INTRODUCTION AND OVERVIEW OF CHAPTER 11

Three hundred years ago, Isaac Newton discovered that the motion of dynamical systems with N degrees of freedom could be described by N second-order differential equations. These differential equations provide us with a mathematical road map, giving directions about the motion of a system for each successive time interval. Since the system's motion in each interval of time is connected smoothly to the motion in the preceding time interval, Newton was convinced that the equations of this motion would have solutions that change smoothly as the initial conditions are varied, that is, would be analytic functions of the time and the initial conditions. Generations of physicists shared Newton's belief that all mechanical problems would have analytic solutions. By the 1830s, Lagrange and Hamilton had improved the analytical techniques for finding the equations most appropriate to a particular physical system. If analytic solutions to a particular problem could not be found, it was thought that only a cleverer, more sophisticated approach was needed. The concept of the “clockwork universe” was accepted after Newton. Such a universe is completely determined by the initial conditions to move along smooth paths for the rest of time, just as the planets seemed to move in perpetual ellipses around the Sun. Laplace was a particular champion of this universal determinism, a view his contemporaries did not hesitate to extend to everything, not only to the problems of mechanics. […]

OVERVIEW OF CHAPTER 10

Reflecting on the past nine chapters, you may realize that we have only solved a few problems in an analytic form. What about the many other problems that one is sure to encounter in physics? Many of the most interesting problems do not have exact analytic solutions. This chapter will introduce a few methods for dealing with problems of this type. Often we start with a problem we already know how to solve, like the Kepler problem or harmonic oscillator. Then we add on a part, known as a “perturbation,” which approximates the more complex problem. To get a more accurate solution, we add on more terms.

If a system in motion is perturbed slightly, does it diverge rapidly from the unperturbed motion or does it oscillate around the unperturbed orbit? In the former case, we say the system is “dynamically unstable”; in the latter case there is “dynamical stability.” If we assume the motion, at least initially, is close to the unperturbed motion, we can subtract the perturbed equations of motion from the unperturbed ones, keeping only terms linear in the difference between the perturbed and unperturbed motion. This is known as “linearizing” the equations of motion. There are two ways to introduce a perturbation of the motion. We can either disturb the initial conditions (known as a one-time perturbation) or else add a small change in the Lagrangian, usually in the potential energy. […]

OVERVIEW OF CHAPTER 6

Canonical transformations are transformations from one set of canonically conjugate variables q, p to another conjugate set Q, P. A transformation is said to be canonical if, after the transformation, Hamilton's equations are still the correct dynamical equations for the time development of the new variables. The new Hamiltonian may look quite different from the old one. It may prove easier to solve the EOM in terms of the new variables Q, P. The concept of a generating function is introduced, which gives an “automatic” method for producing canonical transformations. There are four types of generating functions for canonical transformations. It will be explained how these different generating functions are connected by Legendre transformations.

Poisson brackets will be introduced, which are invariant under canonical transformations. If Hamilton's dynamics is formulated in terms of Poisson brackets, we have a coordinate-free way to express the equations of motion. The close resemblance of Poisson brackets used in classical mechanics to commutators of operators in quantum mechanics is not an accident, since Poisson brackets played a fundamental role in the invention of quantum mechanics.

We proceed from the general notion of a generating function to the special generating function S. which produces a canonical transformation leading to the Hamilton–Jacobi equation. The Hamilton–Jacobi equation leads to a geometric picture of dynamics relating the dynamics to wave motion. […]

OVERVIEW OF CHAPTER 4

One of the greatest advances in science was Newton's discovery that the force of gravity is a universal force that not only causes terrestrial objects to fall but also guides the Moon around the Earth and the planets around the Sun. It was not previously understood that the Moon and planets – indeed the universe – obey the same physical laws as terrestial objects. We take this for granted today, but it was a revolution in human understanding, one from which there has been no turning back.

Our goal in this chapter is to show how this problem – the Kepler problem of planetary orbits – can be solved using the powerful analytical techniques of Lagrangian mechanics. We begin by considering the general solution for motion in a one-dimensional potential V(q). Next, we consider a six-dimensional system of two isolated point masses that interact by a mutual force directed along the line between them. This applies to a wide class of physical problems, with results of general significance. By using symmetry properties we can drastically simplify the problem down to a single equation involving only the radial distance between the two masses. At this stage, by introducing the concept of equivalent potential, the problem is reduced to one with only one degree of freedom. Proceeding further, we restrict our consideration to the force of gravity, a force that diminishes according to the inverse of the square of the distance between the attracting bodies. […]

OVERVIEW OF CHAPTER 3

An oscillator is a system with periodic motion. In mechanical systems, there is a restoring force that can do both positive and negative work as the system moves. Positive work done by this restoring force changes the kinetic energy into potential energy. Negative work done by the force turns the potential energy back into kinetic energy. If the force is linearly proportional to displacement, the oscillator is a linear or simple harmonic oscillator. Linear oscillators have many special properties. In particular, linear oscillators have the important property that the oscillation frequency is independent of amplitude. (This is not true if the oscillator is nonlinear.) The importance of linear oscillators in mechanics lies in the fact that, for small vibration amplitudes, we can approximate the dynamics of most mechanical systems as linear oscillators. Not only mechanical systems like a vibrating airplane wing, but, beyond the realm of mechanics, electrical systems and even an electron bound in an atom can be usefully modeled in this way. To understand large-amplitude oscillatory motion, we have to study nonlinear oscillators. The pendulum is an example of an oscillator that is linear at small amplitudes, yet becomes nonlinear at large amplitudes.

