Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 32E / Solution

A Poisson structure is a smooth antisymmetric matrix field ${\omega }^{ab}$ defining a bracketthat obeys the Jacobi identity. Bilinearity, antisymmetry and the product rule follow from this formula; Jacobi is the remaining condition. Applying Jacobi to the coordinate functions givesThese are also sufficient: expanding Jacobi for arbitrary functions cancels all second-derivative terms by antisymmetry, leaving these coefficients multiplying their first derivatives.

For ${\omega }^{ab}={ϵ}^{abc}{x}^c$, the nonconstant function $f={\sum }_a(x^a{)}^2$ is a Casimir function of a Poisson manifold, becauseby symmetry of $x^b{x}^c$. This is the usual three-dimensional Lie-Poisson bracket.

Since the matrix $M$ is symmetric, ${\partial }_{d}H={\sum }_b{M}^{db}{x}^b$. The Hamilton equations are ${\dot{x}}^a=\{x^a,H\}$, and henceOne valid choice of coefficients is $Q^{abc}={\sum }_d{ϵ}^{adc}{M}^{db}$. If symmetric coefficients in $b,c$ are preferred, the same equations useIn vector notation the evolution is $(Mx)\times x$, consistent with the bracket convention and with preservation of $|x{|}^2$.