Abstract
The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.
Highlights
IntroductionX H with its projection X H via γ onto Q, in the sense that X H and X H are γ-related if and only if (2) holds, provided that γ be closed (or, equivalently, its image be a Lagrangian submanifold of ( T ∗ Q, ωQ )) has opened the possibility to discuss the Hamilton–Jacobi problem in many other scenarios [3,4,5,6]: nonholonomic systems, multisymplectic field theories, and time-dependent mechanics, among others
The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems
The above result is the natural extension of the well-known fact that a section α of the cotangent bundle πQ : T ∗ Q −→ Q is a Lagrangian submanifold with respect to the canonical symplectic structure ωQ = −dθQ on T ∗ Q if and only if α is a closed 1-form
Summary
X H with its projection X H via γ onto Q, in the sense that X H and X H are γ-related if and only if (2) holds, provided that γ be closed (or, equivalently, its image be a Lagrangian submanifold of ( T ∗ Q, ωQ )) has opened the possibility to discuss the Hamilton–Jacobi problem in many other scenarios [3,4,5,6]: nonholonomic systems, multisymplectic field theories, and time-dependent mechanics, among others. We notice that the Hamilton–Jacobi problem has been treated by other authors [10,11], who establish a relationship between the Herglotz variational principle and the Hamilton– Jacobi equation, their interests are analytical rather than geometrical. In. Section 5, we study the relations of the Hamilton–Jacobi problem for a contact Hamiltonian systems and its symplectification.
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