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One normalization of the first Hamiltonian structure uses the constant differential Poisson operator $J=-2{\partial }_x$. For functionals with variational derivatives $f,g$, its Poisson bracket is $\int fJgdx=-2\int f{g}^{\prime }dx$ under periodic or vanishing-boundary conventions. Thus the field bracket is $\{u(x),u(y)\}=-2{\partial }_x\delta (x-y)$. The factor and sign change with normalization; the constant first-order operator is the invariant structural feature.

A field $u(x,t)$ has infinitely many degrees of freedom: its initial state is a function rather than a finite list of coordinates. For a Hamiltonian evolution, the analogue of the finite-dimensional construction seeks an infinite sequence of independent conserved functionals $H_j[u]$ whose Poisson brackets vanish. Their Hamiltonian vector fields then define commuting time evolutions. For example, a Poisson operator $J$ produces the field equation $u_t=J\delta H/\delta u$, where $\delta H/\delta u$ is the variational derivative.

A Lax equation packages many conservation laws into one operator identity. If $L_t=[A,L]$, cyclicity of an appropriate trace givesFor a formal pseudodifferential operator the relevant trace is the Adler trace; suitable fractional powers of a monic differential operator produce further conserved functionals. A Lenard-Magri recursion can prove that such functionals are Poisson-commuting functions, rather than merely conserved quantities. A Lax equation by itself does not establish all the independence and completeness properties needed for integrability.

The Korteweg-De Vries equation illustrates the mechanism. In one time normalization take $L={\partial }^2+u$ and $A={\partial }^3+\frac{3}{2}u\partial +\frac{3}{4}{u}_x$. Direct composition givesIts spectral problem, together with the inverse scattering transform for decaying data or suitable spectral coordinates for periodic data, turns nonlinear evolution into simple evolution of spectral data. Solitary waves arise from discrete spectral data, while continuous data describe dispersive radiation.

Thus infinite-dimensional Hamiltonian integrability is more than an infinite list of formulas: one wants sufficiently complete commuting invariants and a reconstruction mechanism that solves the evolution. Boundary conditions and the function space are essential. A formal hierarchy need not converge, and an infinite family of conserved quantities does not automatically give a compact infinite-dimensional torus or an unrestricted analogue of the Arnold-Liouville theorem.

Use the canonical representatives of functionals of $u$: $P_f=f{\partial }^{-1}$, $P_g=g{\partial }^{-1}$. The formal pseudodifferential composition rule givesand the reversed product has $f,g$ exchanged. ThereforeMultiplying by $L={\partial }^2+u$ on the left, only the leading displayed commutator term can produce a residue. Differentiation of its coefficient produces powers at most ${\partial }^{-2}$, and multiplication by $u$ produces powers at most ${\partial }^{-3}$. Thusand, with the trace convention of part (i),There is no $u$ dependence in this bracket.

Moreover $\mathrm{res}(P_{f}L)=fu$, so $l_{P_f}=\int fudx$ and $\delta l_{P_f}/\delta u=f$. If $K(x,y)=\{u(x),u(y)\}$ is the distributional field bracket, it is characterized bySince $\int {\partial }_x\delta (x-y)g(y)dy=g^{\prime }(x)$, the answer isThe Dirac delta is periodic in the periodic setting, or the ordinary Dirac delta on the real line with admissible test functions. This is the first Hamiltonian structure of the KdV equation, with Poisson operator $-2{\partial }_x$. Its skewness follows by integration by parts; its coefficients are constant, so the functional Jacobi identity has no coefficient-variation terms.

A qualification is needed if the displayed trace formula is interpreted as a bracket on the reduced space for arbitrary operator representatives. Multiplication by a function $h$ gives $l_{m_h}({\partial }^2+u)=0$, yet direct expansion givesWith $h=g=\cos x$ on a period $[0,2\pi ]$, the latter is $\pi$. Hence the formula does not descend through every possible representative of a functional on this reduced affine space. The requested $f{\partial }^{-1},g{\partial }^{-1}$ representatives, used consistently for variational gradients, give the well-defined constant field bracket just computed; a general restriction from the ambient operator algebra needs an appropriate reduction.