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

A Ballesteros, J F Cariñena, F J Herranz, J de Lucas, C Sardón
2013 Journal of Physics A: Mathematical and Theoretical  
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.
more » ... ebra 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.
doi:10.1088/1751-8113/46/28/285203 fatcat:vpmapuwqz5extarltzjz7cb6ai