Difference Equations: The Toda Lattice

The idea of P. Lax [3.1] to construct a nonlinear equation which might be exactly soluble by using a commutation relation like (3.2.1), $${\rm{\dot L(t) = i \{ L,H\} ,}}$$ (7.1.1) is sufficiently general that it can be applied not only to differential operators, but also to (finite or infinite) matrices. It is fascinating to be able to construct in this manner nonlinear difference equations which are soluble in closed form; in the case of finite matrices we can obtain soluble nonlinear dynamical systems with a finite number of degrees of freedom.