Reĭman, A. G. A unified Hamiltonian system on polynomial bundles and the structure of stationary problems. (Russian. English summary) Zbl 0535.58017 Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 131, 118-127 (1983). The paper studies the Hamiltonian structure on the phase space of solutions for nonlinear equations, which are integrable within Zakharov-Shabat scheme with \(U\) and \(V\) operators given in the form \(U(x,\lambda)=\sum^{N}_{i=0}u_ i(x)\lambda^ i\). The author defines the universal phase space \({\mathcal O}^ N\) and the set of equations \(\partial_{t_ n}U=\partial_ xV_ n+[V_ n,U]\), \(n=1,...,N\), on it. Then the universality (in the sense of independence from \(N\)) of the Hamiltonian structure is established. The structure under consideration is restricted to the stationary variety \(\partial_ xV_ n+[V_ n,U]=0\) and the connection between Hamiltonian formalisms for stationary and nonstationary problems is investigated. A symplectic form on the space of stationary solutions, previously defined in terms of the Legendre-Ostrogradsky transformation is shown to coincide with the Kirillov form on the corresponding orbit. Reviewer: Viktor Z. Enol’skij Cited in 1 ReviewCited in 2 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 35Q99 Partial differential equations of mathematical physics and other areas of application Keywords:universal phase space; Legendre-Ostrogradsky transformation; Kirillov form PDFBibTeX XMLCite \textit{A. G. Reĭman}, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 131, 118--127 (1983; Zbl 0535.58017) Full Text: EuDML