scholarly journals Shapovalov Wave-Like Spacetimes

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1372 ◽  
Author(s):  
Konstantin Osetrin ◽  
Evgeny Osetrin
Keyword(s):  
Jacobi Equation ◽  
Eikonal Equation ◽  
Separation Of Variables ◽  
Complete Classification ◽  
Complete Separation ◽  
Einstein Equations ◽  
Coordinate Systems ◽  
Test Particles ◽  
Hamilton Jacobi Equation

A complete classification of space-time models is presented, which admit the privileged coordinate systems, where the Hamilton–Jacobi equation for a test particle is integrated by the method of complete separation of variables with separation of the isotropic (wave) variable, on which the metric depends (wave-like Shapovalov spaces). For all types of Shapovalov spaces, exact solutions of the Einstein equations with a cosmological constant in vacuum are found. Complete integrals are presented for the eikonal equation and the Hamilton–Jacobi equation of motion of test particles.

scholarly journals The spacetime models with dust matter that admit separation of variables in Hamilton–Jacobi equations of a test particle

2016 ◽  
Vol 31 (06) ◽  
pp. 1650027 ◽  
Author(s):  
Konstantin Osetrin ◽  
Altair Filippov ◽  
Evgeny Osetrin
Keyword(s):  
Velocity Field ◽  
Energy Density ◽  
Test Particle ◽  
Jacobi Equation ◽  
Functional Form ◽  
Separation Of Variables ◽  
Complete Separation ◽  
Coordinate Systems ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

The characteristics of dust matter in spacetime models, admitting the existence of privilege coordinate systems are given, where the single-particle Hamilton–Jacobi equation can be integrated by the method of complete separation of variables. The resulting functional form of the 4-velocity field and energy density of matter for all types of spaces under consideration is presented.

Separation of variables for complex Riemannian spaces of constant curvature - I. Orthogonal separable coordinates for S n c and E n c

1984 ◽  
Vol 394 (1806) ◽  
pp. 183-206 ◽  
Keyword(s):  
Separation Of Variables ◽  
Complete Solution ◽  
Complete Classification ◽  
Conformally Flat ◽  
Coordinate Systems ◽  
Spaces Of Constant Curvature ◽  
Orthogonal Coordinate Systems ◽  
Hamilton Jacobi Equation ◽  
Riemannian Spaces

A complete classification of all orthogonal coordinate systems that admit a separation of variables for the null Hamilton-Jacobi equation in conformally flat complex Riemannian spaces is presented. This is a first step towards the complete solution of the problem for complex Riemannian spaces when, in general, the coordinates need not be orthogonal. A detailed prescription for constructing all such orthogonal coordinate systems is presented.

scholarly journals Separation of variables in Hamilton–Jacobi equation for a charged test particle in the Stackel spaces of type (2.1)

2020 ◽  
Vol 17 (14) ◽  
pp. 2050186
Author(s):  
Valeriy Obukhov
Keyword(s):  
Jacobi Equation ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Type Equation ◽  
Parabolic Type ◽  
Complete Separation ◽  
Test Particles ◽  
Charged Test Particle ◽  
Charged Test ◽  
Hamilton Jacobi Equation

We can find all equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that the equations of motion of the classical charged test particles are integrated by the method of complete separation of variables in the Hamilton–Jacobi equation. Separation is carried out using the complete sets of mutually-commuting integrals of motion of type (2.1), whereby in a privileged coordinate system the Hamilton–Jacobi equation turns into a parabolic type equation.

SEPARATION OF VARIABLES FOR THE DIRAC SQUARE EQUATION

1994 ◽  
Vol 03 (04) ◽  
pp. 739-746 ◽  
Author(s):  
V.G. BAGROV ◽  
V.V. OBUKHOV
Keyword(s):  
Jacobi Equation ◽  
Separation Of Variables ◽  
Complete Separation ◽  
Space Time ◽  
Riemann Space ◽  
Necessary Condition ◽  
Separation Procedure ◽  
Electromagnetic Potential ◽  
Curved Space ◽  
Hamilton Jacobi Equation

The problem of separation of variables for the Dirac square equation on a curved space-time in the presence of electromagnetic potential is considered. It is shown that the necessary condition for the separation of variables in the Dirac square equation is the complete separation of variables in the related Hamilton-Jacobi equation, i.e. the Riemann space should be Stäckel. The constructive scheme for separation procedure is presented.

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.

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.

scholarly journals Hamilton–Jacobi Equation for a Charged Test Particle in the Stäckel Space of Type (2.0)

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1289 ◽  
Author(s):  
Valeriy Obukhov
Keyword(s):  
Test Particle ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Killing Vector ◽  
Field Equations ◽  
Tensor Fields ◽  
Charged Test Particle ◽  
Complete Sets ◽  
Charged Test ◽  
Hamilton Jacobi Equation

All electromagnetic potentials and space–time metrics of Stäckel spaces of type (2.0) in which the Hamilton–Jacobi equation for a charged test particle can be integrated by the method of complete separation of variables are found. Complete sets of motion integrals, as well as complete sets of killing vector and tensor fields, are constructed. The results can be used when studying solutions of field equations in the theory of gravity.

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