In Hamiltonian mechanics, a system with $n$ generalized coordinates is represented by a point $$(q_1,\ldots,q_n,p_1,\ldots,p_n)$$ in a $2n$-dimensional phase space. Its evolution is governed by Hamilton's equations: $$\dot q_i=\frac{\partial H}{\partial p_i},\qquad \dot p_i=-\frac{\partial H}{\partial q_i}.$$

For the harmonic oscillator $$H(q,p)=\frac{p^2}{2m}+\frac12m\omega^2q^2,$$ Hamilton's equations give $$\dot q=\frac{p}{m},\qquad \dot p=-m\omega^2q.$$ Combining them yields $\ddot q+\omega^2q=0$, the familiar oscillator equation.

In phase space, constant-energy oscillator trajectories satisfy $$\frac{p^2}{2m}+\frac12m\omega^2q^2=E,$$ which are ellipses. Time evolution moves the state around one such ellipse rather than tracing position alone.

The phase-space viewpoint makes the complete instantaneous state explicit: specifying $q$ without $p$ is generally insufficient to predict future motion. It also reveals structures hidden in second-order Newtonian equations, such as conserved volumes, invariant manifolds and ensemble distributions.

If the Hamiltonian has no explicit time dependence, then $$\frac{dH}{dt}=0$$ along the trajectory. Hamilton's equations thus provide both an equivalent formulation of classical mechanics and the natural language for statistical mechanics, canonical transformations and the transition to quantum theory.