Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings

Abstract

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*TM, and TT*M. We show that there exist an analogous picture in the dynamics of objects for which the configuration space is the vector bundle of n-vectors, if we make use of certain graded bundle structures 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 vector bundle of n-vectors on the bundle of n-covectors, which is canonically a graded bundle of degree n over the bundle of n-vectors. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.