Systems of Two Degrees of Freedom

In this chapter, we focus on systems of 2DOFs. One prototypical example is the mechanical Hamiltonian \(\displaystyle H(y,x):=\frac {1}{2}\langle Ay,y\rangle +V(x):\ T^*\mathbb T^2\to \mathbb R \) where A is positive definite. Since systems of 1DOF are always integrable and can be completely understood using the method in Sect. 1.5.1 of Chap. 1 , systems of 2DOFs are the easiest nontrivial cases. Since the configuration space has dimension 2, the two dimensional topology imposes strong constraints to the dynamics, which makes the dynamics of such systems approachable in various aspects. The study of systems of 2DOFs was initiated by Poincaré, who introduced the restricted three-body problem that is a system with 2DOFs. A few months before his death in 1912, Poincaré formulated the following statement now known as Poincaré’s last geometric theorem.