R-Separation of Variables for the Four-Dimensional Flat Space Laplace and Hamilton-Jacobi Equations

1978 ◽  
Vol 244 ◽  
pp. 241
Author(s):  
E. G. Kalnins ◽  
Willard Miller
Keyword(s):  
Separation Of Variables ◽  
Flat Space ◽  
Jacobi Equations

Symmetry and separation of variables in Hamilton-Jacobi equations. II

1978 ◽  
Vol 21 (9) ◽  
pp. 1130-1132 ◽  
Author(s):  
V. N. Shapovalov
Keyword(s):  
Separation Of Variables ◽  
Jacobi Equations

scholarly journals Universe as Klein–Gordon eigenstates

2021 ◽  
Vol 81 (12) ◽  
Author(s):  
Marco Matone
Keyword(s):  
Linear Form ◽  
Schwarzian Derivative ◽  
Eigenvalue Problems ◽  
Measurement Problem ◽  
Linear Equations ◽  
Flat Space ◽  
Curvature Term ◽  
Klein Gordon ◽  
Linear Differential ◽  
Jacobi Equations

AbstractWe formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the$$\beta $$ β -times $$t_\beta :=\int ^t a^{-2\beta }$$ t β : = ∫ t a - 2 β , where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems $$\begin{aligned} O_{1/2} \Psi =\frac{\Lambda }{12}\Psi , \quad O_1 a =-\frac{\Lambda }{3} a , \end{aligned}$$ O 1 / 2 Ψ = Λ 12 Ψ , O 1 a = - Λ 3 a , which is suggestive of a measurement problem. $$O_{\beta }(\rho ,p)$$ O β ( ρ , p ) are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The $$O_\beta $$ O β ’s are also independent of the spatial curvature, labeled by k, and absorbed in $$\begin{aligned} \Psi =\sqrt{a} e^{\frac{i}{2}\sqrt{k}\eta } . \end{aligned}$$ Ψ = a e i 2 k η . The above pair of equations is the unique possible linear form of Friedmann’s equations unless $$k=0$$ k = 0 , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $$\eta \equiv t_{1/2}$$ η ≡ t 1 / 2 among the $$t_\beta $$ t β ’s, which is the key to absorb the curvature term. An immediate consequence of the linear form is that it reveals a new symmetry of Friedmann’s equations in flat space.

scholarly journals Symmetry operators and separation of variables in the (2 + 1)-dimensional Dirac equation with external electromagnetic field

2018 ◽  
Vol 15 (05) ◽  
pp. 1850085 ◽  
Author(s):  
A. V. Shapovalov ◽  
A. I. Breev
Keyword(s):  
Dirac Equation ◽  
Separation Of Variables ◽  
Flat Space ◽  
External Electromagnetic Field ◽  
Complete Separation ◽  
Electromagnetic Potential ◽  
First Order ◽  
Matrix Coefficients ◽  
Symmetry Operators ◽  
Differential Symmetry

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a [Formula: see text]-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a [Formula: see text]-dimensional Minkowski (flat) space. For each of the sets, we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.

Symmetry and separation of variables in Hamilton-Jacobi equations. I

1978 ◽  
Vol 21 (9) ◽  
pp. 1124-1129 ◽  
Author(s):  
V. N. Shapovalov
Keyword(s):  
Separation Of Variables ◽  
Jacobi Equations

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.

A GEOMETRIC WAY TO OBTAIN DE-SITTER SUBSPACES PERPENDICULAR TO A FINITE SIZE EXTRA-DIMENSION IN CURVED 5D UNIVERSES

2004 ◽  
Vol 19 (20) ◽  
pp. 3377-3394 ◽  
Author(s):  
E. GUENDELMAN ◽  
H. RUCHVARGER
Keyword(s):  
Scalar Field ◽  
Extra Dimension ◽  
Special Form ◽  
Separation Of Variables ◽  
Field Potential ◽  
Finite Size ◽  
Flat Space ◽  
Space Time ◽  
De Sitter ◽  
Geodesic Lines

We define curved five-dimensional (5D) space–time from the embedding of 5D surfaces in a 6D flat space. Demanding that the 6D coordinates satisfy a separation of variables form and that the 5D metric is diagonal, we obtain that each curved 5D surface contains 4D hyperboloid de-Sitter subspaces with maximally symmetry SO (4,1). Therefore, we define a very special form for the curved 5D surface where the extra-dimension is perpendicular to the 4D hyperboloid de-Sitter spaces. By relating to a minimally coupled scalar field with a potential which depends on the extra-dimension only, the curved 5D surface's form is satisfied. A mechanism by means of which the extra-dimension can be of a finite size, is found. The borders of the finite extra-dimension are obtained when the scalar field potential goes to infinity for certain finite values of the scalar field. The geodesic lines' equations show that a particle cannot cross such borders.

Solution of the Hamilton – Jacobi Equations in an Electromagnetic Field Using Separation of Variables Method – Staeckel Boundary Conditions

2020 ◽  
Vol 13 (1) ◽  
pp. 59-65
Keyword(s):  
Electromagnetic Field ◽  
Boundary Conditions ◽  
Separation Of Variables ◽  
Separation Of Variables Method ◽  
Jacobi Equations

This manuscript aims to resolve the Hamilton-Jacobi equations in an electromagnetic field by two methods. The first uses the separation of variables technique with Staeckel boundary conditions, whereas the second uses the Newtonian formalism to solve the same example. Our results demonstrate that the Hamilton-Jacobi variables can be completely detached by using separation of variables technique with Staeckel boundary conditions that correspond to other results using Newtonian formalism.

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