To discuss linear oscillators in a physically realistic way, we must depart from our dealings with conservative systems and introduce a special “damping” force which extracts energy from the oscillator. […]

OVERVIEW OF CHAPTER 7

As it would be viewed by an observer on the Sun, you are racing along at 66,700 miles/hr on an elliptical orbit. A different observer, located at the center of the Earth would see you rotating at 1,038 miles/hr. Yet, in everyday life, we are not normally aware of this. The description of motion depends on the reference frame. Inertial reference frames play a special role.

The Earth we live on is not an inertial frame. It is possible for someone on Earth to detect the Earth's rotation by detecting small deviations from Newton's Laws. While he was still an undergraduate, A. H. Compton invented a table-top experiment which not only demonstrated the Earth's rotation, but also measured the latitude of the laboratory. We need to develop a systematic way of translating back and forth between the description of motion in a rotating frame and the description in an inertial frame. This is a purely geometric or “kinematic” mathematical process, because we assume that the relative motion of the two reference frames is fully specified and is not subject to change by the action of forces, at least within the time period of the experiments we wish to do or during the observations we wish to make.

Motion can take place on a rotating body and be observed either with a reference frame fixed in the body, or from outside (i.e., a coordinate system fixed in “space”). […]

OVERVIEW OF CHAPTER 9

Consider a mechanical system that has N degrees of freedom. Assume also that the system is close to one of its stable equilibrium points. We will show that this system acts like N independent SHOs, usually with N different frequencies. One or more of these independent oscillations can be present depending on the initial conditions. In a state where only a single oscillation frequency is excited, the N different degrees of freedom move synchronously at a common mode frequency. The ratios between the different displacements for each degree of freedom, known as the mode displacement ratios, are an intrinsic characteristic of the normal mode that is oscillating. The amplitude of any particular mode is known as the normal coordinate. Each normal coordinate oscillates in time like a single SHO. All possible movements of the system, for sufficiently small displacements from the equilibrium point, can be described as a linear combination of modes.

Why do we concentrate on “small” vibrations for such a system? By definition, if the differential equations of motion are linear, the system is then said to be a linear system. Taylor's theorem guarantees that most systems are linear if the displacements are small enough. The motion can then be approximately described by a set of linear differential equations very similar to the equation for a simple harmonic oscillator. […]

OVERVIEW OF CHAPTER 2

Nature is found to conspire in just such a way that the time integral of the Lagrangian is smallest if the motion obeys Newton's Laws. Mechanics can be based on the single principle: Minimize the time integral of the Lagrangian. Three laws of motion can be condensed into one universal principle!

The mathematical language needed to provide the framework for this is called variational calculus. The variational calculus can be used as a powerful tool in solving mechanics problems with explicit constraints. It is also the most general means of solving nonholonomic problems with constraints on the velocities such as for rolling motion. This type of problem cannot be solved by choosing coordinates equal to the number of degrees of freedom but must be embedded in a higher-dimensional space.

The well-known theoretical physicist E. P. Wigner refers to the “unreasonable effectiveness of mathematics in theoretical physics.” Mathematical beauty is and should be the chief guiding principle of theorists, according to P. A. M. Dirac, one of the inventors of quantum mechanics. Although it is hard to define exactly what mathematical beauty is, the search for beauty was the guiding principle in the invention of two major advances in physics in the twentieth century: relativistic quantum mechanics and general relativity. In this chapter, we will discover an elegant formulation of classical mechanics. The mathematical techniques uncovered here are not only beautiful, but they have become the language of modern theoretical physics. […]

OVERVIEW OF CHAPTER 1

Joseph Louis Lagrange reformulated Newton's Laws in a way that eliminates the need to calculate forces on isolated parts of a mechanical system. Any convenient variables obeying the constraints on a system can be used to describe the motion. If Lagrangian mechanics rather than Newtonian mechanics is used, it is only necessary to consider a single function of the dynamical variables that describe the motion of the entire system. The differential equations governing the motion are obtained directly from this function without any vector force diagrams. Lagrangian mechanics is extremely efficient: There are only as many equations to solve as there are physically significant variables.

Lagrange did not introduce new physical principles to mechanics. The physical concepts are due to Newton and Galileo. But he succeeded in giving a more powerful and sophisticated way to formulate the mathematical equations of classical mechanics, an approach that has spread its influence over physics far beyond the purely mechanical problems.

We will begin by solving some examples that lead us toward this new formulation of mechanics. We plan to use the concept of virtual work to derive this. We will consider extended rigid bodies to be made up of collections of massive point particles. Summing over the constituent particles will lead to an efficient and general method for obtaining the differential equations of motion for any frictionless mechanical system. […]