Isospectral hamiltonian flows in finite and infinite dimensions
Isospectral hamiltonian flows in finite and infinite dimensions
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DOI：
10.1007/bf01223376
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发表时间：
1988-09
影响因子：
2.4
通讯作者：
M. Adams;J. Harnad;E. Previato
中科院分区：
文献类型：
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作者：
M. Adams;J. Harnad;E. Previato
从固定N × N矩阵A的秩r扰动的Poisson流形A到形式loop代数=gl（r）[[λ，λ−1]]的正部分的对偶构造了一个矩映射.利用Adler-Kostant-Symes定理给出了在上生成可交换等谱流的哈密顿量，这些哈密顿量被矩映射拉回，在上生成可交换等谱哈密顿流.后者可以确定与流动有限维coadjoint轨道和线性化的Jacobi品种的不变谱曲线，一般来说，是一个r-片黎曼曲面。导出了对应于gl（r，n）和sl（r，n）的子代数的n A的约化，确定为对合生成的自同构群的不动点集（即，所有的经典代数），以及减少扭曲的子代数。该理论说明了一些有限维等谱流的例子定义可积哈密顿系统和它们的嵌入有限间隙解决方案的PDE的可积系统。
A moment mapis constructed from the Poisson manifold ℳAof rank-rperturbations of a fixedN×NmatrixAto the dualof the positive part of the formal loop algebra=gl(r)⊗ℂ[[λ, λ−1]]. The Adler-Kostant-Symes theorem is used to give hamiltonians which generate commutative isospectral flows on. The pull-back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in ℳA. The latter may be identified with flows on finite dimensional coadjoint orbits inand linearized on the Jacobi variety of an invariant spectral curveXrwhich, generically, is anr-sheeted Riemann surface. Reductions of ℳAare derived, corresponding to subalgebras ofgl(r, ℂ) andsl(r, ℂ), determined as the fixed point set of automorphism groupes generated by involutions (i.e., all the classical algebras), as well as reductions to twisted subalgebras of. The theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of PDE's.