Superposition Laws Research Articles

Invariants for forced time-dependent oscillators and generalizations

Several methods for deriving invariants for time-dependent systems have been developed. Four of these methods are as follows: (1) dynamical algebra, (2) Noether's theorem, (3) transformation group, and (4) Ermakov's method. These four methods are related by their use in deriving invariants for the forced, time-dependent oscillator. For this system the methods give the same results. We also discuss forced, nonlinear oscillators possessing analogous invariants. The nonlinear superposition laws for the forced oscillator are derived and discussed.

Nonlinear action of Lie groups and superposition principles for nonlinear differential equations

Recently obtained results on superposition principles for systems of ordinary nonlinear equations are reviewed. In particular the general solution of the matrix Riccati equation is expressed in terms of five particular solutions. The study of superposition laws is related to the problem of Bäcklund transformations for nonlinear partial differential equations, to the problem of transforming nonlinear equations into linear ones and to the classification of primitive filtered Lie algebras.

Ermakov systems and quantum-mechanical superposition laws

Ermakov systems are pairs of coupled, time-dependent, nonlinear dynamical equations possessing a joint constant of the motion called an Ermakov invariant. The invariant provides a link between the two equations and leads to a superposition law between solutions to the Ermakov pair. Extensive studies of Ermakov systems in classical mechanics have been carried out. Here we present a detailed study of Ermakov systems from a quantum point of view, and prove that the solution to the Schr\"odinger equation for a general Ermakov system can be reduced to the solution of a time-independent Schr\"odinger equation involving the Ermakov invariant. We thereby arrive at a quantum-mechanical superposition law analogous to the classical superposition law.

Noether’s theorem and Ermakov systems for nonlinear equations of motion

We present a simple but important generalization of the use of Noether’s theorem to generate invariants for time-dependent nonlinear equations of motion. Our main result is that Noether’s theorem produces invariants even when the equation of motion contains arbitrary functions. The results obtained by using Noether’s theorem are the most general results for which one obtains explicit nonlinear superposition laws for the class of equations of motion considered. We also correct a recent paper on this same subject.