In this talk, we introduce Mañé's critical value for a Hamiltonian PDE, specifically the two-component Hunter-Saxton system. We then present the magnetic two-component Hunter-Saxton system (M2HS), which is a magnetic geodesic equation on an infinite-dimensional Lie group. We prove that this magnetic system is magnetically isomorphic to a magnetic system on an infinite-dimensional sphere. Surprisingly, each magnetic geodesic is tangent to the 3-sphere formed by the intersection of the ambient sphere with a complex plane. This geometric description of the M2HS is used to provide explicit criteria for blow-ups and to prove the existence of global weak solutions. In particular, we apply Mañé's critical value to derive an infinite-dimensional magnetic Hopf-Rinow theorem.