2ND-ORDER HAMILTONIAN EQUATIONS ON T-INFINITY AND ALMOST-PERIODIC SOLUTIONS

Motivated by problems arising in nonlinear PDE's with a Hamiltonian st ructure and in high dimensional dynamical systems, we study a suitable generalization to infinite dimensions of second order Hamiltonian equ ations of the type ($) over bar x=partial derivative(x)V, [x is an ele ment of T-N, partial derivative(x)=(partial derivative(x1), ..., parti al derivative(xN))]. Extending methods from quantitative perturbation theory (Kolmogorov-Arnold-Moser theory, Nash-Moser implicit function t heorem, etc.) we construct uncountably many almost-periodic solutions for the infinite dimensional system ($) over bar x(i)=f(i)(x), i is an element of Z(d), x is an element of T-Zd (endowed with the compact to pology); the Hamiltonian structure is reflected by f being a ''general ized gradient.'' Such a result is derived under (suitable) analyticity assumptions on f(i) but without requiring any ''smallness conditions. '' (C) 1995 Academic Press, Inc.