Geometric Hamilton Jacobi Theory Research Articles

A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics

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.

Hamilton–Jacobi theory for nonholonomic and forced hybrid mechanical systems

A hybrid system is a system whose dynamics is given by a mixture of both continuous and discrete transitions. In particular, these systems can be utilized to describe the dynamics of a mechanical system with impacts. Based on the approach by Clark [Invariant measures, geometry and control of hybrid and nonholonomic dynamical systems, Ph.D thesis, University of Mirchigan (2020)], we develop a geometric Hamilton–Jacobi theory for forced and nonholonomic hybrid dynamical systems. We state the corresponding Hamilton–Jacobi equations for these classes of systems and apply our results to analyze some examples.

Implicit contact dynamics and Hamilton-Jacobi theory

In this paper, we introduce implicit Hamiltonian dynamics in the framework of contact geometry in two different ways: first, we introduce classical implicit Hamiltonian dynamics on a contact manifold, followed by evolution Hamiltonian dynamics. In the first case, implicit contact Hamiltonian dynamics is defined as a Legendrian submanifold of a tangent contact space, whilst the implicit evolution dynamic is understood as a Lagrangian submanifold of a certain symplectic space embedded into the tangent contact space. To conclude, we propose a geometric Hamilton-Jacobi theory for both of these formulations.

Time-Dependent Hamiltonian Mechanics on a Locally Conformal Symplectic Manifold

In this paper we aim at presenting a concise but also comprehensive study of time-dependent (t-dependent) Hamiltonian dynamics on a locally conformal symplectic (lcs) manifold. We present a generalized geometric theory of canonical transformations in order to formulate an explicitly time-dependent geometric Hamilton-Jacobi theory on lcs manifolds, extending our previous work with no explicit time-dependence. In contrast to previous papers concerning locally conformal symplectic manifolds, the introduction of the time dependency that this paper presents, brings out interesting geometric properties, as it is the case 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.

Geodesic and Newtonian Vector Fields and Symmetries of Mechanical Systems

Geodesic vector fields and other distinguished vector fields on a Riemann manifold were used in the study of free motions on such a manifold, and we applied the geometric Hamilton–Jacobi theory for the search of geodesic vector fields from Hamilton–Jacobi vector fields and the same for closed vector fields. These properties were appropriately extended to the framework of Newtonian and generalised Newtonian systems, in particular systems defined by Lagrangians of the mechanical type and velocity-dependent forces. Conserved quantities and a generalised concept of symmetry were developed, particularly for Killing vector fields. Nonholonomic constrained Newtonian systems were also analysed from this perspective, as well as the relation among Newtonian vector fields and Hamilton–Jacobi equations for conformally related metrics.

On locally conformally cosymplectic Hamiltonian dynamics and Hamilton–Jacobi theory

Cosymplectic geometry has been proven to be a very useful geometric background to describe time-dependent Hamiltonian dynamics. In this work, we address the globalization problem of locally cosymplectic Hamiltonian dynamics that failed to be globally defined. We investigate both the geometry of locally conformally cosymplectic (LCC) manifolds and the Hamiltonian dynamics constructed on such LCC manifolds. Further, we provide a geometric Hamilton–Jacobi theory on this geometric framework.

Geometric Hamilton–Jacobi theory for systems with external forces

In this paper, we develop a Hamilton–Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems, and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton–Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton–Jacobi problem for a Čaplygin system to the Hamilton–Jacobi problem for a forced Lagrangian system.

Cauchy data space and multisymplectic formulation of conformal classical field theories

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.

Non-commutative integrability, exact solvability and the Hamilton–Jacobi theory

The non-commutative integrability (NCI) is a property fulfilled by some Hamiltonian systems that ensures, among other things, the exact solvability of their corresponding equations of motion. The latter means that an “explicit formula” for the trajectories of these systems can be constructed. Such a construction rests mainly on the so-called Lie theorem on integrability by quadratures. It is worth mentioning that, in the context of Hamiltonian systems, the NCI has been for around 40 years, essentially, the unique criterium for exact solvability expressed in the terms of first integrals (containing the usual Liouville–Arnold integrability criterium as a particular case). Concretely, a Hamiltonian system with n degrees of freedom is said to be non-commutative integrable if we know a set of independent first integrals \(F_{1},\ldots ,F_{l}\) such that: the kernel of the \(l\times l\) matrix with coefficients \(\left\{ F_{i},F_{j}\right\} \), where \(\left\{ \cdot ,\cdot \right\} \) denotes the canonical Poisson bracket, has dimension \(2n-l\) (isotropy); and each bracket \(\left\{ F_{i},F_{j}\right\} \) is functionally dependent on \(F_{1},\ldots ,F_{l}\) (closure). In this paper, we develop two procedures for constructing the trajectories of a Hamiltonian system which only require isotropic first integrals (closure condition is not needed). One of them is based on an extended version of the geometric Hamilton–Jacobi theory, and does not rely on the above mentioned Lie’s theorem. We do all that in the language of functions of several variables.

Hamilton-Jacobi Equation of Time Dependent Hamiltonians

In this work, we apply the geometric Hamilton-Jacobi theory to obtain solution of Hamiltonian systems in classical mechanics that are either compatible with two structures: the first structure plays a central role in the theory of time- dependent Hamiltonians, whilst the second is used to treat classical Hamiltonians including dissipation terms. It is proved that the generalization of problems from the calculus of variation methods in the nonstationary case can be obtained naturally in Hamilton-Jacobi formalism.

