Solution of the Hamilton – Jacobi Equation in a Central Potential Using the Separation of Variables Method with Staeckel Boundary Conditions

2021 ◽  
Vol 14 (4) ◽  
pp. 301-308
Keyword(s):  
Characteristic Function ◽  
Boundary Conditions ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Lagrangian Mechanics ◽  
Principal Function ◽  
Central Potential ◽  
Separation Of Variables Method ◽  
Hamilton Jacobi Equation

Abstract: This manuscript aims at solving Hamilton-Jacobi equation in a central potential using the separation of variables technique with Staeckel boundary conditions. Our results show that the Hamilton – Jacobi variables can be completely separated, which agrees with other results employing different methods. Keywords: Lagrangian mechanics, Hamilton-Jacobi, Staeckel boundary conditions, Staeckel matrix, Staeckel vector, Hamilton's characteristic function, Hamilton's principal function.

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 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 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.

Homogeneous spacetimes and separation of variables in the Hamilton–Jacobi equation

2006 ◽  
Vol 39 (21) ◽  
pp. 6641-6647 ◽  
Author(s):  
Konstantin E Osetrin ◽  
Valery V Obukhov ◽  
Altair E Filippov
Keyword(s):  
Jacobi Equation ◽  
Separation Of Variables ◽  
Hamilton Jacobi Equation

The Hamilton-Jacobi Equation Applied to Continuum

1997 ◽  
Vol 64 (3) ◽  
pp. 658-663 ◽  
Author(s):  
C. M. Leech
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Jacobi Equation ◽  
Principal Function ◽  
Partial Differential ◽  
One Dimensional ◽  
Linear Elastic ◽  
Elastic Bar ◽  
Continuum Systems ◽  
Hamilton Jacobi Equation

The Hamilton-Jacobi partial differential equation is established for continuum systems; to do this a new concept in material distributions is introduced. The Lagrangian and Hamiltonian are developed, so that the Hamilton-Jacobi equation can be formulated and the principal function defined. Finally the principal function is constructed for the dynamics of a one-dimensional linear elastic bar; the solution for its’ vibrations is then established following the differentiation of the principal function.

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