Symmetric Designs on Lie Algebras and Interactions of Hamiltonian Systems

Nonhamiltonian interaction of hamiltonian systems is considered. Dynamical equations are constructed by use of symmetric designs on Lie algebras. The results of analysis of these equations show that some class of symmetric designs on Lie algebras beyond Jordan ones may be useful for a description of almost periodic, asymptotically periodic, almost asymptotically periodic, and possibly, more chaotic systems. However, the behaviour of systems related to symmetric designs with additional identities is simpler than for general ones from different points of view. These facts confirm a general thesis that various algebraic structures beyond Lie algebras may be regarded as certain characteristics for a wide class of dynamical systems. Many important classical hamiltonian dynamical systems are connected with Lie algebras [1,2] or their nonlinear generalizations [3]. An interaction of hamiltonian systems may be of different kinds. First, it may be hamiltonian, i.e. defined by a subsidiary term Hint in the hamiltonian and, possibly, by a certain (maybe rather nonlinear) deformation of initial (”free”) Poisson brackets. Second, it may be nonhamiltonian, i.e. with nonconservative (nonpotential) forces of interaction, however, one may suppose that it is still nondissipative, i.e. the total energy is conserved. To describe algebraic structures governing such interactions is an important unsolved problem of mathematical physics. There exist, at least, two approaches to the problem. The first approch associates the nonhamiltonian interaction with certain deformations or generalizations of the initial algebraic structures (Lie algebras) such as f.e. isotopic pairs [4] or general (nonlinear) I–pairs [5]. The second approach defines such interaction by use of subsidiary algebraic structures on Lie algebras.