As an extended example of Newtonian dynamics, let us examine the motion of a particle under the influence of a force whose magnitude is proportional to the displacement of the particle from an equilibrium position, and whose direction is toward that equilibrium position. If we choose the coordinate system so that the origin is at the point of equilibrium, the force may be written $${\rm{F = - kr,}}$$ (4.1) where k is a positive constant. This linear restoring force is often referred to as a Hooke’s law force, after Robert Hooke, who first observed that the deformation of a solid (such as the stretch of a spring or the bending of a bar) is resisted by a force whose magnitude is approximately proportional to the deformation, as long as the deformation is not so large as to exceed the elastic limit. It is one of the most important forces in the study of physics because it provides a good approximation to many real physical systems and because its equations of motion in both classical and quantum mechanics have analytic solutions.