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Generalized Toda Mechanics Associated with Loop Algebras L(Cr) and L(Dr) and Their Reductions

Generalized Toda Mechanics Associated with Loop Algebras L(C_r) and L(D_r)and Their Reductions
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摘要 We construct a class of integrable generalization of Toda mechanics withlong-range interactions. These systems are associated with the loop algebras L(C_r) and L(D_r) inthe sense that their Lax matrices can he realized in terms of the c = 0 representations of theaffine Lie algebras C_r~((1)) and D_r~((1)) and the interactions pattern involved bears the typicalcharacters of the corresponding root systems. We present the equations of motion and the Hamiltoninnstructure. These generalized systems can be identified unambiguously by specifying the underlyingloop algebra together with an ordered pair of integers (n, m). It turns out that different systemsassociated with the same underlying loop algebra but with different pairs of integers (n_1, m_1) and(n_2, m_2) with n_2 【 n_1 and m_2 【 m_2 can be related by a nested Hamiltonian reduction procedure.For all nontrivial generalizations, the extra coordinates besides the standard Toda variables arePoisson non-commute, and when either n or m ≥ 3, the Poisson structure for the extra coordinatevariables becomes some Lie algebra (i.e. the extra variables appear linearly on the right-hand sideof the Poisson brackets). In the quantum case, such generalizations will become systems withnoncommutative variables without spoiling the integrability.
出处 《Communications in Theoretical Physics》 SCIE CAS CSCD 2005年第1期1-8,共8页 理论物理通讯(英文版)
关键词 TODA many-body system poisson bracket 广义Toda力学 多体系统 泊松括号 非线性偏微分方程
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参考文献12

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