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Applications of Lie-Hamilton systems on the plane: Cayley-Klein Riccati equations and beyond | | F.J. Herranz
; J. de Lucas
; C. Sardon
; | | Date: |
27 Oct 2014 | | Abstract: | A Lie-Hamilton system is a nonautonomous system of first-order ordinary
differential equations describing the integral curves of a $t$-dependent vector
field taking values in a finite-dimensional real Lie algebra of Hamiltonian
vector fields with respect to a Poisson structure. After reviewing the
classification of finite-dimensional real Lie algebras of Hamiltonian vector
fields on $mathbb{R}^2$, we present new Lie-Hamilton systems on the plane with
physical, biological and mathematical applications. New results cover
Cayley-Klein Riccati equations, the hereafter called planar diffusion Riccati
systems and complex Bernoulli equations, all of them with $t$-dependent real
coefficients. Furthermore, we study the existence of local diffeomorphisms
among new and already known Lie-Hamilton systems on the plane. In particular,
we show that the Cayley-Klein Riccati equations describe as particular cases
well-known coupled Riccati equations, second-order Kummer-Schwarz equations,
Milne-Pinney equations, the harmonic oscillator with $t$-dependent frequency
and other systems of physical and mathematical relevance. | | Source: | arXiv, 1410.7336 | | Services: | Forum | Review | PDF | Favorites | |
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