Reviewing the geometric Hamilton-Jacobi theory concerning Jacobi and Leibniz identities

In this survey, we review the classical Hamilton-Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton-Jacobi equation for different geometric structures attending to one particular characterization: whether they fulfill the Jacobi and Leibniz identities simultaneously, or if at least they satisfy one of them. This property reviews the work of the present authors in such a way that it is presented according to a hierarchy of nested brackets. This is a new outlook in which we do not only present the geometric structures themselves, but their hierarchical relations and how they affect their dynamic through the Hamilton-Jacobi equation. In this regard, we review the case of time-dependent (t-dependent in the sequel) and dissipative physical systems as systems that fulfill the Jacobi identity but not the Leibnitz identity, as instances of cosymplectic and contact mechanical problems. Let us remark that dissipative here means dissipation provided by a dissipation parameter included in the Hamiltonian function, for instance, the entropy in some thermodynamical systems. Furthermore, we review the contact-evolution Hamilton-Jacobi theory as a split off the regular contact geometry, and that actually satisfies the Leibniz rule instead of Jacobi. Furthermore, we include a novel result, which is the Hamilton-Jacobi equation for conformal Hamiltonian vector fields as a generalization of the well-known Hamilton-Jacobi on a symplectic manifold, that is retrieved in the case of a zero conformal factor. The interest of a geometric Hamilton-Jacobi equation is the primordial observation that if a Hamiltonian vector field X-H can be projected into a configuration manifold by means of a one-form dW, then the integral curves of the projected vector field X(H)(dW )can be transformed into integral curves of X-H provided that W is a solution of the Hamilton-Jacobi equation. Geometrically, the solution of the Hamilton-Jacobi equation plays the role of a Lagrangian submanifold of a certain bundle. Exploiting these features in different geometric scenarios we propose a geometric theory for multiple physical systems depending on the fundamental identities that their dynamic satisfies. Different examples are pictured to reflect the results provided, being all of them new, except for one that is reassessment of a previously considered example. Finally, let us mention that an important issue in the study of the Hamilton-Jacobi equation is separability, relevant for finding conserved quantities. However, we will not deal with this subject in this survey, although we refer the reader to papers [16, 17, 136].