ABSTRACT
One can predict the tendencies of matter by computing the minima or maxima of certain mathematical functions. These are called extremum (or variational) principles. This chapter shows that balls rolling downhill can be predicted by the minimization of energy. It describes the various types of extrema, called states of equilibrium: stable, unstable, neutral, and metastable states. The chapter illustrates how gases exerting pressure, the mixing and diffusion of molecules, and rubber elasticity can be explained by a maximization principle. A quantity such as the position x of the ball in the valley is called a degree of freedom of the system, because the system is free to change that quantity. The alternative to a degree of freedom is a constraint. The chapter introduces a different extremum principle, one that predicts the distributions of outcomes in statistical systems, such as coin flips or die rolls. This will lead to the concept of entropy and the Second Law of Thermodynamics.
