Stratified Lie systems: theory and applications

Abstract

A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold $M$ described by a $t$-dependent vector field $X=\sum_{\alpha=1}^rg_\alpha X_\alpha$, where $X_1,\ldots,X_r$ are vector fields on $M$ spanning an $r$-dimensional Lie algebra that are tangent to the strata of a stratification $\mathcal{F}$ of $M$ while $g_1,\ldots,g_r:\mathbb{R}\times M\rightarrow \mathbb{R}$ are functions depending on $t$ that are constant along integral curves of $X_1,\ldots,X_r$ for each fixed $t$. We analyse the particular solutions of stratified Lie systems and how their properties can be obtained as generalisations of those of Lie systems. We illustrate our results by studying Lax pairs and a class of $t$-dependent Hamiltonian systems. We study stratified Lie systems with compatible geometric structures. In particular, a class of stratified Lie systems on Lie algebras are studied via Poisson structures induced by $r$-matrices.

Symmetries, Conservation and Dissipation in Time‐Dependent Contact Systems

“…In the last decades, alternative geometric structures and their associated dynamics have been widely studied. In particular, contact geometry has arisen as a geometric solution to model non‐conservative systems, [ 1–8 ] as well as some thermodynamical systems, [ 9–13 ] quantum systems, [ 14 ] nonholonomic systems, [ 15 ] electromagnetism, [ 16 ] gravitation, [ 17 ] Lie systems, [ 18 ] control theory, [ 19 ] dissipative field theories, [ 7,20,21 ] and so on.…”