On the geometrical Cauchy problem for the Hamilton-Jacobi equation

1989 ◽  
Vol 104 (5) ◽  
pp. 525-544 ◽  
Author(s):  
F. Cardin
Keyword(s):  
Cauchy Problem ◽  
Jacobi Equation ◽  
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.

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.

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.

Hamiltonian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Angular Momentum ◽  
Jacobi Equation ◽  
Modern Theory ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Light Waves ◽  
Hamilton Jacobi Equation ◽  
Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

scholarly journals Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

2018 ◽  
Vol 148 (3) ◽  
pp. 559-574
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Jacobi Equation ◽  
Threshold Value ◽  
Monotonicity Property ◽  
Fast Diffusion ◽  
Extinction Time ◽  
One Step ◽  
Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

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