In this article, we provide a Hamilton–Jacobi formalism on locally conformally symplectic (lcs) manifolds. We are interested in the Hamilton–Jacobi as an alternative method for formulating the dynamics, while our interest in the locally conformal character will account for physical theories described by Hamiltonians defined on well-behaved line bundles, whose dynamic takes place in open subsets of the general manifold. We present a lcs Hamilton–Jacobi equation in subsets of the general manifold and then provide a global view by using the Lichnerowicz–deRham differential. We show a comparison between the global and local description of a lcs Hamilton–Jacobi theory, and how actually the local behavior can be glued to retrieve the global behavior of the Hamilton–Jacobi theory. A particular example is the case of Gaussian isokinetic dynamics in which we apply our structure in certain submanifolds of the phase space.

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 d W , then the integral curves of the projected vector field X H d W 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].

Tulczyjew triple for physical systems with configuration manifold equipped with a Lie group structure is constructed and discussed. Systems invariant with respect to group and subgroup actions are considered together with appropriate reductions of the Tulczyjew triple. The theory is applied to free and constrained rigid-body dynamics.

Recently, T. Masson, J. Francois, S. Lazzarini, C. Fournel and J. Attard have introduced a new method of the reduction of gauge symmetries called the dressing field method. In this paper we analyse this method from the fiber bundle point of view and we show the geometric implications for a principal bundle underlying a given gauge theory.We show how the existence of a dressing field satisfying certain conditions naturally leads to the reduction of the principal bundle and, as a consequence, to the reduction of the configuration and phase bundle of the system.