While Newton was still a student at Cambridge University, and before he had discovered his laws of particle motion, the French mathematician Pierre de Fermat proposed a startlingly different explanation of motion. Fermat’s explanation was not for the motion of particles, however, but for light rays. In this chapter we explore Fermat’s approach, and then go on to introduce techniques in variational calculus used to implement this approach, and to solve a number of interesting problems. We then show how Einstein’s special relativity and the principle of equivalence help us demonstrate how variational calculus can be used to understand the motion of particles. All this is to set the stage for applying variational techniques to general mechanics problems in the following chapter.

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.

An enormous advantage of using Lagrangian methods in mechanics is the simplifications that can occur when a system is constrained or if there are symmetries of some kind in the environment of the system. Constraints can be used to reduce the number of generalized coordinates so that solutions become more practicable. In this chapter we will illustrate this fact using the example of contact forces, and demonstrate the use of Lagrange multipliers to learn about the contact forces themselves. Constraints are also typically associated with the breaking of symmetries. Lagrangian mechanics allows us to efficiently explore the relationship between symmetries in a physical situation and dynamical quantities that are conserved. These properties are nicely summarized in a theorem by the German mathematician Emmy Noether (1882--1935), and provide us with deep insight into the physics -- in addition to helping us make important technical simplifications while solving problems. We first discuss constraints and contact forces, and then symmetries and conservation laws.

In this first capstone chapter we aim to set classical mechanics in context. Classical mechanics played a key role in developing modern physics in the first place, and in turn modern physics has given us deeper insights into the meaning and validity of classical mechanics. Classical mechanics, even extended into the realm of special relativity, has its limitations. It arises as a special case of the vastly more comprehensive theory of quantum mechanics. Where does classical mechanics fall short, and why is it limited? The key to understanding this is Hamilton’s principle. We begin with the behavior of waves in classical physics, and then show results of some critical experiments that upset traditional notions of light as waves and atoms as particles. We proceed to give a brief review of Richard Feynman’s sum-over-paths formulation of quantum mechanics, which describes the actual behavior of light and atoms, and then show that Hamilton’s principle emerges naturally in a certain limiting case.

A primer on dimensional analysis, and how to use units to diagnose and solve physics problems.

While gravity was the first of the fundamental forces to be quantified and at least partially understood -- beginning all the way back in the seventeenth century -- it took an additional 200 years for physicists to unravel the secrets of a second fundamental force, the electromagnetic force. Ironically, it is the electromagnetic force that is by far the stronger of the two, and at least as prevalent in our daily lives. The fact that atoms and molecules stick together to form the matter we are made of, the contact forces we feel when we touch objects around us, and virtually all modern technological advances of the twentieth century, all these rely on the electromagnetic force. In this chapter, we introduce the subject within the Lagrangian formalism and demonstrate some familiar as well as unfamiliar aspects of this fascinating fundamental force of Nature.

We will show in this chapter that there is a different framework -- known as the Hamiltonian formalism -- that describes the same fundamental physics as Newtonian mechanics or the Lagrange method. However, just as we found with the Lagrange method, the Hamiltonian description of mechanics gives us a new perspective that opens up a deeper understanding of mechanics, is sometimes advantageous in problem solving, and has also played a crucial role in the emergence of quantum mechanics. Therefore, our goal in this chapter is to develop the Hamiltonian formalism, to explore examples that elucidate the advantages and disadvantages of this new approach, and to develop the powerful related formalisms of canonical transformations, Poisson brackets, and Liouville’s theorem.

A review of Taylor and binomial series.

In this chapter we extend our review of mechanics to include Einstein’s special theory of relativity. We will see that our previous Newtonian framework is a useful description of the mechanical world only when speeds are much less than that of light. We also use this chapter to introduce index notation and general technical tools that will help us throughout the rest of the book. Then, in the following chapter, we will show how relativity provides insights for an entirely different formulation of mechanics -- the so-called variational principle.

A review of Cartesian, polar, and spherical coordinate systems.

In the preceding chapter on rigid-body motion we took a step beyond single-particle mechanics to explore the behavior of a more complex system containing many particles bonded rigidly together. Now we will explore additional sets of many-particle systems in which the individual particles are connected by linear, Hooke’s-law springs. These have some interest in themselves, but more generally they serve as a model for a large number of coupled systems that oscillate harmonically when disturbed from their natural state of equilibrium, such as elastic solids, electric circuits, and multi-atom molecules. We will begin with the oscillations of a few coupled masses and end with the behavior of a continuum of masses described by a linear mass density. The mathematical techniques required to analyze such coupled oscillators are used throughout physics, including linear algebra and matrices, normal modes, eigenvalues and eigenvectors, and Fourier series and Fourier transforms.

A review of Green's theorem, Stokes's theorem, and the divergence theorem.