Dynamics of finite-dimensional mechanical systems on Galilean manifolds

As presented in the related PAMM contribution ‘Kinematics of finite-dimensional mechanical systems on Galilean manifolds’, the state space of a time-dependent finite-dimensional mechanical system is defined as an affine subbundle of the tangent bundle to the Galilean manifold modeling the generalized space-time of the system. The second-order vector field on the state space that describes the system's motion can be associated with a differential two-form called the action form of the mechanical system. In this paper, we postulate the action form for time-dependent finite-dimensional mechanical systems. Moreover, we show that Lagrange's equations of the second kind can be derived as a chart representation of the conditions that define the second-order vector field which describes the motion.