Classical Toda chain

The classical Toda chain describes particles coupled by nearest-neighbor exponential interactions. In the finite open case, its nonlinear motion has $N$ generically independent commuting integrals on a $2N$-dimensional phase space. This record fixes the model and its normalization, works through a three-particle example, and separates this result from the different problems posed by periodic and infinite chains.

Required background. Differentiate a Hamiltonian to obtain equations of motion; derive the open Toda equations develops that calculation. Helpful background. Build the Toda Lax pair explains the matrix representation used below.

The finite open chain

Fix an integer $N\geq 2$. The positions $q_i$, momenta $p_i$, and time $t$ are dimensionless; the equal masses, positive interaction scale, and exponential length scale have been normalized to one. The phase space is

$$(q,p)\in\mathbb R^{2N},\qquad \{q_i,p_j\}=\delta_{ij},\quad \{q_i,q_j\}=\{p_i,p_j\}=0.$$

The particle label is $i=1,\ldots,N$. No ordering restriction on the real coordinates is imposed. The Hamiltonian is

$$H(q,p)=\frac12\sum_{i=1}^{N}p_i^2 +\sum_{i=1}^{N-1}e^{q_i-q_{i+1}}.$$

There are $N-1$ bonds. The first and last particles have one neighbor each, with no bond connecting particle $N$ to particle $1$ and no external end force. This is the nonperiodic system of Moser 1975, § 1, p. 467, equations (1.1)–(1.3), PDF.

Write $u_i=e^{q_i-q_{i+1}}$ for $1\leq i\leq N-1$ and define the endpoint bookkeeping values $u_0=u_N=0$. Hamilton’s equations become

$$\dot q_i=p_i,\qquad \dot p_i=u_{i-1}-u_i.$$

In particular, $\dot p_1=-u_1$ and $\dot p_N=u_{N-1}$. These signs express repulsion: bringing a left neighbor toward a right neighbor increases their interaction energy. The endpoint zeros represent absent bonds; they are not constraints fixing the endpoint positions.

An initial-value problem specifies all $2N$ finite real values $(q(0),p(0))$. Its solution exists for all real time. Indeed, energy conservation gives $|p_i|\leq\sqrt{2H}$ and $u_i\leq H$. Bounded velocities prevent positions from escaping to infinity in finite time, so the smooth Hamiltonian vector field can be continued.

Translation and center-of-mass motion

Only differences of positions enter the potential. Consequently,

$$P=\sum_{i=1}^{N}p_i,\qquad Q=\frac1N\sum_{i=1}^{N}q_i$$

satisfy

$$\dot P=\sum_{i=1}^{N}(u_{i-1}-u_i)=0, \qquad \dot Q=\frac PN.$$

Adding the same constant to every $q_i$ preserves the interaction and all spectral invariants below. The center coordinate is still part of the full phase space; discarding it changes the variable count.

Open, periodic, and infinite are different regimes

The name “Toda chain” alone does not specify boundaries or a spectral problem.

Regime	Defining boundary choice	What changes
Finite open chain	$N-1$ bonds; $u_0=u_N=0$	A finite symmetric tridiagonal matrix describes the spectral invariants used here.
Finite periodic chain	For $N\geq3$, close the chain with $q_{N+1}=q_1+\ell$, $p_{N+1}=p_1$ at fixed shift $\ell$	The added bond is $e^{q_N-q_1-\ell}$. Endpoint equations and spectral data change.
Infinite chain	Sites $i\in\mathbb Z$, with a specified class of initial sequences	Local equations still make sense, but the sum defining a total Hamiltonian may diverge. Operator domains and asymptotic conditions must be supplied.

For the periodic variant displayed here, the potential has $N$ terms instead of $N-1$. In the infinite setting there is no finite list of $N$ eigenvalues to count. The finite open proof on this site establishes neither of those other boundary-value problems. Moser’s original comparison of open and periodic boundaries appears in § 1, pp. 467–469, PDF; his historical discussion is not a statement of today’s research status for the periodic problem.

Spectral representation and integrability

Define the positive bond variables and diagonal entries

$$a_i=e^{(q_i-q_{i+1})/2}\gt0,\qquad b_i=p_i, \qquad a_0=a_N=0.$$

The real symmetric matrix $L$ has entries

$$L_{ii}=b_i,\qquad L_{i,i+1}=L_{i+1,i}=a_i,$$

and all other entries zero. The skew-symmetric matrix $B$ has

$$B_{i,i+1}=-\frac{a_i}{2},\qquad B_{i+1,i}=\frac{a_i}{2},\qquad B_{ii}=0,$$

with other entries zero. Then the Hamilton equations imply

$$\dot L=[B,L]=BL-LB.$$

The minus sign above the diagonal and the factor $1/2$ belong to this normalization. The Lax reference records the convention; the Library derivation verifies every potentially nonzero matrix entry and its conversion to Moser’s convention.

