Spinning Top
Simple example from The Theoretical Minimum I recently re-read The Theoretical Minimum - Classical mechanics , an excellent book. The authors give an example of the use of the Hamiltonian formulation of mechanics to illustrate how much easier it makes things (compared to the Newtonian formulation) and that inspired me to have a go myself. The example they give in the book is of a negligible mass charged sphere spinning in a magnetic field pointing in the z direction. In this case the Hamiltonian $H = T + V$ is proportional to $L_z$ where $\mathbf{L}$ is the angular momentum $L_x \hat{i} + L_y \hat{j} + L_z \hat{k}$. Let's say $H=\omega^2L_z$. The book explains that for any function $G(q,p)$ of the generalized coordinates $q_i$ and the conjugate momenta $p_i$ that $$ \dot{G} = \{G,H\} $$ i.e the time derivative of $G$ is the poisson bracket of $G$ and the Hamiltonian. And indeed this does make the solution simple because the components of angular momentum satisfy $$ \begin{align}