Angle variable (${\theta }_i\in \mathbb{R}/(2\pi \mathbb{Z})$)

A periodic canonical coordinate paired with an action variable. In an integrable background Hamiltonian $H_0(\mathbf{J})$, the angles advance uniformly at ${\dot{\theta }}_i=\partial H_0/\partial J_i$, while the action variables remain constant. The periodicity permits an integer-indexed Fourier series on each regular orbital torus.