General relativity via complete integrals of the Hamilton–Jacobi equation

2005 ◽  
Vol 46 (3) ◽  
pp. 032502 ◽  
Author(s):  
Enrique Montiel-Piña ◽  
Ezra Ted Newman ◽  
Gilberto Silva-Ortigoza
Keyword(s):  
General Relativity ◽  
Jacobi Equation ◽  
Hamilton Jacobi Equation

scholarly journals Solving the Hamilton-Jacobi equation for general relativity

1994 ◽  
Vol 49 (6) ◽  
pp. 2872-2881 ◽  
Author(s):  
J. Parry ◽  
D. S. Salopek ◽  
J. M. Stewart
Keyword(s):  
General Relativity ◽  
Jacobi Equation ◽  
Hamilton Jacobi Equation

scholarly journals Dynamic Path in C4 Space-Time

2018 ◽  
Author(s):  
Dimitris Mastoridis ◽  
K. Kalogirou
Keyword(s):  
General Relativity ◽  
Dispersion Relation ◽  
Jacobi Equation ◽  
Space Time ◽  
Complex Time ◽  
Dynamic Path ◽  
Hamilton Jacobi Equation

After developed the formulation of a "general relativity" in C4 [2], we proceed with the formulation of a Hamilton-Jacobi equation in C4. We argue that in this consideration, the usual problems of the ADM formalism, do not exist, due to the complex time as it exists in our consideration. Specically, we can derive a suitable dispersion relation in order to work with and nd a generalised super Hamiltonian

On the Hamiltonian and Hamilton–Jacobi equations for metric gravity

2020 ◽  
Vol 98 (4) ◽  
pp. 405-412
Author(s):  
Alexei M. Frolov
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Closed System ◽  
Jacobi Equation ◽  
Hamiltonian Equations ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

The closed system of Hamiltonian equations is derived for all tensor components of a free gravitational field gαβ and corresponding momenta πγδ in metric general relativity. The Hamilton–Jacobi equation for a free gravitational field gαβ is also derived and discussed. In general, all methods and procedures based on the Hamiltonian and Hamilton–Jacobi approaches are very effective in actual applications to many problems known in metric general relativity.

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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