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

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.

36 pages, 1 figure

References in corpus (2)

• Lie algebroids and Poisson-Nijenhuis structures
• Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids

Cited by in corpus (16)

• Geometric Hamilton-Jacobi Theory for Nonholonomic Dynamical Systems
• Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization
• Unimodularity and preservation of volumes in nonholonomic mechanics
• Almost Lie structures on an anchored Banach bundle
• Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints
• Gauge momenta as Casimir functions of nonholonomic systems
• A Hamilton-Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
• Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs
• Almost Lie Algebroids and Characteristic Classes
• On the Geometry of the Hamilton-Jacobi Equation and Generating Functions
• Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids
• Integrability on Direct Limit Banach manifolds
• Jacobi algebroids and quasi Q-manifolds
• The ubiquity of the symplectic hamiltonian equations in mechanics
• Geometrical structures of higher-order dynamical systems and field theories
• Simulating Nonholonomic Dynamics