When it comes to solving physics problems, there are two kinds of inputs that are typically drawn upon: First, there are the constraints placed on a problem sometimes referred to as boundary conditions. This includes factors imposed upon the situation by the environment that might not seem, at first glance, to be very deep. Consider a ball, for instance, accelerating down an inclined plane. Without knowing anything about physics, we can say that the ball is going to be somewhere on the inclined plane. These are aspects of the physical phenomena dictated by the environment which can be viewed as constraints. Constraints loom rather large in classical mechanics when one wishes to describe the motion. This is also known as “kinematics.” Second, there are physical laws, such as those formulated by Newton or Einstein, which seem to be much more fundamental. “Dynamics” addresses forces influencing the motions of objects and systems as a whole, where physical laws play a bigger role.
Part of the discussion in this section will focus on what might appear to be the duller side of physics, constraints, but as will hopefully become clear in the course of this chapter that is not necessarily the case. Some of the ideas we are going to take up here can manifest themselves in very deep ways. And we may find, at a very basic level, that the distinction between laws and constraints disappears and that many of the things we attributed to principles could actually emerge from constraints.
Mathematically, we will view topology as an analog of these general physical constraints. Topology describes the global, qualitative aspects of a space, its general features, as opposed to geometry, which delves into the details of a space, involving distances, precise shapes and so forth. Continuity, which is a basic idea in the context of topology, has a natural link to the fact that the laws of physics are continuous: If you change things a bit, the outcome will typically not change drastically. 1
