Generalized Toda Mechanics Associated with Loop Algebra L(Br)

A class of generalization of Toda mechanics with long range interactions is constructed in this paper. These systems are associated with the loop algebras L(Br) in the sense that their Lax matrices can be realized in terms of the c = 0 representations of the affine Lie algebras B(1)r. We adopt a pair of ordered integers (m,n) to describe the Toda mechanics system when we present the equations of motion and the Hamiltonian structure. We also extract the classical r matrix which satisfy the classical Yang-Baxter relation. Such generalizations will become systems with noncommutative variables in the quantum case.