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

In this paper, we propose a geometric Hamilton–Jacobi (HJ) theory on a Nambu–Jacobi (NJ) manifold. The advantage of a geometric HJ theory is that if a Hamiltonian vector field [Formula: see text] can be projected into a configuration manifold by means of a one-form [Formula: see text], then the integral curves of the projected vector field [Formula: see text] can be transformed into integral curves of the vector field [Formula: see text] provided that [Formula: see text] is a solution of the HJ 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 NJ 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 HJ equation on a NJ manifold and apply it to the third-order Riccati differential equation as an example.

Geometry of the discrete Hamilton–Jacobi equation: applications in optimal control

In this paper, we review the discrete Hamilton—Jacobi equation from a geometric point of view. In similarity with the continuous geometric Hamilton—Jacobi theory, we propose two different discrete geometric interpretations for the equation. The first approach is based on the construction of a discrete Hamilton—Jacobi equation using discrete projective flows. For it, we develop some former results on discrete Hamiltonian systems and provide a discrete equation explicitly, which matches some previous results depicted in the literature. The interest of our method is that it retrieves some already known results, but starting from a new outlook. The second approach is formulated in terms of discrete vector fields, whose definition is not straightfoward. For this, we revisit the discrete theory of mechanics by relying on the construction of discrete vector fields taken from optimal control backgrounds. From here, we reconstruct a discrete Hamilton—Jacobi equation in a novel way, and which has not been devised in the literature before. As a last result, both interpretations are proven to be equivalent theoretically, but the numerical results differ slightly. The discrete vector field approach seems fairly more accurate concerning numerical values in the specific example that we show, that is an optimal control problem for a nonlinear system.

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.

Cosymplectic and contact structures for time-dependent and dissipative Hamiltonian systems

In this paper, we apply the geometric Hamilton–Jacobi theory to obtain solutions of classical hamiltonian systems that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent hamiltonians, whilst the second is here used to treat classical hamiltonians including dissipation terms.The interest of a geometric Hamilton–Jacobi equation is the primordial observation that if a hamiltonian vector field XH can be projected into a configuration manifold by means of a 1-form , then the integral curves of the projected vector field can be transformed into integral curves of XH provided that W is a solution of the Hamilton–Jacobi equation. In this way, we use the geometric Hamilton–Jacobi theory to derive solutions of physical systems with a time-dependent hamiltonian formulation or including dissipative terms. Explicit, new expressions for a geometric Hamilton–Jacobi equation are obtained on a cosymplectic and a contact manifold. These equations are later used to solve physical examples containing explicit time dependence, as it is the case of a unidimensional trigonometric system, and two dimensional nonlinear oscillators as Winternitz–Smorodinsky oscillators and for explicit dissipative behavior, we solve the example of a unidimensional damped oscillator.

Geometric Hamilton–Jacobi theory on Nambu–Poisson manifolds

The Hamilton–Jacobi theory is a formulation of classical mechanics equivalent to other formulations as Newtonian, Lagrangian, or Hamiltonian mechanics. The primordial observation of a geometric Hamilton–Jacobi theory is that if a Hamiltonian vector field XH can be projected into the configuration manifold by means of a 1-form dW, then the integral curves of the projected vector field XHdWcan be transformed into integral curves of XH provided that W is a solution of the Hamilton–Jacobi equation. Our aim is to derive a geometric Hamilton–Jacobi theory for physical systems that are compatible with a Nambu–Poisson structure. For it, we study Lagrangian submanifolds of a Nambu–Poisson manifold and obtain explicitly an expression for a Hamilton–Jacobi equation on such a manifold. We apply our results to two interesting examples in the physics literature: the third-order Kummer–Schwarz equations and a system of n copies of a first-order differential Riccati equation. From the first example, we retrieve the original Nambu bracket in three dimensions and from the second example, we retrieve Takhtajan’s generalization of the Nambu bracket to n dimensions.

Structural aspects of Hamilton–Jacobi theory

In our previous papers [J. F. Cariñena, X. Gràcia, G. Marmo, E. Martínez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometric Hamilton–Jacobi theory, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 1417–1458; Geometric Hamilton–Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Meth. Mod. Phys. 7 (2010) 431–454] 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 fibered 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.

Hamilton-Jacobi Theory in Cauchy Data Space

Recently, M. de Le\'on el al. ([9]) have developed a geometric Hamilton-Jacobi theory for Classical Field Theories in the setting of multisymplectic geometry. Our purpose in the current paper is to establish the corresponding Hamilton-Jacobi theory in the Cauchy data space, and relate both approaches.

Unified formalism for the generalized kth-order Hamilton–Jacobi problem

The geometric formulation of the Hamilton–Jacobi theory enables us to generalize it to systems of higher-order ordinary differential equations. In this work we introduce the unified Lagrangian–Hamiltonian formalism for the geometric Hamilton–Jacobi theory on higher-order autonomous dynamical systems described by regular Lagrangian functions.

Geometric Hamilton–Jacobi theory for higher-order autonomous systems

The geometric framework for the Hamilton–Jacobi theory is used to study this theory in the background of higher-order mechanical systems, in both the Lagrangian and Hamiltonian formalisms. Thus, we state the corresponding Hamilton–Jacobi equations in these formalisms and apply our results to analyze some particular physical examples.

GEOMETRIC HAMILTON–JACOBI FIELD THEORY

I review my proposal about how to extend the geometric Hamilton–Jacobi theory to higher derivative field theories on fiber bundles.