Authors: F. Pugliese, A. M. Vinogradov

The geometry of Lagrangian systems, whose Legendre map possesses generic singularities, is studied. On its basis the Transition Principle, prescribing the behaviour of phase trajectories on the singular hypersurface, is proposed. As a by-product, the notion of relative Hamiltonian field associated with an arbitrary Lagrangian is introduced.

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