14 Hamiltonian Mechanics with Time as a Coordinate

Abstract This chapter uses the traditional Hamilton equations as the basis for an extended Hamiltonian theory in which time is treated as a coordinate. The traditional Hamilton equations, including the Hamiltonian form of the generalised energy theorem, will be combined into one set of extended Hamilton equations. The extended Hamilton theory developed in the chapter is of fundamental importance for the more advanced topics in mechanics. It is used to write the relativistically covariant Hamiltonian, which is then used to derive the Klein-Gordon equation of relativistic quantum mechanics. The extended Hamilton equations also provide the basis for the discussion of canonical transformations. The objective of extended Hamiltonian theory is to write the equations of motion in terms of an extended set of phase-space variables.