Phase space

In Chapter 8 we discussed the principles involved in the construction of canonical Lagrangians, action integrals and the derivation of the Euler–Lagrange equations of motion in continuous time (CT) classical mechanics (CM). Then we looked at the analogous situation in discrete time (DT) CM, where the notions of system function and action sum were introduced and used to work out DT equations of motion.

Given the central role of system functions in our approach, the natural question is the following: by what principles do we construct system functions?

The problem we face is that DT CM cannot be logically derived from CT CM, no more than quantum mechanics (QM) can be derived from CM. We have to take a leap and jump into the unknown. Depending on our choice of discretization, we may end up with a system function that does what we expect or one that leads to novel solutions to its equations of motion. This is not necessarily a bad thing.

How we view CT will influence how we discretize time. In Chapter 15, we shall discuss the geometrical approach of Marsden and others to DT CM. Generally, their methodology is aimed at finding variational integrators representing good approximations to CT, because that is their agenda. Our aim is fundamentally different: we want to explore the notion that DT may be a better model of time than CT, rather than an approximation to CT.