Adiabatic Invariants of Herglotz Type for Perturbed Nonconservative Lagrangian Systems

Abstract

Both conservative and nonconservative systems can be studied simultaneously using the differential variational principle of Herglotz type. For a perturbed system, in which parameters change with time, it is useful to find adiabatic invariants. Based on the Herglotz differential variational principle, we study the perturbation of infinitesimal transformations and adiabatic invariants for perturbed nonconservative Lagrangian systems. From the generalized Euler-Lagrange equation and the invariance condition for the Hamilton-Herglotz action under the group of infinitesimal transformations, we obtain an exact invariant of Herglotz type for a holonomic nonconservative system. We propose a definition of higher-order adiabatic invariants of Herglotz type and obtain such adiabatic invariants for nonconservative Lagrangian systems with small perturbations. We prove the corresponding inverse theorem of adiabatic invariants. As examples of using the obtained results, we consider an oscillator with square damping and a system with two degrees of freedom.