Toda lattice

Example

Let $n \ge 2$. The open Toda lattice is the Hamiltonian system on $\R^{2n}$ with positions $q_1, \ldots, q_n$, momenta $p_1, \ldots, p_n$ and Hamiltonian $$H(q, p) = \sum_{k=1}^{n} \frac{p_k^2}{2} + \sum_{k=1}^{n-1} e^{-(q_{k+1} - q_k)}.$$ It describes $n$ particles on a line with exponential forces between neighbours.

It is an integrable system. Flaschka's variables are $a_k = \frac{1}{2} e^{-(q_{k+1} - q_k)/2}$ for $1 \le k \le n - 1$ and $b_k = -\frac{1}{2} p_k$ for $1 \le k \le n$. Let $L$ be the symmetric tridiagonal $n \times n$ matrix with diagonal entries $b_1, \ldots, b_n$ and off-diagonal entries $a_1, \ldots, a_{n-1}$. The equations of motion can be written as a Lax pair equation $L' = [B, L]$, where $B$ is an antisymmetric tridiagonal matrix whose off-diagonal entries are $\pm a_k$. So $L(t)$ is conjugate to $L(0)$ for all $t$, and the $n$ eigenvalues of $L$, or the traces $\tr L^m$ for $m = 1, \ldots, n$, are first integrals. These integrals are independent and Poisson-commute, which gives Liouville integrability.

A check of the scaling: $\tr L^2 = \sum_k b_k^2 + 2 \sum_k a_k^2 = \frac{1}{4} \sum_k p_k^2 + \frac{1}{2} \sum_k e^{-(q_{k+1} - q_k)} = \frac{1}{2} H$. Also $\tr L = -\frac{1}{2} \sum_k p_k$ is a multiple of the total momentum.