k-Symplectic Lie systems: theory and applications

Abstract

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 socalled Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the k-symplectic Lie systems, admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields with respect to the presymplectic forms of a k-symplectic structure. We devise new k-symplectic geometric methods to study their superposition rules, t-independent constants of motion and general properties. Our results are illustrated through examples of physical and mathematical interest. As a byproduct, we find a new interesting setting of application of the k-symplectic geometry: systems of first-order ordinary differential equations.

A new application of k-symplectic Lie systems

“…Our procedures help in deriving superposition rules and constants of motion for such systems. This shows, as done in [3], that k-symplectic structures can be used to analyse systems of first-order ordinary differential equations.…”

“…We now recall the notion of k-symplectic structures and we relate them to various Poisson algebras (see [3] for details). Mathematical structures are assumed to be real, smooth and globally defined.…”

Geometry of Riccati equations over normed division algebras

“…It is worth noting that geometric techniques, e.g. symplectic, Poisson, k-symplectic or Jacobi structures, may be applied to study their properties [7,22,23,30].…”

Multisymplectic structures and invariant tensors for Lie systems

“…This is employed to obtain constants of motion and superposition rules for X [22].Our approach shows that invariants and superposition rules for multisymplectic Lie systems can be obtained through Casimir elements of universal enveloping algebras, which can be understood as symmetric tensors in T (g), or co-cycles of the Chevalley-Eilenberg cohomology of g (see [61]), which are understood as antisymmetric tensors of T (g). Moreover, this method gives rise to obtaining ksymplectic or presymplectic structures compatible with Lie systems, which allows for the application of the techniques in [19,49] to study multisymplectic Lie systems.As an application, our methods are employed to study superposition rules for multisymplectic Lie systems related to locally automorphic Lie systems. In particular, the cases of Schwarz equations and Riccati-type diffusion systems are studied in detail, while control and Darboux-Brioschi-Halphen systems are used to illustrate some results and/or techniques [37,51,57].The structure of the paper goes as follows.…”