The formalism of the previous chapters used a rather arbitrary coordinate system. In the Hamiltonian formalism the coordinates are chosen in quite a different way to reflect more deeply the dynamical properties of the system. In this chapter we derive the Hamiltonian equations of motion. The results of this chapter are later needed mainly to derive some standard results that are the starting point for further studies. The same results can also be obtained in a more traditionalway, but the Hamiltonian approach makes the calculations considerably shorter and more straightforward.

Hamiltonian mechanics and its applications to mechanics in general are explained more extensively in many books on theoretical mechanics. This chapter is based mainly on Goldstein (1950).

Generalised coordinates

We have this far used ordinary Euclidean rectangular coordinates to describe positions and velocities of the objects. They are purely geometric quantities that describe the system in a very simple and understandable way. However, they do not tell us anything about the dynamic properties of the system nor do they utilise any specific features of the system.We now want to find a different kind of description in terms of quantities which do not have these problems.

Motions of bodies may be constrained in various ways. For example, two points of a solid body must always be at the same distance from each other.