Journal ArticleDOI
On the complete symmetry group of the classical Kepler system
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In this paper, a complete symmetry group for the classical Kepler problem is presented, which acts freely and transitively on the manifold of all allowed motions of the system, and the given equations of motion are the only ordinary differential equations that remain invariant under the specified action of the group.Abstract:
A rather strong concept of symmetry is introduced in classical mechanics, in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey Accordingly, a ‘‘complete symmetry group’’ realization in mechanics must be endowed with the following two features: (1) the group acts freely and transitively on the manifold of all allowed motions of the system; (2) the given equations of motion are the only ordinary differential equations that remain invariant under the specified action of the group This program is applied successfully to the classical Kepler problem, since the complete symmetry group for this particular system is here obtained The importance of this result for the quantum kinematic theory of the Kepler system is emphasizedread more
Citations
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The Ermakov equation: A commentary
P. G. L. Leach,S.K. Andriopoulos +1 more
TL;DR: A short history of the Ermakov equation with an emphasis on its discovery by the west and the subsequent boost to research into invariants for nonlinear systems is given in this paper.
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Lie Symmetries of Differential Equations: Classical Results and Recent Contributions
TL;DR: This paper reviews some well known results of Lie group analysis, as well as some recent contributions concerned with the transformation of differential equations to equivalent forms useful to investigate applied problems.
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Symmetry group classification of ordinary differential equations: Survey of some results
TL;DR: In this paper, the salient features of point symmetry group classification of scalar ODEs are presented, including linear nth-order, second-order equations and related results.
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The complete Kepler group can be derived by Lie group analysis
TL;DR: In this paper, it was shown that the complete symmetry group for the Kepler problem can be derived by Lie group analysis, and that the same result is true for any autonomous system.
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The Determination of Nonlocal Symmetries by the Technique of Reduction of Order
Maria Clara Nucci,P. G. L. Leach +1 more
TL;DR: In this article, an autonomous system of ordinary differential equations can be rewritten as a mixed system of first and second order equations for which point symmetries can be automated without having to make an Ansatz on the detailed structure of the symmetry.
References
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Book
Applications of Lie Groups to Differential Equations
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
Journal ArticleDOI
Symmetries and differential equations, by G. W. Bluman and S. Kumei. Pp 412. DM114. 1989. ISBN 3-540-96996-9 (Springer)
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Four-Dimensional Formulations of Newtonian Mechanics and Their Relation to the Special and the General Theory of Relativity
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Conservation Laws for Gauge-Variant Lagrangians in Classical Mechanics
TL;DR: The importance of considering non-invariant Lagrangians and the associated conservation laws is illustrated by several examples: energy conservation, Galilean invariance, dynamical symmetries (harmonic oscillator and Kepler's problem), motion in a uniform electric field as discussed by the authors.