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### Open Access Journals

JGM

We describe the geometric structures involved in the variational formulation of physical theories. In presence of these structures,
the constitutive set of a physical system can be generated by a family of functions. We discuss conditions, under which a family of functions
generates an immersed Lagrangian submanifold. These conditions are given in terms of the Hessian of the family.

JGM

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 $T M$, 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 $\wedge^n T M$, 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 $\wedge^nT M\wedge^nT^*M$, which is canonically a graded bundle of degree $n$ over $\wedge^nT M$. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.

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