A Hamilton-Jacobi theory for implicit differential systems

Abstract

In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of $T{T}^{\ast }Q$ generated by Morse families. The implicit character implies the nonexistence of a Hamiltonian function describing the dynamics. This fact is here amended by a generating family of Morse functions which plays the role of a Hamiltonian. A Hamilton--Jacobi equation is obtained with the aid of this generating family of functions. To conclude, we apply our results to singular Lagrangians by employing the construction of special symplectic structures.

References in corpus (2)

• Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems
• A geometric Hamilton--Jacobi theory for a Nambu--Poisson structure

Cited by in corpus (7)

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• Reviewing the Geometric Hamilton-Jacobi Theory concerning Jacobi and Leibniz identities
• Hamilton-Jacobi theory and integrability for autonomous and non-autonomous contact systems
• Geometric Hamilton-Jacobi theory for systems with external forces
• A Hamilton-Jacobi formalism for higher order implicit systems
• On Locally Conformally Cosymplectic Hamiltonian Dynamics and Hamilton-Jacobi Theory
• Symmetries and Reduction. Part II - Lagrangian and Hamilton-Jacobi picture