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.

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.

The Hamilton-Jacobi Equations for a Relativistic Charged Particle

1963 ◽  
Vol 6 (3) ◽  
pp. 341-350 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Jacobi Equation ◽  
Classical Particle ◽  
Quantum Mechanical ◽  
Order Partial Differential Equation ◽  
First Order ◽  
Relativistic Charged Particle ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.

Hamiltonian Mechanics for Functionals Involving Second-Order Derivatives

2002 ◽  
Vol 69 (6) ◽  
pp. 749-754 ◽  
Author(s):  
B. Tabarrok ◽  
C. M. Leech
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Jacobi Equation ◽  
Equations Of Motion ◽  
Second Order ◽  
Hamiltonian Mechanics ◽  
Energy Functionals ◽  
First Order ◽  
Independent Variable ◽  
Hamilton Jacobi Equation

Hamilton’s principle was developed for the modeling of dynamic systems in which time is the principal independent variable and the resulting equations of motion are second-order differential equations. This principle uses kinetic energy which is functionally dependent on first-order time derivatives, and potential energy, and has been extended to include virtual work. In this paper, a variant of Hamiltonian mechanics for systems whose motion is governed by fourth-order differential equations is developed and is illustrated by an example invoking the flexural analysis of beams. The variational formulations previously associated with Newton’s second-order equations of motion have been generalized to encompass problems governed by energy functionals involving second-order derivatives. The canonical equations associated with functionals with second order derivatives emerge as four first-order equations in each variable. The transformations of these equations to a new system wherein the generalized variables and momenta appear as constants, can be obtained through several different forms of generating functions. The generating functions are obtained as solutions of the Hamilton-Jacobi equation. This theory is illustrated by application to an example from beam theory the solution recovered using a technique for solving nonseparable forms of the Hamilton-Jacobi equation. Finally whereas classical variational mechanics uses time as the primary independent variable, here the theory is extended to include static mechanics problems in which the primary independent variable is spatial.

scholarly journals First order description of static black holes and the Hamilton–Jacobi equation

2010 ◽  
Vol 833 (1-2) ◽  
pp. 1-16 ◽  
Author(s):  
L. Andrianopoli ◽  
R. D'Auria ◽  
E. Orazi ◽  
M. Trigiante
Keyword(s):  
Black Holes ◽  
Jacobi Equation ◽  
First Order ◽  
Hamilton Jacobi Equation

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 COEFFICIENT STABILITY OF OPERATOR–DIFFERENCE SCHEMES

1999 ◽  
Vol 4 (1) ◽  
pp. 135-146
Author(s):  
P. P. Matus ◽  
B. S. Jovanović
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Difference Schemes ◽  
Evolutionary Equation ◽  
Finite Difference Schemes ◽  
Initial Condition ◽  
First Order ◽  
Time Integral ◽  
Priori Estimates ◽  
Coefficient Stability

A priori estimates expressing continuous dependence of the solution of a first order evolutionary equation in Hubert space on initial condition, right hand side and operator perturbations are obtained in time–integral norms. Analogous results hold for corresponding finite difference schemes.

Weak KAM from a PDE point of view: viscosity solutions of the Hamilton–Jacobi equation and Aubry set

2012 ◽  
Vol 142 (6) ◽  
pp. 1193-1236 ◽  
Author(s):  
Albert Fathi
Keyword(s):  
Prior Knowledge ◽  
Jacobi Equation ◽  
Kam Theory ◽  
General Setting ◽  
Point Of View ◽  
Weak Kam Theory ◽  
First Order ◽  
Aubry Set ◽  
The Subject ◽  
Hamilton Jacobi Equation

We introduce the notion of a viscosity solution for the first-order Hamilton–Jacobi equation, in the more general setting of manifolds, to obtain a weak KAM theory using only tools from partial differential equations. This work should be accessible to people with no prior knowledge of the subject.

scholarly journals Sobolev regularity for the first order Hamilton–Jacobi equation

2015 ◽  
Vol 54 (3) ◽  
pp. 3037-3065 ◽  
Author(s):  
Pierre Cardaliaguet ◽  
Alessio Porretta ◽  
Daniela Tonon
Keyword(s):  
Jacobi Equation ◽  
Sobolev Regularity ◽  
First Order ◽  
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.

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