Variational problem of Herglotz type for Birkhoffian system and its Noether’s theorems

Abstract

Herglotz proposed a generalized variational principle through his work on contact transformations and their connections with Hamiltonian systems and Poisson brackets, which provides an effective method to study the dynamics of nonconservative systems. In this paper, the variational problem of Herglotz type for a Birkhoffian system is presented and the differential equations of motion for the system are established. The invariance of the Pfaff–Herglotz action under a group of infinitesimal transformations and its connection with the conserved quantities of the system are studied, and Noether’s theorem and its inverse for the Herglotz variational problem are derived. The variational problem of Herglotz type for a Birkhoffian system reduces to the classical Pfaff–Birkhoff variational problem under classical conditions. Thus, it contains Noether’s theorem for the classical Birkhoffian system as a special case. And since Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics, the results we obtained contain Noether’s theorem of Herglotz variational problems for Hamiltonian systems and Lagrangian systems as special cases. In the end of the paper, we give two examples to illustrate the application of the results.