On extensions, Lie-Poisson systems, and dissipation. (English) Zbl 1495.37050
The authors study Lie-Poisson systems on the dual spaces of unified products of Lie algebras. They show how the extending structure framework allows not only to study the dynamics on 2-cocycle extensions but also to couple mutually interacting Lie-Poisson systems. Moreover, symmetric brackets such as the double bracket, the Cartan-Killing bracket, the Casimir dissipation bracket, and the Hamiltonian dissipation bracket are all explicitly given. Thus, the authors obtain the collective motion of two mutually interacting irreversible dynamical systems as well as the mutually interacting metriplectic flows. Three interesting examples are presented.
Reviewer: Nenad Manojlović (Faro)
MSC:
| 37J37 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures |
| 37J06 | General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
Software:
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