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In this paper, we propose a discrete Hamilton--Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton--Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton--Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton--Jacobi equation is related to the continuous Hamilton--Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton--Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper we provide a numerical example to test the theory., Comment: Hamilton-Jacobi theory; Discrete dynamics; Contact manifolds; Discrete Hamilton--Jacobi

In this article we construct and discuss a new rigorous geometric formalism for gauge field theories. The basis of our work is the notion of the Tulczyjew triple, a geometric structure which successfully solved numerous problems in mathematical description of mechanics and classical field theory. In particular, we construct a Tulczyjew triple for gauge field theories and reduce it for systems that depend only on the value of a connection and curvature instead of the entire first jet of the gauge field. We also introduce new geometric structures such as the vector-affine product of bundles and analyse the connection bundle from a new perspective. Finally, we apply the derived formalism to Yang-Mills theory., Comment: 36 pages

Recently, T. Masson, J. Francois, S. Lazzarini, C. Fournel and J. Attard have introduced a new method of the reduction of gauge symmetry called a 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 the 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., Comment: 17 pages

In this paper we aim at presenting a concise but also comprehensive study of time-dependent (tdependent) Hamiltonian dynamics on a locally conformal symplectic (lcs) manifold. We present a generalized geometric theory of canonical transformations and formulate a time-dependent geometric Hamilton-Jacobi theory on lcs manifolds. In contrast to previous papers concerning locally conformal symplectic manifolds, here the introduction of the time dependency brings out interesting geometric properties, as it is the introduction of contact geometry in locally symplectic patches. To conclude, we show examples of the applications of our formalism, in particular, we present systems of differential equations with time-dependent parameters, which admit different physical interpretations as we shall point out., Comment: 25 pages

In this paper we provide a matched pair decomposition of the space of symmetric contravariant tensors $\mathfrak{T}\mathcal{Q}$. From this procedure two complementary Lie subalgebras of $\mathfrak{T}\mathcal{Q}$ under \textit{mutual} interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to these realizations, Euler-Poincar\'{e} flows on such spaces are decomposed into two subdynamics: one of which is the Euler--Poincar\'{e} formulation of isentropic fluid flows, and the other one corresponds with Euler--Poincar\'{e} equations on higher order contravariant tensors ($n\geq 2$), Comment: arXiv admin note: text overlap with arXiv:2004.12595

Some recent works reveal that there are models of differential equations for the mean and variance of infected individuals that reproduce the SIS epidemic model at some point. This stochastic SIS epidemic model can be interpreted as a Hamiltonian system, therefore we wondered if it could be geometrically handled through the theory of Lie--Hamilton systems, and this happened to be the case. The primordial result is that we are able to obtain a general solution for the stochastic/ SIS-epidemic model (with fluctuations) in form of a nonlinear superposition rule that includes particular stochastic solutions and certain constants to be related to initial conditions of the contagion process. The choice of these initial conditions will be crucial to display the expected behavior of the curve of infections during the epidemic. We shall limit these constants to nonsingular regimes and display graphics of the behavior of the solutions. As one could expect, the increase of infected individuals follows a sigmoid-like curve. Lie--Hamiltonian systems admit a quantum deformation, so does the stochastic SIS-epidemic model. We present this generalization as well. If one wants to study the evolution of an SIS epidemic under the influence of a constant heat source (like centrally heated buildings), one can make use of quantum stochastic differential equations coming from the so-called quantum deformation.

In this article we inspect the dynamics of classical field theories with a local conformal behavior. Our interest in the multisymplectic setting comes from its suitable description of field theories, and the conformal character has been added to account for field theories that are scale invariant, flat spaces, and because some conformal fields can be exactly solved or classified. In particular, we will solve the example of a conformal scalar field using the geometric Hamilton-Jacobi theory that is explicitly proposed for conformal fields on a multisymplectic manifold. To complete the geometric approach to study field theories, we propose the Hamilton-Jacobi theory for conformal fields in a Cauchy data space, in which space and time are split separately and the dynamics is depicted in an infinite-dimensional manifold.

In this paper we aim at addressing the globalization problem of Hamilton-DeDonder-Weyl equations on a local $k$-symplectic framework and we introduce the notion of {\it locally conformal $k$-symplectic (l.c.k-s.) manifolds}. This formalism describes the dynamical properties of physical systems that locally behave like multi-Hamiltonian systems. Here, we describe the local Hamiltonian properties of such systems, but we also provide a global outlook by introducing the global Lee one-form approach. In particular, the dynamics will be depicted with the aid of the Hamilton--Jacobi equation, which is specifically proposed in a l.c.k-s manifold., Comment: 25 pages

In this article we provide a Hamilton-Jacobi formalism in locally conformally symplectic manifolds. Our interest in the Hamilton-Jacobi theory comes from the suitability of this theory as an integration method for dynamical systems, whilst 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 local l.c.s. Hamilton-Jacobi 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 l.c.s. Hamilton--Jacobi theory, and how actually the local behavior can be glued to retrieve the global behavior of the Hamilton-Jacobi theory.

Recently, M. de Le\'on el al. ([8]) have developed a geometrical description of Hamilton-Jacobi theory for multisymplectic field theory. In our paper we analyse in the same spirit a special kind of field theories which are gauge field theories. The Hamilton-Jacobi theory for this kind of fields is shown.

Tulczyjew triple for physical systems with configuration manifold equipped with Lie group structure is constructed and discussed. The case of systems invariant with respect to group acton is considered together with appropriate reduction of the Tulczyjew triple. The theory is applied to free rigid-body dynamics.