Structural aspects of Hamilton-Jacobi theory
arXiv:1511.00288 · doi:10.1142/S0219887816500171
Abstract
In our previous papers [11,13] we showed that the Hamilton-Jacobi problem can be regarded as a way to describe a given dynamics on a phase space manifold in terms of a family of dynamics on a lower-dimensional manifold. We also showed how constants of the motion help to solve the Hamilton-Jacobi equation. Here we want to delve into this interpretation by considering the most general case: a dynamical system on a manifold that is described in terms of a family of dynamics (`slicing vector fields') on lower-dimensional manifolds. We identify the relevant geometric structures that lead from this decomposition of the dynamics to the classical Hamilton-Jacobi theory, by considering special cases like fibred manifolds and Hamiltonian dynamics, in the symplectic framework and the Poisson one. We also show how a set of functions on a tangent bundle can determine a second-order dynamics for which they are constants of the motion.
26 pages. Minor changes (some minor mistakes are corrected)
References in corpus (8)
- Hamilton-Jacobi Theory in k-Symplectic Field Theories
- Hamilton-Jacobi theory in k-cosymplectic field theories
- Geometric Hamilton-Jacobi theory for higher-order autonomous systems
- On the Hamilton-Jacobi Theory for Singular Lagrangian Systems
- Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs
- The Hamilton-Jacobi Formalism for Higher Order Field Theories
- Hamilton-Jacobi Diffieties
- Unified formalism for the generalized kth-order Hamilton-Jacobi problem
Cited by in corpus (6)
- Reviewing the Geometric Hamilton-Jacobi Theory concerning Jacobi and Leibniz identities
- Geometric Hamilton-Jacobi theory for systems with external forces
- Hamilton-Jacobi theory in multisymplectic classical field theories
- An overview of the Hamilton--Jacobi theory: the classical and geometrical approaches and some extensions and applications
- Hamilton-Jacobi theory for gauge field theories
- Symmetries and Reduction. Part II - Lagrangian and Hamilton-Jacobi picture