Section 21–1 Linear differential equations

(Oscillator-like phenomena / Daily-life applications / Mathematical definition)

In this section, Feynman discusses linear differential equations from the perspective of oscillator-like phenomena, daily-life applications, and mathematical definition.

1. Oscillator-like phenomena:

“The harmonic oscillator, which we are about to study, has close analogs in many other fields; … we are really studying a certain differential equation. (Feynman et al., 1963, section 21–1 Linear differential equations).”

Feynman says that a strange thing occurs again and again: the equations which appear in different fields of physics and in other sciences are often almost exactly the same. Many phenomena are modeled using similar differential equations, for example, the propagation of sound waves is analogous to the propagation of light waves. There is a better explanation: “[t]he first is the limited imagination of physicists: when we see a new phenomenon we try to fit it into the framework we already have… (Feynman, 1985, p. 149).” That is, we idealize phenomena that can be described as oscillator-like. In essence, physicists develop models of phenomena using linear differential equations that can be solved and understood.

Feynman suggests that it is best to realize how the study of a phenomenon in one field may permit an extension of our knowledge in another field using linear differential equations. However, Heisenberg (1967) clarifies that “[p]ractically every problem in theoretical physics is governed by nonlinear mathematical equations, except perhaps quantum theory, and even in quantum theory it is a rather controversial question whether it will finally be a linear or nonlinear theory. Therefore by far the largest part of theoretical physics is devoted to nonlinear problems (p. 27).” Generally speaking, we simplify physical problems by ignoring factors such as friction or assuming an object’s displacement is small. Real phenomena are often not modeled by non-linear differential equations because they are either difficult to be solved or insolvable.

2. Daily-life applications:

“… all these phenomena follow equations which are very similar to one another, and this is the reason why we study the mechanical oscillator in such detail (Feynman et al., 1963, section 21–1 Linear differential equations).”

Linear differential equations are applicable to phenomena such as the oscillations of a mass on a spring, the oscillations of charge flowing back and forth in an electrical circuit and the analogous vibrations of the electrons in an atom. Specifically, the simplest linear differential equation is the equation of uniform motion that can be represented by dx/dt = c. In Heisenberg’s (1967) words, “[a]ctually mathematical physics started 300 years ago with the law of inertia, which may be considered to be the solution of the homogeneous linear equation d²x/dt² = 0 where x is the coordinate and t is the time. In the laws of free fall of Galileo, we find that he actually had solved an inhomogeneous linear equation, the force being the inhomogeneous term (p. 28).” In short, linear differential equations are useful in the modeling of physical motions.

There are other applications of differential equations such as a thermostat adjusting a temperature, complicated interactions in chemical reactions, and foxes eating rabbits (rate of change of population). Feynman’s reason for the study of the mechanical oscillator is: these phenomena follow differential equations which are very similar to one another. One may add that “it’s not because Nature is really similar; it’s because the physicists have only been able to think of the same damn thing, over and over again (Feynman, 1985, p. 149).” As a suggestion, we may explain that the differential equations are useful because they can represent the rate of change of an independent variable with respect to a dependent variable. In general, physicists need to find out how an observable would change with respect to time or with respect to distance.

3. Mathematical definition:

“Thus a_ndⁿx/dtⁿ + a_n−1dⁿ⁻¹x/dtⁿ⁻¹ +…+ a₁dx/dt + a₀x = f(t) is called a linear differential equation of order n with constant coefficients (each a_i is constant) (Feynman et al., 1963, section 21–1 Linear differential equations).”

Feynman defines a linear differential equation as a differential equation consisting of a sum of several terms, each term being a derivative of the dependent variable with respect to the independent variable, which is multiplied by some constant. In chapter 25, he adds that “we spend so much time on linear equations: because if we understand linear equations, we are ready, in principle, to understand a lot of things.” However, one may add that a linear differential equation is homogeneous if f(t) = 0 (i.e., the right-hand side of the equation is zero). On the contrary, the linear differential equation that does not fulfill this condition is known as inhomogeneous. Feynman ends this chapter by discussing how an inhomogeneous differential equation is applicable to a forced harmonic oscillator in which f(t) = F₀cos ωt.

In defining the linear differential equation, Feynman uses the phrase “linear differential equation of order n with constant coefficients” and provides an example of the equation. Mathematicians may not be satisfied with his definition because he did not define the order and degree of a differential equation. The order of a differential equation is the order n of the highest derivative dⁿx/dtⁿ present in the equation. The degree of a differential equation is the power of the highest order derivative in the same equation. Furthermore, a linear differential equation is said to be linear if the dependent variable and its derivatives are of first degree, for example, there are no derivatives such as (dy/dx)².

Questions for discussion:

1. Why do we study oscillator-like phenomena?

2. What are the possible applications of linear differential equations?

3. Could a linear differential equation be of order zero or infinity?

The moral of the lesson: oscillator-like phenomena can be modeled by linear differential equations of order n with constant coefficients.

References:

1. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Heisenberg, W. (1967). Nonlinear problems in physics. Physics Today, 20(5), 27-33.