Authors

Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański

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 \( T M \), 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.

Citation

  • Journal: Journal of Geometric Mechanics
  • Year: 2014
  • Volume: 6
  • Issue: 4
  • Pages: 503–526
  • Publisher: American Institute of Mathematical Sciences (AIMS)
  • DOI: 10.3934/jgm.2014.6.503

BibTeX

@article{Grabowski_2014,
  title={{Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings}},
  volume={6},
  ISSN={1941-4897},
  DOI={10.3934/jgm.2014.6.503},
  number={4},
  journal={Journal of Geometric Mechanics},
  publisher={American Institute of Mathematical Sciences (AIMS)},
  author={Grabowski, Janusz and Grabowska, Katarzyna and Urbański, Paweł},
  year={2014},
  pages={503--526}
}

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