Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 33B (Integrable Systems)

Hamilton's equations are ${\dot{x}}_i=p_i$ and ${\dot{p}}_i=-{\partial }_{i}V$. A first integral $F$ satisfies $\{F,H\}=0$. See integrable Hamiltonian system: in three degrees of freedom, integrability requires three independent commuting first integrals.

The conclusion is Arnold-Liouville theorem: a compact connected regular common level is a three-torus, and nearby action-angle coordinates make the flow linear in the angles.

Rotational invariance gives $\{m_y,H\}=\{m_z,H\}=0$. Direct calculation givesTake $G=m_x^2+m_y^2+m_z^2$; angular-momentum brackets imply $\{G,m_x\}=0$, and central symmetry gives $\{G,H\}=0$. Hence $(H,G,m_x)$ are in involution.

At $\mathbf{x}=(1,0,0)$, $m_y=-p_z$, $m_z=p_y$. The differentials of $G$ and $m_x$ have independent components in the $p_y,p_z$ and transverse position directions when $p_y,p_z\ne 0$; $dH$ supplies an independent radial/momentum component. Thus they are locally independent and the system is Liouville integrable.