Chapter 2 reminds the basics of Newtonian and theoretical mechanics, and introduces nonconservative and entropic forces.

The meaning of time reversal on state space is determined by a representation of time translations.

Newton’s remarkable achievements in planetary theory and dynamics were followed by a century of equally remarkable advances in the generalized science of forces acting on bodies in motion. Far from merelyformalizing the Newtonian framework, these advances aimed at solving deep problems left in Newton’s wake. First, Newtonian theory focused on centripetal forces, which are not characteristic of motion under arbitrary constraints or bodily deformation due to pressure and stress. Second, Newton’s attempts at fluid mechanics made clear that media could not be adequately analyzed using strategies developed for point-particles. Both considerations suggested that different forms of body required different analytical and conceptual tools. They required the formulation of generalized principles flexible enough to treat heterogeneous material configurations, but stringent enough to preserve the unity of mechanics as a study of matter and motion. Ultimately, Newton’s successors established a new discipline, analytical mechanics. Its primary object of investigation was the functional representation of the invariant relations behind dynamic phenomena, not the geometric representation of trajectories. The philosophical suppositions behind the new mechanics constituted a new mechanical philosophy, but one that hearkened back to, and completed the project of, the mechanical philosophy of the mid-seventeenth century.

This chapter discusses the geometry of space and the notion of time assumed in Newtonian mechanics. This discussion will also serve to review aspects of mechanics and special relativity that will be important for later developments. Newtonian mechanics assumes a geometry for space and a particular idea for time. The laws of Newtonian mechanics take their standard and simplest forms in inertial frames. Using the laws of mechanics, an observer in an inertial frame can construct a clock that measures the time. Coordinate transformations can make the connection between different inertial frames. Newtonian mechanics assumes there is a single notion of time for all inertial observers. We explore Newtonian gravity and the Principle of Relativity: that identical experiments carried out in different inertial frames give identical results.

We begin our journey of discovery by reviewing the well-known laws of Newtonian mechanics. We set the stage by introducing inertial frames of reference and the Galilean transformation that translates between them, and then present Newton’s celebrated three laws of motion for both single particles and systems of particles. We review the three conservation laws of momentum, angular momentum, and energy, and illustrate how they can be used to provide insight and greatly simplify problem solving. We end by discussing the fundamental forces of Nature, and which of them are encountered in classical mechanics. All this is a preview to a relativistic treatment of mechanics in the following chapter.