Deterministic evolution is a hallmark of classical mechanics. Given a set of exact initial conditions, differential equations evolve the trajectories of particles into the future and can exactly predict the location of every particle at any instant in time. So what happens if our uncertainties in the initial position or velocity of a particle are tiny? Does that mean that our uncertainties about the subsequent motion of the particle are necessarily tiny as well? Or are there situations in which a very slight change in initial conditions leads to huge changes in the later motion? For example, can you really balance a pencil on its point? What has been learned in relatively recent years is that, in contrast to Laplace’s vision of a clock-like universe, deterministic systems are not necessarily predictable. What are the attributes of chaos and how can we quantify it? We begin our discussion with the notion of integrability, which ensures the absence of chaos.

In this chapter we describe motion caused by central forces, especially the orbits of planets, moons, and artificial satellites due to central gravitational forces. Historically, this is the most important testing ground of Newtonian mechanics. In fact, it is not clear how the science of mechanics would have developed if the earth had been covered with permanent clouds, obscuring the moon and planets from view. And Newton’s laws of motion with central gravitational forces are still very much in use today, such as in designing spacecraft trajectories to other planets. Our treatment here of motion in central gravitational forces is followed in the next chapter with a look at motion due to electromagnetic forces, which can also be central in special cases, but are commonly much more varied, partly because they involve both electric and magnetic forces. Throughout this chapter we focus on nonrelativistic regimes. The setting where large speeds are involved and gravitational forces are particularly large is the realm of general relativity, where Newtonian gravity fails to capture the correct physics. We explore such extreme scenarios in the capstone Chapter 10.

Some general features of special polynomials.

In this final chapter we introduce Hamilton--Jacobi theory along with its special insights into classical mechanics, and then go on to show how Erwin Schrödinger used the Hamilton--Jacobi equation to learn how to write his famous quantum-mechanical wave equation. In doing so, we will have introduced the reader to two of the ways classical mechanics served as a stepping stone to the world of quantum mechanics. Back in Chapter 5 we showed how Feynman’s sum-over-paths method is related to the principle of least action and the Lagrangian, and here we will show how Schrödinger used the Hamilton--Jacobi equations to invent wave mechanics. These two approaches, along with a third approach developed by Werner Heisenberg called “matrix mechanics,” turn out to be quantum-mechanical analogues of the classical mechanical theories of Newton, Lagrange, Hamilton, and Hamilton and Jacobi, in that they are describing the same thing in different ways, each with its own advantages and disadvantages.

As we saw in Chapter 1, Newton’s laws are valid only for observers at rest in an inertial frame of reference. But to an observer in a non-inertial frame, like an accelerating car or a rotating carnival ride, the same object will generally move in accelerated curved paths even when no forces act upon it. How then can we do mechanics from the vantage point of actual, non-inertial frames? In many tabletop situations, the effects of the non-inertial perspective are small and can be neglected. Yet even in these situations we often still need to quantify how small these effects are. Furthermore, learning how to study dynamics from the non-inertial vantage point turns out to be critical in understanding many other interesting phenomena, including the directions of large-scale ocean currents, the formation of weather patterns -- including hurricanes and tornados, life inside rotating space colonies or accelerating spacecraft, and rendezvousing with orbiting space stations. There is an infinity of ways a frame might accelerate relative to an inertial frame. Two stand out as particularly interesting and useful: linearly uniformly accelerating frames, and rotating frames.

Basics of fractal dimension.

Watching a shoe tumble erratically as it flies through mid-air may be entertaining, but -- to anyone without a background in rigid-body dynamics -- it can look quite troubling. There is no net torque acting on the shoe, yet the rotational motion looks and is rather complicated. However, with the powerful tools provided by the Lagrangian formalism, we are well equipped to tackle this subject, and go beyond it to more complicated examples of rotational motion. We start with a definition of a rigid body, and then proceed to introduce the Euler angles that can be used to describe the orientation of an object in three-dimensional space. With this scaffolding established, we can go on to describe torque-free dynamics, and then full rotational evolution with nonzero torque. For simplicity, throughout this chapter we restrict our discussion to nonrelativistic dynamics.

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.