GEOMETRY OF LAGRANGIAN AND HAMILTONIAN FORMALISMS IN THE DYNAMICS OF STRINGS

The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is TM, is based on the existence of canonical symplectic isomorphisms of double vector bundles T*TM, T*T*M, and TT*M, where the symplectic structure on TT*M is the tangent lift of the canonical symplectic structure T*M. We show that there exists an analogous picture in the dynamics of objects for which the configuration space is Lambda(TM)-T-n, if we make use of certain structures of graded bundles of degree n, i.e. objects generalizing vector bundles (for which n = 1). For instance, the role of TT*M is played in our approach by the manifold Lambda(TM)-T-n Lambda T-n*M, which is canonically a graded bundle of degree n over Lambda(TM)-T-n. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.