We use the geometric approach to the theory of Lie systems of differential equations in order to study dissipative Ermakov systems. We prove that there is a superposition rule for solutions of such equations. This fact enables us to express the general solution of a dissipative Milne--Pinney equation in terms of particular solutions of a system of second-order linear differential equations and a set of constants.

2009-02-12math-phmath.MP

Relativistic Ermakov-Milne-Pinney Systems and First Integrals

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 relativisti…

2021-02-18math-phmath.MPnlin.SI

Transverse Waves in Nonlinearly Elastic Solids and the Milne-Pinney (or Ermakov) Equation

We establish a connection between the general equations of nonlinear elastodynamics and the nonlinear ordinary differential equation of Pinney [Proc. Amer. Math. Soc. 1 (1950) 681]. As a starting point, we use the exact travelling wave solutions of nonlinear elasticity discovered by Carroll [Acta Mechanica 3 (1967) 167]. The connection provides a method for finding new exact and approximate dynamic solutions for neo-Hookean and Mooney-Rivlin solids, and for the general third- and fourth-order elasticity models of incompressible solids.

2013-02-07cond-mat.softmath-phmath.MP

Nonlinear Supersymmetric (Darboux) Covariance of the Ermakov-Milne-Pinney Equation

It is shown that the nonlinear Ermakov-Milne-Pinney equation $ρ^{\prime\prime}+v(x)ρ=a/ρ^3$ obeys the property of covariance under a class of transformations of its coefficient function. This property is derived by using supersymmetric, or Darboux, transformations. The general solution of the transformed equation is expressed in terms of the solution of the original one. Both iterations of these transformations and irreducible transformations of second order in derivatives are considered to obtain the chain of mutually related Ermakov-Milne-Pinney equations. The behaviour of the Lewis invarian…

2002-09-06gr-qcmath-phmath.MP

Nonlocal transformations of the Generalized Liénard type equations and dissipative Ermakov-Milne-Pinney systems

We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Liénard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We obtain the corresponding new first integrals of these derived equations, this method yields a natural generalization of the construction of Ermakov-Lewis invariant for a time dependent oscillator to (coupled) Liénard and Liénard type equations. We also study the linearization problem for the coupled Liénard equation using nonlocal transformations an…

2019-05-02math-phmath.MPnlin.SI

Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping

We introduce a special type of dissipative Ermakov-Pinney equations of the form v_{ζζ}+g(v)v_ζ+h(v)=0, where h(v)=h_0(v)+cv^{-3} and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel equation. When h_0(v) is a linear function, h_0(v)=λ^2v, general solutions are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type. In addition, the same kinds of general solutions are constructed for the cases of higher-order Reid nonlinearities. The Chiellini dissipat…

2015-03-05math-phmath.MP

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Milne quantization for non-Hermitian systems

We generalize the Milne quantization condition to non-Hermitian systems. In the general case the underlying nonlinear Ermakov-Milne-Pinney equation needs to be replaced by a nonlinear integral differential equation. However, when the system is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and one may employ the original energy integral to determine its quantization. We illustrate the working of the general framework with the Swanson model and two explicit examples for pairs of supersymmetric Hamiltonians. In one case both partner Hamiltonians are Hermitian and in the other …

2015-07-14math-phmath.MPquant-ph

Cosmological aspects of the Eisenhart-Duval lift

A cosmological extension of the Eisenhart-Duval metric is constructed by incorporating a cosmic scale factor and the energy-momentum tensor into the scheme. The dynamics of the spacetime is governed the Ermakov-Milne-Pinney equation. Killing isometries include spatial translations and rotations, Newton--Hooke boosts and translation in the null direction. Geodesic motion in Ermakov-Milne-Pinney cosmoi is analyzed. The derivation of the Ermakov-Lewis invariant, the Friedmann equations and the Dmitriev-Zel'dovich equations within the Eisenhart--Duval framework is presented.

2018-04-10astro-ph.COgr-qchep-th

Gambier方程在宇宙学中的出现

Emergence of the Gambier equation in cosmology

本文揭示Gambier方程在FLRW宇宙学与暗物质状态方程中的出现，建立Friedmann方程与具Painlevé性质Gambier方程的对应，并探讨广义Pinney及Milne-Pinney等特例，连接扩展FLRW模型至参数化Gambier方程。

2023-05-15gr-qcmath-phmath.MP

Ermakov-Lewis dynamic invariants with some applications

Contents: Introduction(3).The method of Ermakov(4).The method of Milne(7). Pinney's result(8).Lewis' results(8). The interpretation of Eliezer and Gray(14). The connection of the Ermakov invariant with Nöther's theorem(17). Possible generalizations of Ermakov's method(20). Geometrical angles and phases in Ermakov's problem(22). Application to the minisuperspace cosmology(26). Application to physical optics(42).Conclusions(47). Appendix A: Calculation of the integral of I(48). Appendix B: Calculation of <\hat{H}> in eigenstates of \hat{I}(49).References(50).

2000-03-18math-phmath.MP

Recent Applications of the Theory of Lie Systems in Ermakov Systems

We review some recent results of the theory of Lie systems in order to apply such results to study Ermakov systems. The fundamental properties of Ermakov systems, i.e. their superposition rules, the Lewis-Ermakov invariants, etc., are found from this new perspective. We also obtain new results, such as a new superposition rule for the Pinney equation in terms of three solutions of a related Riccati equation.

2008-03-12math-phmath.CAmath.MP

Lie point symmetry analysis of a second order differential equation with singularity

By using Lie symmetry methods, we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the Ermakov-Pinney equation. In this context, nonlinear superposition rule for second order Kummer-Schwarz equation is rediscovered. Invariance under one-dimensional symmetry group is also used to obtain first integrals (Ermakov-Lewis invariants). Our motivation is a type of equations with singular term that arises in many applications, in particular in the study of general NLS (nonlinear Schrödinger) equations.

2017-03-22math-phmath.MPnlin.SI

Generalized EMP and nonlinear Schrodinger-type reformulations of some scalar field cosmological models

We show that Einstein's gravitational field equations for the Friedmann-Robertson-Lemaître-Walker (FRLW) and for two conformal versions of the Bianchi I and Bianchi V perfect fluid scalar field cosmological models, can be equivalently reformulated in terms of a single equation of either generalized Ermakov-Milne-Pinney (EMP) or (non)linear Schrödinger (NLS) type. This work generalizes or presents an alternative to similar reformulations published by the authors who inspired this thesis: R. Hawkins, J. Lidsey, T. Christodoulakis, T. Grammenos, C. Helias, P. Kevrekidis, G. Papadopoulos and F. Wi…

2010-05-09gr-qcmath-phmath.MP