Maximal Almost-Periodic Solutions for Lagrangian Equations on Infinite Dimensional Tori

Abstract

We shall briefly discuss the generalization to infinite dimensions of the existence of maximal quasi-periodic solutions for the following Euler-Lagrange equations on T ^N ≡ (R /2πZ ) ^N:

$${\ddot x_i} = {V_{xi}}(x),i = 1,...,N$$

(1.1)

associated to the Lagrangian

$$L(\dot x,x) = \frac{1}{2}\sum\limits_{i = 1}^N {\dot x_i^2} + V(x)$$

(1.2)

V(x) = V(x ₁,…, x _N ) being a smooth function 2π-periodic in x _i; ${x_i};{( \cdot )_{{x_i}}}$ denotes partial differentiation: ${V_{xi}} \equiv \frac{{\partial V}}{{\partial {x_i}}}.$

Partially supported by CNR-GNFM grant, n. 2398/91