Lie–hamilton Systems: Theory and Applications

Abstract

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie-Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearisability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.

k-Symplectic Lie systems: theory and applications

“…It can be proved that r X : x ∈ N → dim D X x ∈ N ∪ {0} must only be constant on the connected components of an open and dense subset U X of N (see [21]), where D X becomes a regular, involutive and integrable distribution. Since dim V X x = dim N − r X (x), then V X becomes a regular co-distribution on each connected component of U X also.…”

“…Meanwhile, the associated co-distribution appears in the study of constants of motion for Lie systems [21]. For instance, the following proposition described in [21] shows that (locally defined) t -independent constants of motion of t-dependent vector fields are determined by (locally defined) exact one-forms taking values in its associated co-distribution. Then, V X is what really matters in the calculation of such constants of motion for a system X.…”

“…Nevertheless, Lie systems appear in important physical and mathematical problems and enjoy relevant geometric properties [4,20,23,25,26,30,51,53], 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 [7,8,21]. Surprisingly, studying these particular types of Lie systems led to investigate much more Lie systems and applications than before.…”

Lie–Scheffers Systems

“…As another example consider the differential equation of an n-dimensional Smorodinsky-Winternitz oscillator of the form [WSUP67,CLS13] ⎧ ⎨ ⎩ẋ…”

“…An important case of Lie-Scheffers systems is when (M, ) is a symplectic manifold and the vector fields in M arising in the expression of the t-dependent vector field describing a Lie-Scheffers system are Hamiltonian vector fields closing on a finitedimensional real Lie algebra g [CLS13]. When these vector fields are complete, they are fundamental vector fields of a symplectic action of a Lie group G with Lie algebra g on the symplectic manifold (M, ).…”