Regularity of generating families of functions
Włodzimierz M. Tulczyjew Paweł Urbański
Journal of Geometric Mechanics 2010, 2(2): 199-221 doi: 10.3934/jgm.2010.2.199
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
Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings
Janusz Grabowski Katarzyna Grabowska Paweł Urbański
Journal of Geometric Mechanics 2014, 6(4): 503-526 doi: 10.3934/jgm.2014.6.503
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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