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.

A MECHANICS FOR THE RICCI FLOW

2009 ◽  
Vol 06 (05) ◽  
pp. 759-767 ◽  
Author(s):  
S. ABRAHAM ◽  
P. FERNÁNDEZ DE CÓRDOBA ◽  
JOSÉ M. ISIDRO ◽  
J. L. G. SANTANDER
Keyword(s):  
Ricci Flow ◽  
Jacobi Equation ◽  
Classical Mechanics ◽  
Riemannian Metric ◽  
Flow Equation ◽  
Time Dependent ◽  
Dimensional Manifold ◽  
Conformally Flat ◽  
Hamilton Jacobi Equation

We construct the classical mechanics associated with a conformally flat Riemannian metric on a compact, n-dimensional manifold without boundary. The corresponding gradient Ricci flow equation turns out to equal the time-dependent Hamilton–Jacobi equation of the mechanics so defined.

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 Canonical transformations and the Hamilton-Jacobi theory in quantum mechanics

1999 ◽  
Vol 77 (6) ◽  
pp. 411-425 ◽  
Author(s):  
J -H Kim ◽  
H -W Lee
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Generating Functions ◽  
Jacobi Equation ◽  
Classical Case ◽  
Time Dependent ◽  
Canonical Transformations ◽  
Hamilton Jacobi Equation ◽  
Number Form

Canonical transformations using the idea of quantum generating functions are applied to construct a quantum Hamilton-Jacobi theory, based on the analogy with the classical case. An operator and a c-number form of the time-dependent quantum Hamilton-Jacobi equation are derived and used to find dynamical solutions of quantum problems. The phase-space picture of quantum mechanics is discussed in connection with the present theory.PACS Nos.: 03.65-w, 03.65Ca, 03.65Ge

scholarly journals Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Albert Fathi
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Jacobi Equation ◽  
Compact Manifold ◽  
Complete Riemannian Manifold ◽  
Time Dependent ◽  
Jacobi Equations ◽  
Uniformly Continuous ◽  
Hamilton Jacobi Equation

AbstractIf $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

Estimation of the regions of attraction for autonomous nonlinear systems

2018 ◽  
Vol 41 (1) ◽  
pp. 97-106
Author(s):  
Guoqiang Yuan ◽  
Yinghui Li
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Jacobi Equation ◽  
Time Dependent ◽  
Implicit Function ◽  
Region Of Attraction ◽  
Sublevel Set ◽  
Initial Domain ◽  
Regions Of Attraction ◽  
Hamilton Jacobi Equation

A methodology for estimating the region of attraction for autonomous nonlinear systems is developed. The methodology is based on a proof that the region of attraction can be estimated accurately by the zero sublevel set of an implicit function which is the viscosity solution of a time-dependent Hamilton–Jacobi equation. The methodology starts with a given initial domain and yields a sequence of region of attraction estimates by tracking the evolution of the implicit function. The resulting sequence is contained in and converges to the exact region of attraction. While alternative iterative methods for estimating the region of attraction have been proposed, the methodology proposed in this paper can compute the region of attraction to achieve any desired accuracy in a dimensionally independent and efficient way. An implementation of the proposed methodology has been developed in the Matlab environment. The correctness and efficiency of the methodology are verified through a few examples.

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.

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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