Kinematics of finite-dimensional mechanical systems on Galilean manifolds

In 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.