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Keywords = Ermakov–Milne–Pinney equation

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19 pages, 800 KiB  
Article
Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems
by Qiliang Zhao, Pengming Zhang and Peter A. Horvathy
Symmetry 2021, 13(10), 1866; https://doi.org/10.3390/sym13101866 - 3 Oct 2021
Cited by 8 | Viewed by 2239
Abstract
The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” [...] Read more.
The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)
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12 pages, 346 KiB  
Article
Relativistic Ermakov–Milne–Pinney Systems and First Integrals
by Fernando Haas
Physics 2021, 3(1), 59-70; https://doi.org/10.3390/physics3010006 - 12 Feb 2021
Cited by 4 | Viewed by 2903
Abstract
The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding [...] Read more.
The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov–Milne–Pinney equation and associated first integral. The special relativistic extension of the Ray–Reid system and invariant is obtained. General properties of the relativistic Ermakov–Milne–Pinney are analyzed. The conservative case of the relativistic Ermakov–Milne–Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered to be well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov–Milne–Pinney equation has additional nonlinearities, due to the relativistic effects. Full article
(This article belongs to the Section Classical Physics)
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14 pages, 254 KiB  
Article
Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations
by Rutwig Campoamor-Stursberg
Symmetry 2016, 8(3), 15; https://doi.org/10.3390/sym8030015 - 17 Mar 2016
Cited by 6 | Viewed by 4258
Abstract
A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints [...] Read more.
A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems. Full article
(This article belongs to the Special Issue Symmetry and Integrability)
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