Isotropic submanifolds and the inverse problem for mechanical constrained systems

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We give a new characterization of the inverse problem of the calculus of variations that is easily extended to constrained systems, both in the autonomous and non-autonomous cases. The transition from unconstrained to constrained systems is given by passing from Lagrangian submanifolds to isotropic ones. If the constrained system is variational we use symplectic techniques to extend these isotropic submanifolds to Lagrangian ones and describe the solutions of the constrained system as solutions of a variational problem without constraints. Mechanical examples such as the rolling disk are provided to illustrate the main results.