Symmetries and periodic orbits in simple hybrid Routhian systems

Abstract

Symmetries are ubiquitous in a wide range of nonlinear systems. Particularly in systems whose dynamics is determined by a Lagrangian or Hamiltonian function. For hybrid systems which possess a continuous-time dynamics determined by a Lagrangian function, with a cyclic variable, the degrees of freedom for the corresponding hybrid Lagrangian system can be reduced by means of a method known as hybrid Routhian reduction. In this paper we study sufficient conditions for the existence of periodic orbits in hybrid Routhian systems which also exhibit a time-reversal symmetry. Likewise, we explore some stability aspects of such orbits through the characterization of the eigenvalues for the corresponding linearized Poincaré map. Finally, we apply the results to find periodic solutions in underactuated hybrid Routhian control systems.

A note on Hybrid Routh reduction for time-dependent Lagrangian systems

“…This work attempt to go one step further and discuss Routh reduction for nonautonomous hybrid systems. We remark that some of the techniques outlined here have been recently applied to the study of periodic orbits in reduced hybrid Lagrangian systems [6,7,8,13].…”