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arXiv · 1305.6272

From constants of motion to superposition rules for Lie-Hamilton systems

A. Ballesteros·J. F. Cariñena·F. J. Herranz·J. de Lucas·C. Sardón
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Abstract

A Lie system is a nonautonomous system of first-order differential equations possessing a superposition rule, i.e. a map expressing its general solution in terms of a generic finite family of particular solutions and some constants. Lie-Hamilton systems form a subclass of Lie systems whose dynamics is governed by a curve in a finite-dimensional real Lie algebra of functions on a Poisson manifold. It is shown that Lie-Hamilton systems are naturally endowed with a Poisson coalgebra structure. This allows us to devise methods to derive in an algebraic way their constants of motion and superposition rules. We illustrate our methods by studying Kummer-Schwarz equations, Riccati equations, Ermakov systems and Smorodinsky-Winternitz systems with time-dependent frequency.

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A. Ballesteros, J. F. Cariñena, F. J. Herranz, J. de Lucas, C. Sardón. 2013-05-27. From constants of motion to superposition rules for Lie-Hamilton systems. https://doi.org/10.1088/1751-8113%2F46%2F28%2F285203

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