Geometry of the discrete Hamilton--Jacobi equation. Applications in optimal control

In this paper, we review the discrete Hamilton--Jacobi theory from a geometric point of view. In the discrete realm, the usual geometric interpretation of the Hamilton--Jacobi theory in terms of vector fields is not straightforward. Here, we propose two alternative interpretations: one is the interpretation in terms of projective flows, the second is the temptative of constructing a discrete Hamiltonian vector field renacting the usual continuous interpretation. Both interpretations are proven to be equivalent and applied in optimal control theory. The solutions achieved through both approaches are sorted out and compared by numerical computation

A geometric Hamilton--Jacobi theory on a Nambu-Jacobi manifold

In this paper we propose a geometric Hamilton--Jacobi theory on a Nambu--Jacobi manifold. The advantange of a geometric Hamilton--Jacobi theory is 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 the vector field $X_H$ provided that $W$ is a solution of the Hamilton--Jacobi equation. This procedure allows us to reduce the dynamics to a lower dimensional manifold in which we integrate the motion. On the other hand, the interest of a Nambu--Jacobi structure resides in its role in the description of dynamics in terms of several Hamiltonian functions. It appears in fluid dynamics, for instance. Here, we derive an explicit expression for a geometric Hamilton--Jacobi equation on a Nambu--Jacobi manifold and apply it to the third-order Riccati differential equation as an example

Miura-reciprocal transformations for non-isospectral Camassa-Holm hierarchies in 2+12+1 dimensions

We present two hierarchies of partial differential equations in $2+1$ dimensions. Since there exist reciprocal transformations that connect these hierarchies to the Calogero-Bogoyavlenski-Schiff equation and its modified version, we can prove that one of the hierarchies can be considered as a modified version of the other. The connection between them can be achieved by means of a combination of reciprocal and Miura transformations

Non-isospectral 1+1 hierarchies arising from a Camassa--Holm hierarchy in 2+1 dimensions

The non-isospectral problem (Lax pair) associated with a hierarchy in 2+1 dimensions that generalizes the well known Camassa-Holm hierarchy is presented. Here, we have investigated the non-classical Lie symmetries of this Lax pair when the spectral parameter is considered as a field. These symmetries can be written in terms of five arbitrary constants and three arbitrary functions. Different similarity reductions associated with these symmetries have been derived. Of particular interest are the reduced hierarchies whose $1+1$ Lax pair is also non-isospectral

A Hamilton-Jacobi formalism for higher order implicit systems

In this paper, we present a generalization of a Hamilton--Jacobi theory to higher order implicit differential equations. We propose two different backgrounds to deal with higher order implicit Lagrangian theories: the Ostrogradsky approach and the Schmidt transform, which convert a higher order Lagrangian into a first order one. The Ostrogradsky approach involves the addition of new independent variables to account for higher order derivatives, whilst the Schmidt transform adds gauge invariant terms to the Lagrangian function. In these two settings, the implicit character of the resulting equations will be treated in two different ways in order to provide a Hamilton--Jacobi equation. On one hand, the implicit differential equation will be a Lagrangian submanifold of a higher order tangent bundle and it is generated by a Morse family. On the other hand, we will rely on the existence of an auxiliary section of a certain bundle that allows the construction of local vector fields, even if the differential equations are implicit. We will illustrate some examples of our proposed schemes, and discuss the applicability of the proposal

On Lie systems and Kummer-Schwarz equations

A Lie system is a system of first-order differential equations admitting a superposition rule, i.e., a map that expresses its general solution in terms of a generic family of particular solutions and certain constants. In this work, we use the geometric theory of Lie systems to prove that the explicit integration of second- and third-order Kummer--Schwarz equations is equivalent to obtaining a particular solution of a Lie system on SL(2,R). This same result can be extended to Riccati, Milne--Pinney and other related equations. We demonstrate that all the above-mentioned equations associated with exactly the same Lie system on SL(2,R) can be integrated simultaneously. This retrieves and generalizes in a unified and simpler manner previous results appearing in the literature. As a byproduct, we recover various properties of the Schwarzian derivative.Comment: 29 pages. A relevant error and several typos correcte

A Hamilton-Jacobi theory for implicit differential systems

In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of $TT^*Q$ generated by Morse families. The implicit character implies the nonexistence of a Hamiltonian function describing the dynamics. This fact is here amended by a generating family of Morse functions which plays the role of a Hamiltonian. A Hamilton--Jacobi equation is obtained with the aid of this generating family of functions. To conclude, we apply our results to singular Lagrangians by employing the construction of special symplectic structures

Lie--Hamilton systems: theory and applications

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearisability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.Comment: 20 page

Lie systems, Lie symmetries and reciprocal transformations

This work represents a PhD thesis concerning three main topics. The first one deals with the study and applications of Lie systems with compatible geometric structures, e.g. symplectic, Poisson, Dirac, Jacobi, among others. Many new Lie systems admitting Vessiot--Guldberg Lie algebras of Hamiltonian vector fields relative to the above mentioned geometric structures are analyzed and their importance is illustrated by their appearances in physical, biological and mathematical models. The second part details the study of Lie symmetries and reductions of relevant hierarchies of differential equations and their corresponding Lax pairs. For example, the Cammasa-Holm and Qiao hierarchies in 2+1 dimensions. The third and last part is dedicated to the study of reciprocal transformations and their application in differential equations appearing in mathematical physics, e.g. Qiao and Camassa-Holm equations equations appearing in the second part.Comment: PhD thesi

Integrable 1+11+1 dimensional hierarchies arising from the reduction of a nonisospectral problem in 2+12+1 dimensions

This work presents a classical Lie point symmetry analysis of a two-component, non-isospectral Lax pair of a hierarchy of partial differential equations in $2+1$ dimensions, which can be considered as a modified version of the Camassa-Holm hierarchy in $2+1$ dimensions. A classification of reductions for this spectral problem is performed. Non-isospectral reductions in $1+1$ dimensions are considered of remarkable interest