Generalized Toda Mechanics Associated with Classical Lie Algebras and Their Reductions

For any classical Lie algebra , we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers . The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for with are also given. For all , it is shown that the dynamics of the - and the -Toda chains are natural reductions of that of the -chain, and for , there is also a family of symmetrically reduced Toda systems, the -Toda systems, which are also integrable. In the quantum case, all -Toda systems with 1$' SRC='http://ej.iop.org/images/0253-6102/41/3/339/ctp_41_3_339_12.gif'/> or 1$' SRC='http://ej.iop.org/images/0253-6102/41/3/339/ctp_41_3_339_13.gif'/> describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all -Toda systems survive after quantization.