The Journal of Geometric Mechanics (JGM)

Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics

Pages: 159 - 198, Volume 2, Issue 2, June 2010      doi:10.3934/jgm.2010.2.159

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Manuel de León - 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)
David Martín de Diego - Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, UAM, C/Nicolás Cabrera, 15, 28049 Madrid, Spain (email)

Abstract: In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. The proposed formalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vector bundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system). From them, it is possible to formulate the Hamilton-Jacobi equation, obtaining as a particular case, the classical theory. The main application in this paper is to nonholonomic mechanical systems. For it, we first construct the linear almost Poisson structure on the dual space of the vector bundle of admissible directions, and then, apply the Hamilton-Jacobi theorem. Another important fact in our paper is the use of the orbit theorem to symplify the Hamilton-Jacobi equation, the introduction of the notion of morphisms preserving the Hamiltonian system; indeed, this concept will be very useful to treat with reduction procedures for systems with symmetries. Several detailed examples are given to illustrate the utility of these new developments.

Keywords:  Hamilton-Jacobi equation, linear almost Poisson structure, almost differential, skew-symmetric algebroid, orbit theorem, Hamiltonian morphism, nonholonomic mechanical system.
Mathematics Subject Classification:  17B66, 37J60, 53D17, 70F25, 70G45, 70H20.

Received: April 2010;      Revised: June 2010;      Available Online: June 2010.