Mal’tsev, A. Ya. The averaging of Hamiltonian structures in the discrete variant of Whitham’s method. (English. Russian original) Zbl 0928.35170 Russ. Math. Surv. 53, No. 1, 214-216 (1998); translation from Usp. Mat. Nauk 53, No. 1, 219-220 (1998). The discrete variant of Whitnam’s method concerns the quasiperiodic \(m\)-phase solutions \[ \varphi_n^i(t) =\Phi^i\bigl(k(U)n +\omega (U)t+ \vartheta_0, U\bigr) \] \((k(U)\) and \(\omega(U)\) are \(m\)-dimensional vectors, \(U=(U^1, \dots, U^N)\) are parameters, \(\Phi^i(\vartheta,U)\) are 2-periodic functions of the \(m\) variables \(\vartheta)\) of the nonlinear system \[ \partial\varphi^i_n/ \partial t=f^i(\varphi_n, \varphi_{n-1},\varphi_{n+1}, \dots) \tag{1} \] \((\varphi_n= (\varphi^1_n, \dots,\varphi^p_n)\), \(n\in\mathbb{N}\), \(f^i\) depend on finite many arguments). Assuming the system (1) to be Hamiltonian with \(N\) involutory first integrals \(I^\nu=\sum P^\nu (\varphi_n, \varphi_{n-1}, \varphi_{n+1}, \dots)\), the Whitnam system describing the evolution of parameters \(U\) can be written as \[ {\partial\over\partial T}\langle P^\nu\rangle \bigl(U(X) \bigr)= \sum m{\partial \over\partial X}\langle Q^\nu_{ (m)} \rangle\bigl(U(X) \bigr) \tag{2} \] where \(\langle\cdot\rangle\) denotes the averaging. Then (2) again is Hamiltonian with a Poisson bracket of hydrodynamical type and Hamiltonian \(H=\int\langle P_H\rangle(X)dX\), the averaging of the primary Hamiltonian. Reviewer: Jan Chrastina (Brno) Cited in 1 Document MSC: 35Q58 Other completely integrable PDE (MSC2000) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Keywords:completely integrable systems; Hamiltonian structure; averaging; Whitnam’s method PDFBibTeX XMLCite \textit{A. Ya. Mal'tsev}, Russ. Math. Surv. 53, No. 1, 214--216 (1998; Zbl 0928.35170); translation from Usp. Mat. Nauk 53, No. 1, 219--220 (1998) Full Text: DOI arXiv