Many equations we encounter in mathematical physics are too complicated to solve analytically. One of the reasons can be that an equation contains too many terms to handle. In practice, however, these terms may vary in size. Ignoring the smaller terms may simplify the problem to the extent that it can be solved in closed form. Moreover, by deleting terms that are relatively small, we can focus on the terms that contain the significant physics. In this sense, ignoring the smaller terms can actually give a better physical insight into the processes that really do matter.

Scale analysis is a technique in which one estimates the different terms in an equation by considering the scale over which the relevant parameters vary. This is an extremely powerful tool for simplifying problems. A comprehensive overview of this technique with many applications is given by Kline (1986) and in Chapter 6 of Lin and Segel (1974). Interesting examples of the application of scaling arguments to biology are given by Vogel (1998).

With the application of scale analysis one caveat must be made. One of the major surprises of classical physics of the twentieth century was the discovery of chaos in dynamical systems (Tabor, 1989). In a chaotic system small changes in the initial conditions lead to a change in the time evolution of the system that grows exponentially with time. Deleting (initially) small terms from the equation of motion of such a system can have a similar effect; this can lead to changes in the system that may grow exponentially with time. This means that for chaotic systems one must be careful in omitting small terms from the equations.

The principle of scale analysis is first applied to the problem of determining the sense of rotation of a vortex in an emptying bathtub. Many of the equations that are used in physics are differential equations. For this reason it is crucial in scale analysis to be able to estimate the order of magnitude of derivatives.

From this book and most other books on mathematical physics, you may have obtained the impression that most equations in the physical sciences can be solved. This is actually not true; most textbooks (including this book) give an unrepresentative state of affairs by only showing the problems that can be solved in closed form. It is an interesting paradox that as our theories of the physical world become more accurate, the resulting equations become more difficult to solve. In classical mechanics the problem of two particles that interact with a central force can be solved in closed form, but the three-body problem in which three particles interact has no analytical solution. In quantum mechanics, the one-body problem of a particle that moves in a potential can be solved for a limited number of situations only: for the free particle, the particle in a box, the harmonic oscillator, and the hydrogen atom. In this sense the one-body problem in quantum mechanics has no general solution. This shows that as a theory becomes more accurate, the resulting complexity of the equations makes it often more difficult to actually find solutions.

One way to proceed is to compute numerical solutions of the equations. Computers are a powerful tool and can be extremely useful in solving physical problems. Another approach is to find approximate solutions to the equations. In Chapter 11, scale analysis was used to drop from the equations terms that are of minor importance. In this chapter, a systematic method is introduced to account for terms in the equations that are small but that make the equations difficult to solve. The idea is that a complex problem is compared to a simpler problem that can be solved in closed form, and to consider these small terms as a perturbation to the original equation. The theory of this chapter then makes it possible to determine how the solution is perturbed by the perturbation in the original equation; this technique is called perturbation theory.

Many physical systems can only oscillate at certain specific frequencies. As a child (and hopefully also as an adult), you observed that a swing in a playground moves only with a specific natural period, and that the force pushing the swing is only effective when the period of the force matches the period of the swing. The patterns of motion at which a system oscillates are called the normal modes of the system. A swing has one normal mode (Section 17.1), but you have seen in Section 12.5 that a simple model of a tri-atomic molecule has three normal modes. An example of the normal modes of a system is given in Figure 19.1, which shows the pattern of oscillation of a metal plate driven by an oscillator at six different frequencies. The screw in the middle of the plate shows the point at which the force on the plate is applied, and before any force is applied, sugar is evenly sprinkled on the plate. When an external force is applied at the frequency of one of the normal modes of the plate, oscillations of the plate result in a pattern of motion with nodal lines. These nodal lines define where the motion vanishes and as a result the sugar on the plate collects at these lines.

In this chapter, the normal modes of a variety of systems are analyzed. Normal modes play an important role in many applications, because the frequencies of normal modes provide important information about physical systems. Examples include the spectral lines of the light emitted by atoms, which have led to the advent of quantum mechanics and its description of the structure of atoms. In another example, the normal modes of the Earth provide information about the internal structure of our planet. In addition, normal modes are used in this chapter to introduce some properties of special functions, such as Bessel and Legendre functions.

In this chapter several elements of linear algebra are treated that have important applications in physics or that serve to illustrate methodologies used in other areas of mathematical physics. For example, linear algebra provides a foundation for the inverse theory presented in Chapter 22.

12.1 Projections and the completeness relation

In mathematical physics, projections play an important role. This is true not only in linear algebra, but also in the analysis of linear systems such as linear filters in data processing (Section 14.10), and in the analysis of vibrating systems such as the normal modes of the Earth (Section 19.7). Let us consider a vector v that we want to project along a unit vector n (Figure 12.1). In the examples in this section we work in a three-dimensional space, but the arguments presented here can be generalized to any number of dimensions.

We denote the projection of v along n as Pv, where P stands for the projection operator. In a three-dimensional space this operator can be represented by a 3 × 3 matrix. It is our goal to find the operator P in terms of the unit vector n, as well as the matrix form of this operator. By definition the projection of v is directed along n; hence,

Pv = Cn.

This means that we know the projection operator once the constant C is known.

Problem a Show that with the variables defined in Figure 12.1 the length of the vector Pv is |Pv| = |v| cos ϕ. Use (n · v) ≡ n|v| cosφ = |v| cos φ to show that C = (n · v).

Inserting this expression for the constant C in (12.1) leads to an expression for the projection

Pv = n (n · v).

Problem b Show that the component v⊥ perpendicular to n as defined in Figure 12.1 is given by:

v⊥ = v − n(n · v).

In this book we have explored many methods of mathematics as used in the physical sciences. Mathematics plays a central role in the physical sciences because it is the only language we have for expressing quantitative relations in the world around us. In fact, mathematics not only allows us to express phenomena in a quantitative way, it also has a remarkable predictive power in the sense that it allows us to deduce the consequences of natural laws in terms of measurable quantities. In fact, we do not quite understand why mathematics gives such an accurate description of the world around us (Wigner, 1960).

It is truly stunning how accurate some of the predictions in (mathematical) physics have been. The orbits of the planetary bodies can now be computed with extreme accuracy. Stephenson and Morrison (1995) compared the path of a solar eclipse at 181 BC with historic descriptions made in a city in eastern China that was located in the path of the solar eclipse. According to the computations, the path of the solar eclipse passed 50 degrees west of the site of this historic observation. This eclipse took place about 2000 years ago; this means that the Earth has rotated through about 2.8 × 108 degrees since the eclipse. The relative error in the path of the eclipse over the Earth is thus only 1.7 × 10−7. In fact, this discrepancy of 50 degrees can be explained by the observed deceleration of the Earth due to the braking effect of the Earth's tides.

1. Two-matrix model: introduction

The Hermitian two-matrix model is the probability measure

defined on pairs (M1> , M2) of n× n Hermitian matrices. Here V and W are two polynomial potentials, τ ≠ 0 is a coupling constant, and

is a normalization constant in order to make (1-1) a probability measure.

In recent works of the authors and collaborators [Duits et al. 2011; 2012;

Duits and Kuijlaars 2009; Mo 2009] the model was studied with the aim to gain

understanding in the limiting behavior of the eigenvalues of M1 as n→∞, and

to find and describe new types of critical behaviors.