Structural aspects of Hamilton–Jacobi theory
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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.
The final publication is available at Springer via http://dx.doi.org/10.1142/S0219887816500171
CitationCariñena, J.F., Gràcia, Xavier, Marmo, G., Martínez, E., Muñoz-Lecanda, Miguel C., Roman-Roy, N. Structural aspects of Hamilton–Jacobi theory. "International journal of geometric methods in modern physics", 1 Febrer 2016, vol. 13, núm. 2, p. 1-26.