The eigenvalues of $L$ are real, distinct, and constant along the motion. Reality follows from symmetry. Distinctness follows from nonzero off-diagonal entries: the first component of an eigenvector determines all subsequent components by the three-term equation, and it cannot vanish unless the whole eigenvector vanishes. Each eigenspace is therefore one-dimensional. Constancy is a consequence of the Lax equation.

Define

$$I_k=\frac1k\operatorname{tr}L^k, \qquad k=1,2,\ldots.$$

The first three expressions are

$$\begin{aligned} I_1&=P,\\ I_2&=H,\\ I_3&=\frac13\sum_{i=1}^{N}p_i^3 +\sum_{i=1}^{N-1}(p_i+p_{i+1})e^{q_i-q_{i+1}}. \end{aligned}$$

For this finite open Hamiltonian, $\{I_j,I_k\}=0$, and the first $N$ integrals are functionally independent on an open dense subset of $\mathbb R^{2N}$. These facts establish Liouville integrability. Conservation by itself would leave two gaps: Poisson commutation and independence. The full proof fills both directly. When $N=2$, only $I_1,I_2$ are needed; $I_3$ is then a dependent conserved quantity.

A three-particle example

Take

$$q(0)=(0,0,0),\qquad p(0)=(1,0,-1).$$

Both initial bond strengths are $a_1=a_2=1$. Thus

$$L(0)= \begin{pmatrix} 1&1&0\\ 1&0&1\\ 0&1&-1 \end{pmatrix},\qquad B(0)=\frac12 \begin{pmatrix} 0&-1&0\\ 1&0&-1\\ 0&1&0 \end{pmatrix}.$$

Direct substitution gives

$$P=0,\qquad H=3,\qquad I_3=0.$$

The characteristic polynomial is

$$\det(\lambda\mathbf1-L(0))=\lambda^3-3\lambda,$$

so the constant spectrum is $(-\sqrt3,0,\sqrt3)$. These eigenvalues are not the three momenta at $t=0$.

The equations of motion also give a local check independent of eigenvalue calculations:

$$\dot q(0)=(1,0,-1),\qquad \dot p(0)=(-1,0,1),\qquad \dot a_1(0)=\dot a_2(0)=\frac12.$$

The initial particles move inward, while the endpoint forces oppose that motion. The three-particle project follows the trajectory and compares numerical motion with an exact solution available for these symmetric initial data. The Library example checks the Poisson brackets and a nonzero Jacobian minor at this same point.

What the spectral data determine

The conserved eigenvalues determine all traces of powers of $L$, including $P$ and $H$. They do not specify an initial configuration. Many matrices have the same eigenvalues, and the common translation of all positions is invisible to $L$.

For this open Jacobi matrix, inverse spectral reconstruction supplements the eigenvalues with positive endpoint spectral weights whose sum is one. Those data recover the matrix; recovering all positions also requires a center coordinate. The precise finite inverse construction is in Moser 1975, § 3, pp. 475–477, PDF.

The open motion has a scattering interpretation, rather than periodic recurrence: the endpoint momentum obeys $\dot p_1=-a_1^2\lt0$ at every finite phase-space point. Moreover, every common spectral level contains the unbounded line obtained by translating all positions. Liouville integrability here therefore makes no claim that those levels are compact tori. Moser’s asymptotic analysis gives further scattering information; that is a stronger result than conservation alone (§ 2, pp. 471–474, PDF).

No equilibrium ensemble, thermodynamic limit, quantum spectrum, or general correlation-function formula is defined by the finite initial-value problem above. Each would require additional objects and assumptions.

Continue with the open Toda sequence

The open Toda learning sequence connects the Hamiltonian calculation, Lax pair, spectral invariants, and integrability checks. Use extract spectral invariants for matrix practice, check involution and independence for the two remaining tests, and the Library article for the complete finite-$N$ proof. The bibliography provides the shared source record.

References

• Moser, Jürgen. “Finitely many mass points on the line under the influence of an exponential potential—an integrable system.” In J. Moser (ed.), Dynamical Systems, Theory and Applications, Lecture Notes in Physics 38, Springer, 1975, pp. 467–497. DOI. Open PDF.