Define a Poisson structure on an open set $U \subset \mathbb{R}^{n}$ in terms of an anti-symmetric matrix $\omega^{a b}: U \longrightarrow \mathbb{R}$, where $a, b=1, \cdots, n$. By considering the Poisson brackets of the coordinate functions $x^{a}$ show that

$$\sum_{d=1}^{n}\left(\omega^{d c} \frac{\partial \omega^{a b}}{\partial x^{d}}+\omega^{d b} \frac{\partial \omega^{c a}}{\partial x^{d}}+\omega^{d a} \frac{\partial \omega^{b c}}{\partial x^{d}}\right)=0$$

Now set $n=3$ and consider $\omega^{a b}=\sum_{c=1}^{3} \varepsilon^{a b c} x^{c}$, where $\varepsilon^{a b c}$ is the totally antisymmetric symbol on $\mathbb{R}^{3}$ with $\varepsilon^{123}=1$. Find a non-constant function $f: \mathbb{R}^{3} \longrightarrow \mathbb{R}$ such that

$$\left\{f, x^{a}\right\}=0, \quad a=1,2,3$$

Consider the Hamiltonian

$$H\left(x^{1}, x^{2}, x^{3}\right)=\frac{1}{2} \sum_{a, b=1}^{3} M^{a b} x^{a} x^{b}$$

where $M^{a b}$ is a constant symmetric matrix and show that the Hamilton equations of motion with $\omega^{a b}=\sum_{c=1}^{3} \varepsilon^{a b c} x^{c}$ are of the form

$$\dot{x}^{a}=\sum_{b, c=1}^{3} Q^{a b c} x^{b} x^{c},$$

where the constants $Q^{a b c}$ should be determined in terms of $M^{a b}$.