Section 23–1 Complex numbers and harmonic motion

(Real force / Real part of x / Right frequency)

In this section, Feynman discusses the real force, the real part of x, and the right frequency of a forced oscillation.

1. Real force:

“The complex number F that we have so defined is not a real physical force, because no force in physics is really complex; actual forces have no imaginary part, only a real part (Feynman et al., 1963, section 23–1 Complex numbers and harmonic motion).”

Feynman says that the real force, F = F₀cos ωt, can be written as the real part of a complex number F = F₀e^iωt. Physicists use complex functions to represent oscillatory functions because it is easier to work with an exponential function than with a cosine. In other words, the use of complex functions is based on pragmatic grounds or a matter of convenience. One may confess that this trick involves guessing x = e^iωt or x = te^ωt. On the other hand, Schrödinger struggled in determining between a real and complex wave function for his quantum theory. However, Wigner (1990) argues that the use of complex numbers in quantum mechanics is not a calculational trick. On the use of complex numbers, Wilczek (2015) explains that some follow “the advice of Father Jim Malley — ‘It is more blessed to ask forgiveness than permission’—and used them (p. 348).”

Feynman suggests a mathematical trick to represent a force that is out of phase by a delayed phase Δ. It refers to the real part of F₀e^i(ωt−Δ), but we may write e^i(ωt−Δ) = e^iωte^−iΔ. An advantage of the trick is that we can simplify the general solution of x using xe^iωt instead of xe^i(ωt−Δ). Some students may have difficulties understanding the term F₀e^i(ωt−Δ) because of the symbol F_0. It is worth mentioning that the complex force F is assumed to be out of phase to the displacement x by a phase difference Δ. One should realize that F₀ is the complex force at time t = 0 and it is not necessarily the maximum force. Importantly, a multiplication of F₀ by e^−iΔ can increase the magnitude of force to its maximum value. On the contrary, some physicists may prefer to let the factor e^−iΔ to appear in the displacement as xe^i(ωt−Δ).

2. Real part of x:

“… we deduce that the real part of x satisfies the equation with the real part of the force (Feynman et al., 1963, section 23–1 Complex numbers and harmonic motion).”

Feynman mentions that the real part of x satisfies the equation with the real part of the force, but the method is valid only for equations which are linear. That is, he refers to equations in which x appears in every term only in the first power or zeroth power. Feynman’s description of linear differential equation is potentially misleading. For example, Lorenz equations dx/dt = –sx + sy, dy/dt = – xz + rx – y, dz/dt = xy – bz are three ordinary differential equations that do not have any term that is in the second power or higher. After Feynman delivered this lecture, Lorenz (1963) discovered this system that is non-linear, non-periodic, and deterministic, but it leads to chaotic results. One feature of non-linearity is: we cannot combine two independent solutions to get a general solution (i.e., the whole is not equal to the sum of the parts).

Feynman elaborates that if there were a term λx² in the equation, then when we substitute x_r+ix_i, we would get λ(x_r+ix_i)² and it would yield λ(x_r²−x_i²) as the real part and 2iλx_rx_i as the imaginary part. In short, it introduces an artificial thing (imaginary part) in the analysis. Perhaps the following simplification may help some students: e.g., let the differential equation be dx/dt + x² = 0. If the two possible solutions are x₁ and x₂, this means that dx₁/dt + x₁² = 0 and dx₂/dt + x₂² = 0. To check the linear property of the equation, we need to find out whether d(x₁+x₂)/dt + (x₁+x₂)² = 0. By expanding the equation, d(x₁+x₂)/dt + (x₁+x₂)² = (dx₁/dt + dx₂/dt) + (x₁² + 2x₁x₂ + x₂²) = (dx₁/dt + x₁²) + (dx₂/dt + x₂²) + 2x₁x₂ = 2x₁x₂. It introduces a cross-term 2x₁x₂ that prevents the equation to be definitely equal to zero, and thus, x₁ + x₂ is not a general solution.

3. Right frequency:

“So we get a very strong response when we apply the right frequency ω (Feynman et al., 1963, section 23–1 Complex numbers and harmonic motion).”

Feynman explains that we get a very strong response when we apply the right frequency ω. He provides a simple example: if we hold a pendulum on the end of a string and shake it at just the right frequency, then it will swing very high. Alternatively, the explanations could include basic physical principles. Firstly, if the forcing frequency ω is exactly equal to natural frequency ω₀, the swing would oscillate at an infinite amplitude is an idealization. Specifically, it is about shaking at the right time (or right phase) instead of only right frequency. If the force and the motion of the pendulum are in the same direction (or same phase), then there is a maximum energy transfer in accordance with the work-energy theorem. However, one should realize that the phase difference between the force and displacement is p/2 radians (see Fig. 23-3).

According to Feynman, the magnitude of x is related to the size of the F by the factor 1/m(ω₀²−ω²). In section 21–5, he uses a trigonometric method to get ω₀² = k/m. To use an exponential method, we may first guess x = Ce^wt. By substituting it into md²x/dt² + kx = 0, we have m(w²)(Ce^wt) + k(Ce^wt) = mw²+ k = 0 whose solution is w = ±Ö(-k/m). Feynman has explained in section 21–5 that w₀ = Ö(k/m) is the natural frequency and this allows us to write w = ±iw₀. Thus, the general solution is x = Ce^(iw₀t) + C*e^(-iw₀t), whereby the two arbitrary constants C and C* should be complex numbers such that we have real solution (Symon, 1971). By solving the equations, we will get x = (C/2)e^(iw₀t + q) + (C/2)e^(-iw₀t + q) = C (cos w₀t + q).

Questions for discussion:

1. Would you prefer using a mathematical trick by letting F = F₀e^i(ωt−Δ) or x = x₀e^i(ωt−Δ)?

2. Is the imaginary part of the equation definitely an artificial thing without meaning?

3. Why do we get a very strong response if we apply the right frequency ω?

The moral of the lesson: The complex number F that we have defined is not a real physical force, because no force in physics is really complex; actual forces have no imaginary part, but only a real part.

References:

1. 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.

2. Lorenz, E. (1963). Deterministic Non-Periodic Flow. Journal of the Atmospheric Sciences, 20(2), 130–141.

3. Symon, K. R. (1971). Mechanics (3rd ed.). Reading, MA: Addison-Wesley.

4. Wigner, E.P. (1990). The unreasonable effectiveness of mathematics in the natural sciences. In R. E. Mickens (ed.). Mathematics and Science (pp. 291–306). Singapore: World Scientific.

5. Wilczek, F. (2015). A Beautiful Question: Finding Nature’s Deep Design. New York: Penguin Press.