Second Order Hamiltonian Equations on ${\bf T}^\infty$ and Almost-Periodic Solutions (87K, Plain TeX) Feb 5, 93
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Abstract. Motivated by problems arising in nonlinear PDE's with a Hamiltonian structure and in high dimensional systems, we study a suitable generalization to infinite dimensions of second order Hamiltonian equations of the type $\ddot x=\partial_xV,$ [$x\in {\bf T}^N,$ $\partial_x\equiv(\partial_{x_1}, \dots,\partial_{x_N})$]. Extending methods from quantitative perturbation theory (Kolmogorov-Arnold-Moser theory, Nash-Moser implicit function theorem, etc.) we construct uncountably many almost-periodic solutions for the infinite dimensional system $\ddot x_i=f_i(x), i\in{\bf Z}^d, x\in{\bf T}^{\bf Z}^d$ (endowed with the compact topology); the Hamiltonian structure is reflected by $f$ being a "generalized gradient". Such result is derived under (suitable) analyticity assumptions on $f_i$ but without requiring any "smallness conditions".

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