## Nambu dynamics, $$n$$-Lie algebras and integrability.(English)Zbl 1196.37110

The generalization of a bracket Poisson on a case several Hamiltonians is offered. The connection of this theory with Nambu’s dynamics and analogy to brackets of $$n$$-Lie algebras is shown. As an example the system of spin particles is considered.

### MSC:

 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 22E70 Applications of Lie groups to the sciences; explicit representations 17A42 Other $$n$$-ary compositions $$(n \ge 3)$$