Geometry of spaces with the Jacobi metric

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Abstract
The generalized Maupertuis principle is formulated for systems with the natural Lagrangian and an indefinite form of the kinetic energy. The generalization is applied to the theory of gravity and cosmology. For such systems, the metric determined by the kinetic energy form has a Lorentz signature. The theorem is proved concerning the behavior of trajectories in a neighborhood of the boundary of the region admissible for motion. This region is not a smooth manifold but turns out to be a differential space of constant differential dimension. This fact allows us to use geometric methods analogous to those elaborated for smooth manifolds. It is shown that singularities of the Jacobi metric are not dangerous for the motion; its trajectories are smooth in the sense of the theory of differential spaces.