Multisymplectic Lie Systems
Presenting author: Silvia Vilariño
Authors: X. Gràcia, J. de Lucas, M. Muñoz-Lecanda, N. Román-Roy and S. Vilariño
A Lie system is a 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 vector fields: a so-called Vessiot--Guldberg Lie algebra.
This condition is so stringent that just few systems of differential equations can be considered as Lie
systems [DIS]. Nevertheless, Lie systems appear in important physical and
mathematical problems and enjoy relevant geometric properties [ADR12,DIS,
Clem06,Ru08,Ru10,Ib00,WintSecond], which strongly prompt their analysis.
Some attention has lately been paid to Lie systems admitting
a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields with respect to several geometric structures
[BBHLS13,BCHLS13,CLS13]. Surprisingly, studying these particular types of Lie systems led to investigate much more
Lie systems and applications than before. The first attempt in this direction was performed by
Marmo, Cariñena and Grabowski [CGM00], who briefly studied Lie systems with Vessiot--Guldberg Lie algebras of Hamiltonian
vector fields relative to a symplectic structure. This line of research was posteriorly followed by several researchers
[ADR12,Ru10].
The study of Lie systems with associated geometric structures is very important. For instance Lie-Hamilton systems, Dirac-Lie systems,
k-symplectic- Lie systems, etc. Using these geometric structures one can obtain time-independent constants of motions and superpositions rules.
In this talk we present a particular type of Lie systems, those admitting a Vessiot-Gulberg Lie algebra of Hamiltonian vector field relative to a multisymplectic structure. This new type of Lie systems has interesting properties. In this poster we want to present some properties of these systems of differential equations.
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