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To describe time-dependent finite-dimensional mechanical systems, their generalized space-time is modeled as a Galilean manifold. On this basis, we present a geometric mechanical theory that unifies Lagrangian and Hamiltonian mechanics. Moreover, a general definition of force is given, such that the theory is capable of treating nonpotential forces acting on a mechanical system. Within this theory, we elaborate the interconnections between classical equations known from analytical mechanics such as the principle of virtual work, Lagrange's equations of the second kind, Hamilton's equations, Lagrange's central equation, Hamel's generalized central equation as well as Hamilton's principle.

The principle of virtual work and Hamilton's principle on Galilean manifolds