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Lagrangian Lie Subalgebroids Generating Dynamics for Second-Order Mechanical Systems on Lie Algebroids

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Abstract

The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles, Lie algebras, principal bundles, reduced systems, and constrained are included in such description. In this paper, we investigate how to derive the dynamics associated with a Lagrangian system defined on the set of admissible elements of a given Lie algebroid using Tulczyjew’s triple on Lie algebroids and constructing a Lagrangian Lie subalgebroid of a symplectic Lie algebroid, by building on the geometric formalism for mechanics on Lie algebroids developed by M. de León, J.C. Marrero and E. Martínez on “Lagrangian submanifolds and dynamics on Lie algebroids”.

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Authors and Affiliations

  1. Center for Research and Development in Mathematics and Applications, ISCA, Universidade de Aveiro, 3810-500, Aveiro, Portugal

    Lígia Abrunheiro

  2. Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049, Madrid, Spain

    Leonardo Colombo

Authors
  1. Lígia Abrunheiro
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  2. Leonardo Colombo
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Correspondence to Leonardo Colombo.

Additional information

The work of Leonardo Colombo was partially supported by Ministerio de Economía, Industria y Competitividad (MINEICO, Spain) under Grant MTM2016-76702-P and ”Severo Ochoa Programme for Centres of Excellence” in R & D (SEV-2015-0554). The work of Lígia Abrunheiro was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013. Part of the results of this work corresponds with the Ph.D. thesis of the corresponding author [9].

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Abrunheiro, L., Colombo, L. Lagrangian Lie Subalgebroids Generating Dynamics for Second-Order Mechanical Systems on Lie Algebroids. Mediterr. J. Math. 15, 57 (2018). https://doi.org/10.1007/s00009-018-1108-x

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  • DOI: https://doi.org/10.1007/s00009-018-1108-x

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