The Ermakov equation: A commentary

Abstract

We present 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 although recognizing some of the significant developments in the East. We present the modern context of the Ermakov Equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete Math., 2 (2008), 123-145) for an English translation of Ermakov's original paper. THE LEWIS INVARIANT AND PINNEY'S SOLUTIONIn 1950 the late Edmund Pinney [2] presented in a very succinct paper [85] the solution of the equationin which the overdot represents differentiation with respect to the independent variable, t, which in many applications is the time (See also [7]). The solution which he gave is(2) x(t) = Au 2 + 2Buv + Cv 2 1/2 , where u(t) and v(t) are any two linearly independent solutions of the equation0 2000 Mathematics Subject Classification. 34A05, 01A75, 70F05, 22E70. The Ermakov Equation: history and impact 147 and the constants, A, B and C, are related according to B 2 − AC = 1/W 2 with W being the constant Wronskian of the two linearly independent solutions.In 1966 the late Ralph Lewis, while he was on sabbatical at the University of Heidelberg, commenced the calculation of an invariant for the Hamiltonian corresponding to (3), videlicetusing Kruskal's asymptotic method [54]. An adiabatic invariant for (4) had been proposed by Lorentz at the Solvay Congress of 1911 [98], but this was not satisfactory for the application of interest to Lewis which was the motion of a charged particle in an electromagnetic field expected to be rapidly varying in the context of plasma confinement. Kruskal's method involves an asymptotic expansion in terms of a parameter, ε, and for the first term in the expansion Lewis obtainedwhere ρ(t) is a solution of (1) and so is given by (2). To the surprise of Lewis the second and third terms in the asymptotic expansion were zero. He then essayed 1 a direct calculation of the first derivative of I 0 and found it to be zero! His results are found in [65][66][67].Although this result could scarcely be considered to be of use in classical problems, the reduction of the solution of the corresponding time-dependent Schrödinger equation [68] to the solution of (3) could be regarded as a distinct advantage since any numerical computation was then deferred until almost the last line. A clear example of this is found in the calculation of Berry's Phase for the time-dependent linear oscillator [55]. The utility of the result was enhanced when it was found that (1) occurred as an auxiliary equation in the calculation of invariants for nonquadratic Hamiltonian systems [16,17,37,38,26,[69][70][71][72][73][74]. ERMAKOV AND HIS INVARIANTIn Ermakov's paper the roles of (1), (3) and (5) are interchanged by comparison with the work of Lewis 2 . Ermakov introduces (1) as an equation auxiliary to (3), multiplies by an integr...

Cosmological Einstein-Skyrme solutions with nonvanishing topological charge

“…1 [39,40], which is also found in various physical systems (see for instance [41,42]). One of its features is that it is invariant under a larger than expected symmetry, SL(2, R) in this case.…”

On a hidden symmetry of quantum harmonic oscillators

“…The maximal kinematical invariance group of the simple harmonic oscillator [50] provides the six-parameter family of solutions, namely (1.2) and (1.3)-(1.9), for an arbitrary choice of the initial data (of the corresponding Ermakov-type system [17], [39], [41], [44]). These "hidden parameters" usually disappear after evaluation of matrix elements and cannot be observed from the spectrum.…”

“…We establish a (hidden symmetry revealing) connection with certain Ermakov-type system which allows us to bypass a complexity of the traditional Lie algebra approach [44] (see [17], [41] and the references therein regarding the Ermakov equation). (A general procedure of obtaining new solutions by acting on any set of given ones by enveloping algebra of generators of the Heisenberg-Weyl group is described in [15].)…”

“…More applications include, but are not limited to charged particle traps and motion in uniform magnetic fields, molecular spectroscopy and polyatomic molecules in varying external fields, crystals through which an electron is passing and exciting the oscillator modes, and other mode interactions with external fields. Quadratic Hamiltonians have particular applications in quantum electrodynamics because the electromagnetic field can be represented as a set of generalized driven harmonic oscillators [13], [20].The maximal kinematical invariance group of the simple harmonic oscillator [50] provides the six-parameter family of solutions, namely (1.2) and (1.3)-(1.9), for an arbitrary choice of the initial data (of the corresponding Ermakov-type system [17], [39], [41], [44]). These "hidden parameters" usually disappear after evaluation of matrix elements and cannot be observed from the spectrum.…”

Quantum operations: technical or fundamental challenge?

“…The exact solutions of (1) were widely explored by Lewis and Riesenfeld [27,28,23], then by Malkin, Manko, Trifonov [29,30] in terms of adiabatic invariants, interrelated also with an important techniques of quantum tomography [6,7,9]. The mathematical algorithms, though elementary, are not immediate to apply (they require anyhow the solution of the 2-nd order differential eq.…”