An introduction to Lie systems

Cristina Sardon (University of Salamanca, Spain)

Thursday 5th November, 2015 16:00-17:00 Maths 522

Abstract

A Lie system is a system of ordinary dierential equations (ODE)
admitting a superposition principle or superposition rule, i.e., a map
allowing us to express the general solution of the system of ODEs in
terms of a family of particular solutions and a set of constants related
to initial conditions. These superposition principles are, in general,
nonlinear.
From the point of view of their applications, Lie systems play a
relevant role in Cosmology, Financial Mathematics, Control Theory,
Quantum Mechanical problems and Biology.
We focus on a particular type of Lie system: the so-called Lie{
Hamilton systems on the plane. A complete classication of these systems
is provided on R2, their solutions in terms of superposition principles
are achieved through the Poisson coalgebra theory and physical
interpretations are contemplated within a number of explicit examples.
For these matters, a brief review of Dierential Geometry fundamentals
in modern language is followed along this seminar.
References
[1] A. Ballesteros, A. Blasco, F.J. Herranz, J. de Lucas, C. Sardon, Lie{
Hamilton systems on the plane: theory, classication and applications,
J. Dierential Equations 258, 2873{2907 (2015).
[2] J.F. Cari~nena, J. de Lucas, C. Sardon, A new Lie system's approach to
second{order Riccati equations, Int. J. Geom. Methods Mod. Phys. 9,
1260007 (2012).
[3] J.F. Cari~nena, J. de Lucas, C. Sardon, Lie{Hamilton systems: theory
and applications, Int. Geom. Methods Mod. Phys. 10, 0912982 (2013).
[4] J. de Lucas, C. Sardon, On Lie systems and Kummer{Schwarz equations,
J. Math. Phys. 54, 033505 (2013).

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