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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4181a14259d1344524534bda8087d90b82420718)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/769ce56b13c72565b3bd4240d7ad5cb8cde9e8dc)
where
is the displacement of the
-th particle from its equilibrium position,
and
is its momentum (mass
),
and the Toda potential
.