ABSTRACT
Associated with each one-parameter group of symmetries of a Hamiltonian system is a conserved quantity. For a group of symmetries we get thereby a vector-valued conserved quantity called the momentum. We shall discuss the properties of the momentum and how to construct it in Sect. 4.2, after summarizing the necessary topics from Lie group theory in Sect. 4.1. When symmetries are present the phase space can be reduced; that is, a number of variables eliminated. This topic is the subject of Sect. 4.3. Mechanical systems on Lie groups and the rigid body are discussed in Sect. 4.4. Smale's topological program for a mechanical system with symmetry is presented in Sect. 4.5 and this is applied to the rigid body problem in Sect. 4.6. A number of results presented in this chapter are new.
