Lax & zero-curvature equations

A Lax representation encodes evolution through compatibility of linear equations. For finite matrices it implies isospectral motion. For a field-dependent connection, the corresponding local condition is zero curvature; global conserved quantities also depend on boundaries. Neither representation alone proves that a Hamiltonian system has enough independent commuting integrals.

Required background. Matrix multiplication, differentiation and cyclicity of the trace are used below. The Toda Lax lesson develops the finite-matrix calculation in a concrete system.

Finite matrix Lax equations

Let $L(t)$ be a differentiable $n\times n$ matrix and $B(t)$ continuous on a time interval. Use

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

For any positive integer $k$, the product rule and cyclicity give

$$\begin{aligned} \frac{d}{dt}\operatorname{tr}L^k &=k\operatorname{tr}(L^{k-1}\dot L)\\ &=k\operatorname{tr}(L^{k-1}BL-L^kB)=0. \end{aligned}$$

This calculation requires a finite matrix trace. It does not license cyclic permutation of arbitrary unbounded differential operators.

There is also a direct similarity argument. Solve $\dot S=BS$ with $S(t_0)=\mathbf1$. The fundamental solution is invertible, and differentiation shows

$$\frac{d}{dt}(S^{-1}LS)=S^{-1}(\dot L-[B,L])S=0.$$

Consequently $L(t)=S(t)L(t_0)S(t)^{-1}$. Its characteristic polynomial and eigenvalues, including algebraic multiplicities, are constant. No assumption of distinct eigenvalues or diagonalizability is needed. If $B$ is real skew-symmetric, $\frac{d}{dt}(S^{\mathsf T}S)=0$ and $S$ is orthogonal. For this convention and the trace argument, compare Torrielli 2016, §3.1, p. 9, equations (3.1)–(3.3), PDF.

Auxiliary eigenvectors and gauge changes

The compatible equations are $L\psi=\lambda\psi$ and $\dot\psi=B\psi$, with $\lambda$ constant. For a differentiable invertible matrix $g(t)$, change the auxiliary vector to $\widetilde\psi=g\psi$. Then

$$\widetilde L=gLg^{-1},\qquad \widetilde B=\dot g\,g^{-1}+gBg^{-1}.$$

The derivative term is essential when $g$ depends on time, including dependence through evolving phase-space coordinates. Substituting these expressions gives $\dot{\widetilde L}=[\widetilde B,\widetilde L]$. Conjugating $B$ alone would omit the derivative of the moving basis.

Replacing $B$ by $B+C$ also leaves the equation unchanged if $[C,L]=0$. Replacing $L$ by $L+c\mathbf1$ preserves it only when $c$ is constant; otherwise $\dot c\,\mathbf1$ remains in the derivative.

The open Toda normalization

For the dimensionless finite open Toda model, set $a_i=e^{(q_i-q_{i+1})/2}$, $a_0=a_N=0$. The nonzero entries are

$$\begin{aligned} L_{ii}&=p_i,&L_{i,i+1}=L_{i+1,i}&=a_i,\\ B_{i,i+1}&=-\frac{a_i}{2},& B_{i+1,i}&=\frac{a_i}{2}. \end{aligned}$$

The commutator gives

$$\dot p_i=a_{i-1}^2-a_i^2,\qquad \dot a_i=\frac{a_i}{2}(p_i-p_{i+1}).$$

Here $I_k=\operatorname{tr}(L^k)/k$ has $I_1=\sum_i p_i$ and $I_2=H$. To translate Moser 1975, §2, p. 470, equation (2.1), and pp. 472–473, PDF, use $a_i^{\mathrm M}=a_i/2$, $b_i^{\mathrm M}=-p_i/2$. With $D=\operatorname{diag}(1,-1,1,\ldots)$, his matrices satisfy $L^{\mathrm M}=-DLD/2$ and $B^{\mathrm M}=DBD$. Time is unchanged. A sign or scale taken from another convention must be translated throughout the pair and its invariants.

The Library proof establishes the additional involution and independence claims on the original canonical phase space.

