scholarly journals The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1993
Author(s):  
Manuel de León ◽  
Manuel Lainz ◽  
Álvaro Muñiz-Brea
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Jacobi Equation ◽  
Vector Fields ◽  
Hamiltonian Function ◽  
Hamilton Jacobi Equation

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.

scholarly journals Hamilton-Jacobi Equation of Time Dependent Hamiltonians

2020 ◽  
Vol 5 (1-2) ◽  
pp. 09-15
Author(s):  
Anoud K. Fuqara ◽  
Amer D. Al-Oqali ◽  
Khaled I. Nawafleh
Keyword(s):  
Hamiltonian Systems ◽  
Jacobi Equation ◽  
Classical Mechanics ◽  
Time Dependent ◽  
Calculus Of Variation ◽  
Jacobi Formalism ◽  
Equation Of Time ◽  
Hamilton Jacobi Equation

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.

scholarly journals The Lax–Oleinik semi-group: a Hamiltonian point of view

2012 ◽  
Vol 142 (6) ◽  
pp. 1131-1177 ◽  
Author(s):  
Patrick Bernard
Keyword(s):  
Hamiltonian Systems ◽  
Viscosity Solutions ◽  
Jacobi Equation ◽  
Kam Theory ◽  
Companion Paper ◽  
Point Of View ◽  
Invariant Sets ◽  
Evolution Operators ◽  
Weak Kam Theory ◽  
Hamilton Jacobi Equation

The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on ℝd and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications.

XXVI.—The Universal Integral Invariants of Hamiltonian Systems and Application to the Theory of Canonical Transformations

1948 ◽  
Vol 62 (3) ◽  
pp. 237-246 ◽  
Author(s):  
Hwa-Chung Lee
Keyword(s):  
Hamiltonian System ◽  
Differential Equations ◽  
Hamiltonian Systems ◽  
Hamiltonian Function ◽  
Integral Invariant ◽  
Canonical Transformations ◽  
System Of Differential Equations ◽  
Integral Invariants ◽  
Image Position ◽  
Universal Integral

I. Introduction.—Consider a Hamiltonian system of differential equationswhere H is a function of the 2n variables qi and pi involving in general also the time t. For each given Hamiltonian function H the system (1.1) possesses infinitely many absolute and relative integral invariants of every order r = 1,…, 2n, which can all be written out when (1.1) is integrated. Our interest now is not in these integral invariants, which are possessed by one Hamiltonian system, but in those which are possessed by all Hamiltonian systems. Such an integral invariant, which is independent of the Hamiltonian H, is said to be universal.

scholarly journals Structural aspects of Hamilton–Jacobi theory

2016 ◽  
Vol 13 (02) ◽  
pp. 1650017 ◽  
Author(s):  
J. F. Cariñena ◽  
X. Gràcia ◽  
G. Marmo ◽  
E. Martínez ◽  
M. C. Muñoz-Lecanda ◽  
...  
Keyword(s):  
Jacobi Equation ◽  
Vector Fields ◽  
Hamiltonian Dynamics ◽  
Dimensional Manifold ◽  
Special Cases ◽  
Fibered Manifolds ◽  
Hamilton Jacobi Equation ◽  
Structural Aspects ◽  
Lower Dimensional ◽  
Nonholonomic Dynamical Systems

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.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

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