Let $\mathbf{x}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$. Consider a Lagrangian

$$\mathcal{L}=\frac{1}{2} \dot{\mathbf{x}}^{2}+y \dot{x}$$

of a particle constrained to move on a sphere $|\mathbf{x}|=1 / c$ of radius $1 / c$. Use Lagrange multipliers to show that

$$\ddot{\mathbf{x}}+\ddot{y} \mathbf{i}-\dot{x} \mathbf{j}+c^{2}\left(|\dot{\mathbf{x}}|^{2}+y \dot{x}-x \dot{y}\right) \mathbf{x}=0$$

Now, consider the system $(*)$ with $c=0$, and find the particle trajectories.