【摘 要】
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For any classical Lie algebra g, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers (m, n). The universal
【机 构】
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Institute of Modern Physics
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For any classical Lie algebra g, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers (m, n). The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets areprovided, and explicit examples for g = Br, Cr,Dr with m, n ≤ 3 are also given. For all m, n, it is shown that the dynamics of the (m, n - 1)- and the (m - 1, n)-Toda chains are natural reductions of that of the (m, n)-chain,and for m = n, there is also a family of symmetrically reduced Toda systems, the (m, m)sym-Toda systems, which are also integrable. In the quantum case, all (m,n)-Toda systems with m ＞ 1 or n ＞ 1 describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all (m, n)-Toda systems survive after quantization.
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