Hamilton-Jacobi theory in multisymplectic classical field theories

Abstract

In this paper we extend the geometric formalism of the Hamilton-Jacobi theory for hamiltonian mechanics to the case of classical field theories in the framework of multisymplectic geometry and Ehresmann connections.

Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems

“…It is based on a principal idea: a hamiltonian vector field X H can be projected into the configuration manifold by means of a 1-form dW , then the integral curves of the projected vector field X dW H can be transformed into integral curves of X H provided that W is a solution of the Hamilton-Jacobi equation [3,21,26,30,43,54]. In the last decades, the Hamilton-Jacobi theory has been interpreted in modern geometric terms [10,11,32,47,52] and has been applied in multiple settings: as nonholonomic [10,11,32,37], singular lagrangian mechanics [38,40] and classical field theories [36,49]. The construction of a Hamilton-Jacobi theory often relies in the existence of lagrangian/legendrian submanifolds, a notion that has gained a lot of attention given its applications in dynamics since their introduction by Tulczyjew [58,59].…”

A geometric Hamilton–Jacobi theory on a Nambu–Jacobi manifold

“…Using the approach discussed above, the HJ theory has also been extended to nonholonomic mechanics, geometric mechanics on Lie algebroids, singular systems, control theory, classical field theories and different geometric backgrounds [5,6,22,18,19,21]. This proves the wide applicability of the geometric interpretation of the HJ theory and its recent interest among the scientific community.…”

Geometric Hamilton–Jacobi theory on Nambu–Poisson manifolds

“…In particular, the HJ theory has been widely studied from a geometric point of view. A lot of geometric results for solving the HJ equation were obtained in [9,10,25,27,28,30,34]. The primordial observation for a HJ theory is that if a Hamiltonian vector field X H can be projected into the configuration manifold by means of a 1-form dW , then the integral curves of the projected vector field X dW H can be transformed into integral curves of X H provided that W is a solution of (3).…”

“…Here, E = T * Q for instance. This kind of diagram has been applied to multiple theories, as nonhonolomic [9,10,27,29], singular Lagrangian Mechanics [26,30] and classical field theories [28] in different geometric settings [9,10,27,34,36,38].…”