The evolution operator solution of the Cauchy problem for the Hamilton-Jacobi equation

1977 ◽  
Vol 26 (3-4) ◽  
pp. 232-264 ◽  
Author(s):  
Michael B. Tamburro
Keyword(s):  
Cauchy Problem ◽  
Jacobi Equation ◽  
Evolution Operator ◽  
Operator Solution ◽  
The Cauchy Problem ◽  
Hamilton Jacobi Equation

scholarly journals Spherically symmetric T-solution of the equations of 5-dimensional Kaluza–Klein theory

2020 ◽  
Vol 28 (2) ◽  
pp. 51-56
Author(s):  
V. D. Gladush
Keyword(s):  
Cauchy Problem ◽  
Cosmological Model ◽  
Configuration Space ◽  
Jacobi Equation ◽  
Spherically Symmetric ◽  
Klein Theory ◽  
The Cauchy Problem ◽  
Hamilton Jacobi Equation ◽  
Hilbert Action ◽  
Kaluza Klein

A geometrodynamical approach to the five-dimensional (5D) spherically symmetric cosmological model in the Kaluza–Klein theory is constructed. After dimensional reduction, the 5D Hilbert action is reduced to the Einstein form describing the gravitational, electromagnetic, and scalar interacting fields. The subsequent transition to the configuration space leads to the supermetric and the Einstein–Hamilton–Jacobi equation, with the help of which the trajectories in the configuration space are found. Then the evolutionary coordinate is restored, and the Cauchy problem is solved to find the time dependence of the metric and fields. The configuration corresponds to a cosmological model of the Kantovsky–Sachs type, which has a hypercylinder topology and includes scalar and electromagnetic fields with contact interaction.

scholarly journals ON REPRESENTATION FORMULA FOR SOLUTIONS OF HAMILTON‐JACOBI EQUATION FOR SOME TYPES OF INITIAL CONDITIONS

2001 ◽  
Vol 6 (2) ◽  
pp. 241-250
Author(s):  
G. Gudynas
Keyword(s):  
Cauchy Problem ◽  
Jacobi Equation ◽  
Initial Function ◽  
Initial Conditions ◽  
Representation Formula ◽  
Fundamental Solutions ◽  
The Cauchy Problem ◽  
Hamilton Jacobi Equation

This article investigates the representation formula for the semiconcave solutions of the Cauchy problem for Hamilton‐Jacobi equation with the convex Hamiltonian and the unbounded lower semicontinous initial function. The formula like Hopf ‘s formula is given by forming envelope of some fundamental solutions of the equation.

scholarly journals Variational and viscosity operators for the evolutionary Hamilton–Jacobi equation

2019 ◽  
Vol 21 (04) ◽  
pp. 1850018 ◽  
Author(s):  
Valentine Roos
Keyword(s):  
Jacobi Equation ◽  
A Priori ◽  
Initial Condition ◽  
First Order ◽  
Momentum Variable ◽  
Lipschitz Estimates ◽  
The Cauchy Problem ◽  
Local Lipschitz Estimates ◽  
Variational Operator ◽  
Hamilton Jacobi Equation

We study the Cauchy problem for the first-order evolutionary Hamilton–Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax–Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable.

Representation Formula for Solutions of Eikonal Type Equations

2001 ◽  
Vol 6 (1) ◽  
pp. 57-65
Author(s):  
Gintautas Gudynas
Keyword(s):  
Cauchy Problem ◽  
Fluid Mechanics ◽  
Boundary Problem ◽  
Jacobi Equation ◽  
Type Equation ◽  
Representation Formula ◽  
Fundamental Solutions ◽  
Material Sciences ◽  
And Control ◽  
Hamilton Jacobi Equation

Equations of an eikonal type ones arise in many areas of applications, including optics, fluid mechanics, material sciences, and control theory. This article investigates the representation formula for semiconcave solutions of the boundary problem for eikonal type equation. The formula is given by forming envelope of some fundamental solutions of the equation. Using these fundamental solutions we obtain the representation formula for solutions of Cauchy problem for appropriate Hamilton – Jacobi equation.

Canonical & Gauge Transformations

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Jacobi Equation ◽  
Liouville Integrability ◽  
Wave Mechanics ◽  
Integrability Theorem ◽  
Gauge Transformations ◽  
The Cauchy Problem ◽  
Jacobi Equations ◽  
Jacobi Formulation ◽  
Hamilton Jacobi Equation ◽  
De Broglie Bohm

In this chapter, the Hamilton–Jacobi formulation is discussed in two parts: from a generating function perspective and as a variational principle. The Poincaré–Cartan 1-form is derived and solutions to the Hamilton–Jacobi equations are discussed. The canonical action is examined in a fashion similar to that used for analysis in previous chapters. The Hamilton–Jacobi equation is then shown to parallel the eikonal equation of wave mechanics. The chapter discusses Hamilton’s principal function, the time-independent Hamilton–Jacobi equation, Hamilton’s characteristic function, the rectification theorem, the Maupertius action principle and the Hamilton–Jacobi variational problem. The chapter also discusses integral surfaces, complete integral hypersurfaces, completely separable solutions, the Arnold–Liouville integrability theorem, general integrals, the Cauchy problem and de Broglie–Bohm mechanics. In addition, an interdisciplinary example of medical imaging is detailed.

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.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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