Elementary Newtonian Mechanics

Abstract

This chapter deals with the kinematics and the dynamics of a finite number of mass points that are subject to internal, and possibly external, forces, but whose motions are not further constrained by additional conditions on the coordinates. Constraints such as requiring some mass points to follow given curves in space, to keep their relative distance fixed, or the like, are introduced in Chap. 2. Unconstrained mechanical systems can be studied directly by means of Newton’s equations and do not require the introduction of new, generalized coordinates that incorporate the constraints and are dynamically independent. This is what is meant by “elementary” in the heading of this chapter — though some of its content is not elementary at all. In particular, at an early stage, we shall discover an intimate relationship between invariance properties under coordinate transformations and conservation laws of the theory, which will turn out to be a basic, constructive element for all of mechanics and which, for that matter, will be felt like a cantus firmus ¹ throughout the whole of theoretical physics. The first, somewhat deeper analysis of these relations already leads one to consider the nature of the spatial and temporal manifolds that carry mechanical motions, thereby entering a discussion that is of central importance in present-day physics at both the smallest and the largest dimensions.