Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 57 / 3

A Poisson manifold is a smooth manifold $P$ with a bilinear bracket on $C^{\infty }(P)$ that is antisymmetric, satisfies the Jacobi identity and obeys $\{f,gh\}=\{f,g\}h+g\{f,h\}$. Equivalently it has a bivector $\Pi$ defining $\{f,g\}=\Pi (df,dg)$ with vanishing Schouten bracket $[\Pi ,\Pi ]$; nondegeneracy is not required.

For a symplectic manifold, define the Hamiltonian vector field by ${\iota }_{X_f}\omega =df$ and put $\{f,g\}=\omega (X_f,X_g)$. Nondegeneracy gives a unique $X_f$, antisymmetry is inherited from $\omega$, and $d(gh)=gdh+hdg$ proves the Leibniz rule. Moreover ${\mathcal{L}}_{X_f}\omega =d{\iota }_{X_f}\omega +{\iota }_{X_f}d\omega =0$ by Cartan's magic formula. Since $X_{f}g=-\{f,g\}$,For completeness, evaluate $d\omega =0$ on $X_f,X_g,X_h$. Its three derivative terms sum to minus the cyclic sum of $\{f,\{g,h\}\}$, and its three bracket terms sum to minus the same cyclic sum, using the displayed identity. Hence that sum is zero. This proves the Jacobi identity for the Poisson bracket, not merely the Jacobi identity modulo constants. The inverse of a closed nondegenerate two-form therefore defines a Poisson manifold.

The Symplectic Darboux theorem says that near every point of a $2n$-dimensional symplectic manifold there are coordinates $(q^1,\dots ,q^n,p_1,\dots ,p_n)$ withThus every symplectic structure has the same local normal form, despite possible global topological differences.

For the graph embedding $\iota (q)=(q,p(q))$, pull back the symplectic form in Darboux coordinates:The coordinate two-forms are linearly independent. The graph has dimension $n$, so it is a Lagrangian submanifold exactly when this pullback vanishes, equivalently when its displayed coefficient matrix is symmetric.

Let ${\theta }_L={\sum }_i{p}_i(q)d{q}^i$. Then $d{\theta }_L=-{\iota }^{\ast }\omega$, so the same condition says that ${\theta }_L$ is a closed differential form. On a sufficiently small contractible coordinate neighborhood the Poincaré lemma supplies $S$ with ${\theta }_L=dS$. ConsequentlyOne explicit primitive on a star-shaped neighborhood of $q_0$ is $S(q)=S(q_0)+{\int }_0^1{p}_i(q_0+t(q-q_0))(q^i-q_0^i)dt$. Differentiate it and use ${\partial }_j{p}_i={\partial }_i{p}_j$ to recognize the derivative of $t{p}_j(q_0+t(q-q_0))$; the result is ${\partial }_{j}S=p_j(q)$. Conversely $p=dS$ has symmetric mixed derivatives and gives a Lagrangian graph. This construction requires the local projection to the $q$ coordinates to be nonsingular. A general Lagrangian submanifold need not be such a graph in a preassigned chart, and a closed one-form on a larger domain need not be globally exact.