The Toda lattice is a simple model for a nonlinear one-dimensional crystal that describes the motion of a chain of particles with nearest neighbor interactions. The equation of motion for such a system is given by

(1)

where m denotes the mass of each particle, the displacement of the nth particle from its equilibrium position, V(r) is the interaction potential, and . As first discovered by Toda, this system becomes particularly interesting in the case of exponential restoring force, which is often implicitly assumed (Flaschka 1974). The equations of motion are derivative from the Hamiltonian

(2)

where is the displacement of the nth mass from equilibrium and is the corresponding momentum.

Ford et al. (1973) considered the three-particle Toda lattice on a ring (i.e., with periodic boundary conditions, so that ) with exponential interactions, which has a Hamiltonian of the form

(3)

Because , this can be reduced to the two-dimensional form

(4)

(Ford et al. 1973; Tabor 1989, p. 123). For the case of equal masses , Ford et al. (1973) perform numerical experiments which showed that the system's surface of section contained only smooth curves, suggesting it was in fact integrable. Hénon (1974) subsequently found the corresponding integral of motion, given by

	(5)

In the limit of small x and y, this reduces to

(6)

which is just the angular momentum of the system.

Unlike the case of equal masses, Casati and Ford (1975) showed that chaotic behavior is obtained if .

Flaschka (1974) showed that Toda lattice is a finite-dimensional analog of the Korteweg-de Vries equation.

Korteweg-de Vries Equation