Some contributions to k-contact Lagrangian field equations, symmetries and dissipation laws

Abstract

It is well known that $k$-contact geometry is a suitable framework to deal with non-conservative field theories. In this paper, we study some relations between solutions of the $k$-contact Euler–Lagrange equations, symmetries, dissipation laws and Newtonoid vector fields. We review the $k$-contact Euler–Lagrange equations written in terms of $k$-vector fields and sections and provide new results relating the solutions in both approaches. We also study different kinds of symmetries depending on the structures they preserve: natural (preserving the Lagrangian function), dynamical (preserving the solutions), and $k$-contact (preserving the underlying geometric structures) symmetries. For some of these symmetries, we provide Noether-like theorems relating symmetries and dissipation laws. We also analyze the relation between $k$-contact symmetries and Newtonoid vector fields. Throughout the paper, we will use the damped vibrating string as our main illustrative example.