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The Journal of Geometric Mechanics (JGM)
 

The ubiquity of the symplectic Hamiltonian equations in mechanics

Pages: 1 - 34, Volume 1, Issue 1, March 2009      doi:10.3934/jgm.2009.1.1

 
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P. Balseiro - Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 123, 28006 Madrid, Spain and CONICET, Argentina (email)
M. de León - Unidad Asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 123, 28006 Madrid, Spain (email)
Juan Carlos Marrero - Unidad Asociada ULL-CSIC "Geometría Diferencial y Mecánica Geométrica", Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain (email)
D. Martín de Diego - Unidad Asociada ULL-CSIC Geometría Diferencial y Mecánica Geométrica, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 123, 28006 Madrid, Spain (email)

Abstract: In this paper, we derive a "Hamiltonian formalism" for a wide class of mechanical systems, that includes, as particular cases, classical Hamiltonian systems, nonholonomic systems, some classes of servomechanisms... This construction strongly relies on the geometry characterizing the different systems. The main result of this paper is to show how the general construction of the Hamiltonian symplectic formalism in classical mechanics remains essentially unchanged starting from the more general framework of algebroids. Algebroids are, roughly speaking, vector bundles equipped with a bilinear bracket of sections and two vector bundle morphisms (the anchors maps) satisfying a Leibniz-type property. The bilinear bracket is not, in general, skew-symmetric and it does not satisfy, in general, the Jacobi identity. Since skew-symmetry is related with preservation of the Hamiltonian, our Hamiltonian framework also covers some examples of dissipative systems. On the other hand, since the Jacobi identity is related with the preservation of the associated linear Poisson structure, then our formalism also admits a Hamiltonian description for systems which do not preserve this Poisson structure, like nonholonomic systems.
   Some examples of interest are considered: gradient extension of dynamical systems, nonholonomic mechanics and generalized nonholonomic mechanics, showing the applicability of our theory and constructing the corresponding Hamiltonian formalism.

Keywords:  Algebroids, Hamiltonian mechanics, gradient extension, nonholonomic mechanics, generalized nonholonomic systems, exact symplectic algebroid, closed symplectic sections.
Mathematics Subject Classification:  Primary: 70H05; Secondary: 70G45; 53D05; 37J60.

Received: December 2008;      Revised: March 2009;      Available Online: April 2009.