Invariants of Multi-Component Ermakov Systems

It is shown that certain multi-component Ermakov systems admit Lewis-Ray-Reid invariants. This extends the result to the two-component Ermakov system.     

受到与速度平方成正比的力的谐振子的普遍解

将Ermakov系统加以推广，得出受到与速度平方成正比的力的变频率谐振子的不变量，求出其普遍解．     

四维相空间中非中心势动力学系统的Poisson 结构和Casimir 函数

将非中心势动力学系统的运动微分方程写成Ermakov形式，得到Ermakov不变量. 运用Hamilton理论，把Ermakov不变量当作Hamiltonian 函数，在四维相空间中建立了非中心势动力学系统的Poisson 结构。结果表明：此Poisson 结构是一退化的结构，而系统具有四个不变量，即Hamiltonian 函数，Ermakov不变量及两个Casimir函数。     

The parametric orbits and the form invariance of three-body in one-dimension

In this paper, the differential equations of motion of a three-body interacting pairwise by inverse cubic forces(“centrifugal potential”) in addition to linear forces (“harmonical potential”) are expressed in Ermakov formalism in two-dimension polar coordinates, and the Ermakov invariant is obtained. By rescaling of the time variable and the space coordinates, the parametric orbits of the three bodies are expressed in terms of relative energy H1 and Ermakov invariant. The form invariance of the transformations of two conserved quantities are also studied.     

哈密顿Ermakov系统的形式不变性

把形式不变性的方法用于研究哈密顿Ermakov系统，从哈密顿Ermakov系统的形式不变性出发，运用比较系数法得到与形式不变性相应的点对称变换生成元的表达式及势能所满足的偏微分方程.结果表明，在点对称变换下，只有自治的哈密顿Ermakov系统才具有形式不变性. 关键词： 哈密顿Ermakov系统 拉格朗日函数 点对称变换 形式不变性     

非中心力场中经典粒子的轨道参数方程与对称性

把非中心力场中经典粒子运动微分方程写成Ermakov方程的形式，得到Ermakov不变量.用改变时间坐标标度的方法得到用能量H和Ermakov不变量表示的轨道参数方程，并研究两守恒量（能量和Ermakov不变量）相应的无限小变换的Noether对称性、Lie对称性和形式不变性.研究结果表明：与两守恒量相应的无限小变换既具有Noether对称性，也具有Lie对称性和形式不变性. 关键词： 非中心力场 轨道参数方程 守恒量 对称性     

Going to Spontaneous Boiling-Up Onset

The article describes the experimental approach to elucidate the characteristics of the initial spontaneous boiling (spontaneous boiling-up) and the related effect of attainable liquid superheat. Presented is the analysis of the pioneering works on this subject carried out by G.V. Ermakov in the 60ies under the leadership of V.P. Skripov. They were the “healthy stimulus” for the revival of interest to liquid superheat in the scientific community. The article is devoted to the 80ies anniversary of Ermakov (1938–2012), who has been recognized for a series of investigations on thermodynamic properties of superheated liquids and the kinetics of liquid boiling-up [1]. The article presents discussion of the most striking results obtained in Ermakov’s team and also the previously unpublished results. Selection of issues for discussion was dictated by the preferences of the authors who collaborated with Ermakov.     

On a canonical nine-body problem. Integrable Ermakov decomposition via reciprocal transformations

Here, a recently introduced nine-body problem is shown to be decomposable via a novel class of reciprocal transformations into a set of integrable Ermakov systems. This Ermakov decomposition is exploited to construct more general integrable nine-body systems in which the canonical nine-body system is embedded.     

New Ermakov systems from Lie symmetry theory

We derive and discuss new Ermakov systems from the point of view of the Lie symmetry of differential equations.     

Solutions to N-dimensional time-dependent Schrödinger equations

We extend the Lewis-Riesenfeld technique of solving the time-dependent Schrödinger equation to N-dimensional Ermakov systems.     

Nonlinear superposition law for generalized Ermakov systems

We discuss a generalized Ermakov system for which a general nonlinear superposition law exists. We give an explicit example of the use of the new superposition law.     

Ermakov System and Plane Curve Motions in Affine Geometries

It is shown that the Pinney equation, Ermakov systems, and their higher-order generalizations describe self-similar solutions of plane curve motions in centro-affine and affine geometries.     

On Hybrid Ermakov-Painlevé Systems. Integrable Reduction

Hybrid Ermakov-Painlevé II-IV systems are introduced here in a unified manner. Their admitted Ermakov invariants together with associated canonical Painlevé equations are used to establish integrability properties.     

A new approach to Ermakov systems and applications in quantum physics

Milne–Pinney equation [(x)\ddot]=-w²(t)x+ k/x³\ddot x=-\omega^2(t)x+ k/{x^3} is usually studied together with the time-dependent harmonic oscillator [(y)\ddot]+w²(t) y=0\ddot y+\omega^2(t) y=0 and the system is called Ermakov system, and actually Pinney showed in a short paper that the general solution of the first equation can be written as a superposition of two solutions of the associated harmonic oscillator. A recent generalization of the concept of Lie systems for second order differential equations and the usual techniques of Lie systems will be used to study the Ermakov system. Several applications of Ermakov systems in Quantum Mechanics as the relation between Schroedinger and Milne equations or the use of Lewis–Riesenfeld invariant will be analysed from this geometric viewpoint.     

Ermakov systems,Noether's theorem and the Sarlet-Bahar method

We derive invariants for a nonlinear equation of motion containing arbitrary functions. The method employed is the recently discussed direct method of Sarlet and Bahar. The resulting invariants are a special case of Ermakov invariants. We compare these results to the results obtained by applying Noether's theorem to the same equation of motion.     

Symmetry-preserving perturbations of the Bateman Lagrangian and dissipative systems

Perturbations of the classical Bateman Lagrangian preserving a certain subalgebra of Noether symmetries are studied, and conservative perturbations are characterized by the Lie algebra sl(2, ?) ⊕ so(2). Non-conservative albeit integrable perturbations are determined by the simple Lie algebra sl(2,?), showing further the relation of the corresponding non-linear systems with the notion of generalized Ermakov systems.     

Exact wave functions for generalized harmonic oscillators

We transform the time-dependent Schrödinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time evolution of exact wave functions of generalized harmonic oscillators is determined in terms of the solutions of certain Ermakov and Riccatitype systems. In addition, we show that the classical Arnold transform is naturally connected with Ehrenfest’s theorem for generalized harmonic oscillators.     

Quantum Analogue of Ermakov Systems and the Phase of the Quantum Wave Function

Based on the multidimensional Ermakov theory, a general result that relates the Schrodinger equation and the Milne equation in terms of a space invariant is established. Using this result not only the role of phase in the Wigner function approach to quantum mechanics is demonstrated but also a better explanation for the Aharonov–Bohm effect is sought in terms of a fundamental phase and the matter-field-coupling current. The existence of a similar space invariant is also emphasized for the nonlinear Schrodinger equation.