On the geometry of singular Lagrangians

The geometry of Lagrangian systems, whose Legendre map possesses generic si ngularities, is studied. On its basis the Transition Principle, prescribing the behaviour of phase trajectories on the singular hypersurface, is propo sed. The notion of relative Hamiltonian vector field associated with an arb itrary Lagrangian is studied and in particular applied to the constraint al gorithm. (C) 2000 Elsevier Science B.V. All rights reserved.