Mechanics is a vast and difficult subject. It is virtually impossible to provide a thorough introduction in a couple of sections. Here, the purpose instead is to overview some of the main concepts and to provide some models that may be used with the planning algorithms in Chapter 14. The presentation in this section and in Section 13.4 should hopefully stimulate some further studies in mechanics (see the suggested literature at the end of the chapter). On the other hand, if you are only interested in using the differential models, then you can safely skip their derivations. Just keep in mind that all differential models produced in this section end with the form , which is ready to use in planning algorithms.
There are two important points to keep in mind while studying mechanics:
Several formulations of mechanics arrive at the same differential constraints, but from different mathematical reasoning. The remainder of this chapter overviews three schools of thought, each of which is more elegant and modern than the one before. The easiest to understand is Newton-Euler mechanics, which follows from Newton's famous laws of physics and is covered in this section. Lagrangian mechanics is covered in Section 13.4.1 and arrives at the differential constraints using very general principles of optimization on a space of functions (i.e., calculus of variations). Hamiltonian mechanics, covered in Section 13.4.4, defines a higher dimensional state space on which the differential constraints can once again be obtained by optimization.