An overview of the Hamilton--Jacobi theory: the classical and geometrical approaches and some extensions and applications
arXiv:2101.03830 · doi:10.3390/math9010085
Abstract
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, and 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.
arXiv comment22 pp
References in corpus (27)
- Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories
- Geometric Hamilton-Jacobi Theory
- A unified framework for mechanics. Hamilton-Jacobi equation and applications
- Geometric Hamilton-Jacobi Theory for Nonholonomic Dynamical Systems
- Geometry of Lagrangian First-order Classical Field Theories
- Nonholonomic Hamilton-Jacobi Theory via Chaplygin Hamiltonization
- A contact geometry framework for field theories with dissipation
- Discrete Hamilton-Jacobi Theory
- Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems
- A -contact Lagrangian formulation for nonconservative field theories
- A Geometric Hamilton-Jacobi Theory for Classical Field Theories
- Hamilton-Jacobi Theory in k-Symplectic Field Theories
- Nonholonomic Hamilton-Jacobi equation and Integrability
- Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints
- Hamilton-Jacobi theory in k-cosymplectic field theories
- Geometric Hamilton-Jacobi theory for higher-order autonomous systems
- A review on contact Hamiltonian and Lagrangian systems
- Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs
- A Hamilton-Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds
- Extended Hamilton-Jacobi theory, contact manifolds and integrability by quadratures
- The Hamilton-Jacobi Formalism for Higher Order Field Theories
- Hamilton-Jacobi Diffieties
- Hamilton-Jacobi theory in Cauchy data space
- Geometric Hamilton-Jacobi Field Theory
- Structural aspects of Hamilton-Jacobi theory
- Hamilton-Jacobi theory in multisymplectic classical field theories
- Unified formalism for the generalized kth-order Hamilton-Jacobi problem