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June  2010, 2(2): 159-198. doi: 10.3934/jgm.2010.2.159

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

Manuel de León 1, , Juan Carlos Marrero 2, and  David Martín de Diego 3,

1. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 123, 28006 Madrid, Spain
2. 

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
3. 

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

Received  April 2010 Revised  June 2010 Published  June 2010

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, almost differential, Hamiltonian morphism, orbit theorem, skew-symmetric algebroid, nonholonomic mechanical system., linear almost Poisson structure.
Mathematics Subject Classification: 17B66, 37J60, 53D17, 70F25, 70G45, 70H2.
Citation: Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
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