Kinematics of finite‐dimensional mechanical systems on Galilean manifolds

Abstract

AbstractIn a coordinate‐free description of time‐independent finite‐dimensional mechanical systems the configuration manifold plays a central role. In the case of time‐dependent mechanical systems, time needs to be included in the space on which the related physical theory is formulated. In this respect, we show that a so‐called Galilean manifold not only provides a ‘generalized space‐time’ but that it allows the coordinate‐free presentation of a physical theory for time‐dependent finite‐dimensional mechanical systems. The motion of a mechanical system is interpreted as an integral curve of a second‐order vector field on the state space related to the Galilean manifold of the system. Second‐order vector fields, which are the coordinate‐free equivalent of second‐order differential equations, are in one‐to‐one correspondence with the action forms introduced by Loos [4,5]. Because of this bijective relation, the kinetic part of the theory can be formulated by postulating the action form governing the motion of a finite‐dimensional mechanical system.