Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 8 / 1

A Poisson structure is a real bilinear operation on smooth functions on $M$,which is antisymmetric, obeys the Jacobi identity, and is a derivation in each argument. Explicitly,Antisymmetry supplies the corresponding Leibniz rule in the first argument. Equivalently it is a smooth Poisson bivector $\Pi$ with $\{f,g\}=\Pi (df,dg)$ and $[\Pi ,\Pi ]=0$ for the Schouten-Nijenhuis bracket. In local coordinates,The last equation is the coordinate Jacobi identity. Nondegeneracy is not part of the definition: a Poisson structure may have Casimir functions and symplectic leaves of different dimensions.

A Hamiltonian system consists of such a Poisson manifold, a Hamiltonian $H\in C^{\infty }(M)$, and the flow of its Hamiltonian vector field. Choose the conventionFor an observable with no explicit time dependence, $df/dt=\{f,H\}$. In particular $H$ is conserved because $\{H,H\}=0$. For the canonical Poisson bracket, this convention gives ${\dot{q}}_i=\partial H/\partial p_i$ and ${\dot{p}}_i=-\partial H/\partial q_i$.

For the Arnold-Liouville theorem, work on a symplectic manifold of dimension $2d$. Let $F_1=H,F_2,\dots ,F_d$ be smooth functions with pairwise vanishing Poisson brackets, and supposeon a connected component $N$ of a common level. If $N$ is compact, then $N$ is a $d$-dimensional torus. Its commuting Hamiltonian vector fields generate translations, and the Hamiltonian motion on it is linear in angular coordinates. Compactness ensures completeness of these restricted vector fields.

Moreover a neighbourhood of $N$ admits action-angle variables $(I_1,\dots ,I_d,{\theta }_1,\dots ,{\theta }_d)$, with each angle defined modulo $2\pi$, for whichOne compatible symplectic form convention is $\omega ={\sum }_{i}d{\theta }_i\wedge d{I}_i$. The equations becomeThe resulting motion is periodic when the frequencies are commensurable and otherwise quasiperiodic on a subtorus. This is Liouville integrability with $d$ independent first integrals in involution. Compactness and regularity cannot be discarded from the torus conclusion. On a general Poisson manifold, apply this theorem on a regular symplectic leaf of dimension $2d$; the ambient dimension alone does not specify the required number of integrals.

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.