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  "body": "\n## Example\n\nLet $n \\ge 2$.\nThe open *Toda lattice* is the [[Hamiltonian system|0EDKGA2FAK]] on $\\R^{2n}$ with positions $q_1, \\ldots, q_n$, momenta $p_1, \\ldots, p_n$ and Hamiltonian\n$$H(q, p) = \\sum_{k=1}^{n} \\frac{p_k^2}{2} + \\sum_{k=1}^{n-1} e^{-(q_{k+1} - q_k)}.$$\nIt describes $n$ particles on a line with exponential forces between neighbours.\n\nIt is an [[integrable system|N0N7CMRCMH]].\nFlaschka'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 $1 \\le k \\le n$.\nLet $L$ be the symmetric tridiagonal $n \\times n$ matrix with diagonal entries $b_1, \\ldots, b_n$ and off-diagonal entries $a_1, \\ldots, a_{n-1}$.\nThe equations of motion can be written as a [[Lax pair|G3A4S7CSB3]] equation $L' = [B, L]$, where $B$ is an antisymmetric tridiagonal matrix whose off-diagonal entries are $\\pm a_k$.\nSo $L(t)$ 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.\nThese integrals are independent and Poisson-commute, which gives Liouville integrability.\n\nA check of the scaling: $\\tr L^2 = \\sum_k b_k^2 + 2 \\sum_k a_k^2 = \\frac{1}{4} \\sum_k p_k^2 + \\frac{1}{2} \\sum_k e^{-(q_{k+1} - q_k)} = \\frac{1}{2} H$.\nAlso $\\tr L = -\\frac{1}{2} \\sum_k p_k$ is a multiple of the total momentum.\n\n## Remarks\n\nMorikazu Toda introduced the lattice in 1967.\nIts integrability was shown by Hénon and by Flaschka in 1974.\nThere is also a periodic Toda lattice, with $q_{n+1} = q_1$, which is integrable as well.\n\n## Explanations\n\n### L1 human\n\nThe Toda lattice is a row of particles pushed apart by forces that grow exponentially as neighbours come close.\nAlthough the forces are nonlinear, the system has as many conserved quantities as particles, so its motion is completely regular.\nThe conserved quantities are hidden in a matrix built from positions and momenta: as time goes on the matrix changes, but its eigenvalues do not.\n",
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