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Journal of Nonlinear Science

, Volume 25, Issue 1, pp 203–246 | Cite as

Unimodularity and Preservation of Volumes in Nonholonomic Mechanics

  • Yuri N. Fedorov
  • Luis C.  García-Naranjo
  • Juan C. MarreroEmail author
Article

Abstract

The equations of motion of a mechanical system subjected to nonholonomic linear constraints can be formulated in terms of a linear almost Poisson structure in a vector bundle. We study the existence of invariant measures for the system in terms of the unimodularity of this structure. In the presence of symmetries, our approach allows us to give necessary and sufficient conditions for the existence of an invariant volume, which unify and improve results existing in the literature. We present an algorithm to study the existence of a smooth invariant volume for nonholonomic mechanical systems with symmetry and we apply it to several concrete mechanical examples.

Keywords

Nonholonomic mechanical systems Linear almost Poisson structure Modular vector field Unimodularity Invariant volume forms Symmetries Reduction 

Mathematics Subject Classification

37C40 37J60 70F25 70G45 70G65 

Notes

Acknowledgments

This work has been partially supported by MEC (Spain) Grants MTM2009-13383, MTM2011-15725-E, MTM2012-34478, MTM2012-31714 and the project of the Canary Government ProdID20100210. All the authors are grateful to their institutions for funding our research visits, which allowed the completion of the present article.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Yuri N. Fedorov
    • 1
  • Luis C.  García-Naranjo
    • 2
  • Juan C. Marrero
    • 3
    Email author
  1. 1.Department de Matematica Aplicada IUniversitat Politecnica de CatalunyaBarcelonaSpain
  2. 2.Departamento de Matemáticas y MecánicaIIMAS-UNAMMexico CityMexico
  3. 3.ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Departamento de Matemática Fundamental, Facultad de MatemáticasUniversidad de La LagunaCanary IslandsSpain

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