar to the k-symplectic or the k-cosymplectic formulations. This was developed by some of the authors in [46,48,61,83]. Other less-general approaches to these kinds of geometric structures were also introduced in [8,19,44,79,87].…”