Toda lattice

The Toda lattice, introduced by Morikazu Toda (1967), is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system.

It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian

${\displaystyle {\begin{aligned}H(p,q)&=\sum _{n\in \mathbb {Z} }\left({\frac {p(n,t)^{2}}{2}}+V(q(n+1,t)-q(n,t))\right)\end{aligned}}}$

and the equations of motion

${\displaystyle {\begin{aligned}{\frac {d}{dt}}p(n,t)&=-{\frac {\partial H(p,q)}{\partial q(n,t)}}=e^{-(q(n,t)-q(n-1,t))}-e^{-(q(n+1,t)-q(n,t))},\\{\frac {d}{dt}}q(n,t)&={\frac {\partial H(p,q)}{\partial p(n,t)}}=p(n,t),\end{aligned}}}$

where ${\displaystyle q(n,t)}$ is the displacement of the ${\displaystyle n}$-th particle from its equilibrium position,

and ${\displaystyle p(n,t)}$ is its momentum (mass ${\displaystyle m=1}$),

and the Toda potential ${\displaystyle V(r)=e^{-r}+r-1}$.