JGM
Regularity of generating families of functions
Włodzimierz M. Tulczyjew Paweł Urbański
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.
keywords: Lagrangian submanifold Liouville structure Hessian. symplectic reduction
JGM
Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings
Janusz Grabowski Katarzyna Grabowska Paweł Urbański
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.
keywords: minimal surfaces. Tulczyjew triples Lagrange formalism Hamiltonian formalism double vector bundles variational calculus

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