Zero curvature as a compatibility condition

Let $U(x,t;\lambda)$ and $V(x,t;\lambda)$ be smooth $n\times n$ matrices on a simply connected coordinate patch, with fixed spectral parameter $\lambda$. Adopt the auxiliary system

$$\partial_x\psi=U\psi,\qquad \partial_t\psi=V\psi.$$

Differentiating in the two orders gives

$$(\partial_t\partial_x-\partial_x\partial_t)\psi =(U_t-V_x+UV-VU)\psi.$$

Compatibility for an invertible fundamental matrix of solutions therefore requires

$$U_t-V_x+[U,V]=0.$$

Conversely, this smooth flatness condition gives local compatible fundamental solutions from an initial frame. Global single-valuedness on a domain with nontrivial loops requires separate consideration. This is the convention of Torrielli 2016, §3.2, p. 12, equations (3.14)–(3.15), PDF, with his spatial and temporal matrices denoted here by $U,V$.

Under $\widetilde\psi=g(x,t;\lambda)\psi$,

$$\begin{aligned} \widetilde U&=g_xg^{-1}+gUg^{-1},\\ \widetilde V&=g_tg^{-1}+gVg^{-1}. \end{aligned}$$

The curvature transforms to $g(U_t-V_x+[U,V])g^{-1}$. Thus vanishing curvature is invariant under a smooth invertible change of frame. Stating instead $(\partial_x+U)\psi=0$ changes the signs; derive the condition from the auxiliary equations rather than memorizing a detached formula.

Transport, monodromy and boundaries

Define the spatial transport matrix by

$$\partial_b T(b,a;t,\lambda)=U(b,t;\lambda)T(b,a;t,\lambda), \qquad T(a,a;t,\lambda)=\mathbf1.$$

For fixed endpoints and a flat connection, its time derivative is

$$\partial_tT(b,a)=V(b)T(b,a)-T(b,a)V(a).$$

One way to verify this is to differentiate $T_t-V(b)T+TV(a)$ with respect to $b$. Zero curvature makes it satisfy the homogeneous transport equation, with zero initial value at $b=a$, so it vanishes. The endpoint identity is Torrielli 2016, §3.2, pp. 12–13, equations (3.17)–(3.18), PDF.

On a circle of length $\ell$, a periodic frame with $V(a+\ell)=V(a)$ makes the monodromy $T(a+\ell,a)$ satisfy a commutator equation. Its spectrum and traces are then conserved. On a finite interval with unequal endpoint matrices, the displayed derivative is generally not a commutator. Decay, reflection or other boundary conditions need their own analysis; local zero curvature does not remove these terms.

Under the gauge change above,

$$\widetilde T(b,a)=g(b)\,T(b,a)\,g(a)^{-1}.$$

For periodic $g$, monodromy changes by similarity. Unequal endpoint gauges do not generally preserve its spectrum.

What a Lax representation establishes

Given structure	Consequence under the stated assumptions	Additional question
Finite $\dot L=[B,L]$	Constant characteristic polynomial	Are the induced integrals independent and Poisson-commuting?
Local zero curvature	Compatible auxiliary equations	Which boundary conditions give conserved transport data?
Conserved spectral family	Candidate conserved quantities	How many are independent on the physical phase space?
Numerical spectral drift near zero	A tested diagnostic of a calculation	Does the trajectory satisfy the original initial-value problem?

For Liouville integrability on a $2N$-dimensional symplectic phase space, establish $N$ commuting first integrals that are functionally independent on the specified regular set, with $H$ among them or a function of them. Compact invariant tori require further hypotheses. The Toda integrals lesson makes these distinctions explicit.

Check your understanding

Let $U(x,t)=t\mathbf1$ and $V(x,t)=x\mathbf1$ on an interval. Verify flatness and decide whether $T(b,a)$ has time-independent eigenvalues.

Solution

$U_t=V_x=\mathbf1$ and the commutator vanishes, so the connection is flat. However, $T(b,a)=e^{t(b-a)}\mathbf1$, whose eigenvalues change with time when $b\ne a$. The endpoint formula gives $T_t=(b-a)T$. This is a direct example of the boundary term obstructing isospectral transport despite local compatibility.

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.
• Torrielli, Alessandro. Lectures on Classical Integrability. arXiv:1606.02946v1 [hep-th], 2016. Version record. Open PDF. The locators above refer to this version.