Ermakov's Superintegrable Toy and Nonlocal Symmetries

Abstract

Abstract. We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the algebra sl(2, R). The number of point symmetries is insufficient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the complete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The problem illustrates the difficulties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.

Recent Applications of the Theory of Lie Systems in Ermakov Systems

“…Such a problem has being receiving these last years very much attention because of its very important applications in physics and mathematics, see [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. This approach has recently been revisited from a more geometric perspective in [18] where the rôle of the superposition function is played by an appropriate connection.…”

On the frames of spaces of finite-dimensional Lie algebras of dimension at most 6

“…This is a well-known nonlinear superposition property of the so-called Ermakov systems (see, for example, [10,18,21], [40][41][42], [45,51,65,67,69,74] and references therein). Here we have obtained this 'nonlinear superposition principle' (or Pinney's solution) in an operator form by multiplication and addition of the linear dynamical invariants together with an independent characterization of all quantum quadratic invariants, which seems to be missing, in general, in the available literature (see also [43,45] for an important classical case).…”

Dynamical invariants for variable quadratic Hamiltonians

“…(more details are given in appendix A). This is a well-known nonlinear superposition property of the so-called Ermakov systems (see, for example, [10,18,21], [40][41][42], [45,51,65,67,69,74] and references therein). Here we have obtained this 'nonlinear superposition principle' (or Pinney's solution) in an operator form by multiplication and addition of the linear dynamical invariants together with an independent characterization of all quantum quadratic invariants, which seems to be missing, in general, in the available literature (see also [43,45] for an important classical case).…”