Consider the Poisson bracket structure on $\mathbb{R}^{3}$ given by

$$\{x, y\}=z, \quad\{y, z\}=x, \quad\{z, x\}=y$$

and show that $\left\{f, \rho^{2}\right\}=0$, where $\rho^{2}=x^{2}+y^{2}+z^{2}$ and $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ is any polynomial function on $\mathbb{R}^{3}$.

Let $H=\left(A x^{2}+B y^{2}+C z^{2}\right) / 2$, where $A, B, C$ are positive constants. Find the explicit form of Hamilton's equations

$$\dot{\mathbf{r}}=\{\mathbf{r}, H\}, \quad \text { where } \quad \mathbf{r}=(x, y, z)$$

Find a condition on $A, B, C$ such that the oscillation described by

$$x=1+\alpha(t), \quad y=\beta(t), \quad z=\gamma(t)$$

is linearly unstable, where $\alpha(t), \beta(t), \gamma(t)$ are small.