Garnier System

Abstract

In the present chapter we study several closely related integrable mechanical systems with cubic non-linearities, along with their integrable discretizations. The first one is the so-calledanharmonic oscillator:

$$ \ddot x_k = - \omega _k x_k - 2x_k \left\langle {x,x} \right\rangle , 1 \leqslant k \leqslant N, $$

(22.1.1)

Where

$$\left\langle {x,x} \right\rangle = \sum\limits_{{j = 1}}^{N} {x_{j}^{2}} . $$