An Overview of the Hamilton-Jacobi Theory: the Classical and Geometrical Approaches and Some Extensions and Applications

This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton-Jacobi theory. The relation with the classical Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, as well as how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation.

Automated summary

This work reviews the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton-Jacobi theory, analyzing the relation with canonical transformations. It also briefly explains a more general framework for the theory, showing how the Lagrangian and Hamiltonian cases of dynamical systems can be recovered from this generic framework and how the model can be extended to other types of physical systems.