The motion of a particle on the hyperbolic plane under the influence of a central potential \(U\) is an interesting and old problem. By equipping \(\mathbb R^3\) with the Lorenz inner product \(\langle, \rangle_L\), the authors obtain on \(\mathbb R^{2, 1}\) a pseudo-Riemannian metric and pull-back the canonical symplectic 2-form on \(T^* \mathbb R^{2,1}\).
They study the motion of a particle on:
\[ \mathbb{H}^2= \{ x \in \mathbb{R}^{2,1}: \langle x,x \rangle_L+1=0, \, x_3> 0\} \]
with the Hamiltonian
\[ H(x, y)= \frac{1}{2}\langle y, y\rangle_L + U(x). \]
By the principal axis theorem for the Lorenz group, the authors reduce the problem and discuss the situation for three subgroups: \(G_e \) (elliptic rotations), \(G_h\) (hyperbolic rotations), and \( G_p\) (parabolic rotations), which leave respectively the \(x_3\)-axis, \(x_1\)-axis and \(x_1=0, x_2= x_3\) invariant.