As we have seen in the preceding chapter, every solution curve of an autonomous Hamiltonian system lies inside an energy hypersurface, i.e. a level set of the Hamiltonian function (Propositions 9.17 and 9.24). In order to understand the dynamics of a given system, such as the question of whether there are any periodic solutions, it is helpful and often essential to understand the topology of these level sets

In this chapter I shall describe the so-called geodesic flow on the 2-sphere as a Hamiltonian system. The connection with the Kepler problem is provided by the discussion in Section 8.3; this connection can also be established via the methods of Chapter 9. Another reason why this model example from Riemannian geometry is of special interest in the context of celestial mechanics lies in the fact that it gives rise to energy hypersurfaces with the same topology as certain energy levels in (PCR3B).

Notably, this discussion will lead us to consider the three-dimensional real projective space ℝP3, and I shall present various equivalent descriptions of that space. A homeomorphism between ℝP3 and the special orthogonal group SO(3) is constructed with the help of Hamilton's quaternions; an introduction to the quaternion algebra is contained in Section 10.4.

The geodesic flow on the 2-sphere

As discussed in Section 8.3, the geodesics on the 2-sphere S2 are the great circles, parametrised proportionally to arc length. For simplicity, we shall now always assume unit speed parametrisation. The set of all unit speed geodesics can be interpreted as the collection of flow lines of a dynamical system on the so-called unit tangent bundle of S2: given an initial condition comprising a starting point x0 ∈ S2 and an initial velocity vector y0 of length 1 in the tangent space Tx0S2 (see Definition 8.21 and Exercise 8.9), […]