A particle of mass $m$ has position vector $\mathbf{r}(t)$ in a frame of reference that rotates with angular velocity $\boldsymbol{\omega}(t)$. The particle moves under the gravitational influence of masses that are fixed in the rotating frame. Explain why the Lagrangian of the particle is of the form

$$L=\frac{1}{2} m(\dot{\mathbf{r}}+\boldsymbol{\omega} \times \mathbf{r})^{2}-V(\mathbf{r}) .$$

Show that Lagrange's equations of motion are equivalent to

$$m(\ddot{\mathbf{r}}+2 \boldsymbol{\omega} \times \dot{\mathbf{r}}+\dot{\boldsymbol{\omega}} \times \mathbf{r}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r}))=-\boldsymbol{\nabla} V$$

Identify the canonical momentum $\mathbf{p}$ conjugate to $\mathbf{r}$. Obtain the Hamiltonian $H(\mathbf{r}, \mathbf{p})$ and Hamilton's equations for this system.