Symmetries, Conservation and Dissipation in Time-Dependent Contact Systems

In contact Hamiltonian systems, the so-called dissipated quantities are akin to conserved quantities in classical Hamiltonian systems. In this article, a Noether's theorem for non-autonomous contact Hamiltonian systems is proved, characterizing a class of symmetries which are in bijection with dissipated quantities. Other classes of symmetries which preserve (up to a conformal factor) additional structures, such as the contact form or the Hamiltonian function, are also studied. Furthermore, making use of the geometric structures of the extended tangent bundle, additional classes of symmetries for time-dependent contact Lagrangian systems are introduced. The results are illustrated with several examples. In particular, the two-body problem with time-dependent friction is presented, which could be interesting in celestial mechanics.