The Role of the Jacobi Last Multiplier in Nonholonomic Systems and Locally Conformal Symplectic Structure

Abstract

In this pedagogic article we study the geometrical structure of nonholonomic system and elucidate the relationship between Jacobi’s last multiplier (JLM) and nonholonomic systems endowed with the almost symplectic structure. In particular, we present an algorithmic way to describe how the two form and almost Poisson structure associated to nonholonomic system, studied by L. Bates and his coworkers (Rep Math Phys 42(1–2):231–247, 1998; Rep Math Phys 49(2–3):143–149, 2002; What is a completely integrable nonholonomic dynamical system, in Proceedings of the XXX symposium on mathematical physics, Torun, 1998; Rep Math Phys 32:99–115, 1993), can be mapped to symplectic form and canonical Poisson structure using JLM. We demonstrate how JLM can be used to map an integrable nonholonomic system to a Liouville integrable system. We map the toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system.