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- Title
GEOMETRY OF LAGRANGIAN AND HAMILTONIAN FORMALISMS IN THE DYNAMICS OF STRINGS.
- Authors
GRABOWSKI, JANUSZ; GRABOWSKA, KATARZYNA; URBAŃSKI, PAWEŁ
- 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 in_nitesimal con_guration space is TM, 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 ΛnTM, 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 ΛnTMΛnT*M, which is canonically a graded bundle of degree n over ΛnTM. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.
- Subjects
HAMILTON'S equations; HAMILTONIAN mechanics; ISOMORPHISM (Mathematics); LAGRANGIAN mechanics; PLATEAU'S problem; VECTOR bundles
- Publication
Journal of Geometric Mechanics, 2014, Vol 6, Issue 4, p503
- ISSN
1941-4889
- Publication type
Article
- DOI
10.3934/jgm.2014.6.503