Why we take only the real part of a solution as the actual motion

classical-mechanicscomplex numbersdifferential equationslinear systemsoscillators

I am taking Analytical Mechanics, and in Goldstein's book, chapter 6 (page 241) about linear oscillations, he says the following:

"… $\eta_i=Ca_ie^{-i\omega t}$ (6.11) … It is understood of course that it is the real part of (6.11) that is to correspond to the actual motion."

Equation 6.11 is the solution for the equation $T_i\ddot{\eta_i}+V_i\eta_i=0$ (1).

I understand why we chose the solution to be of the form 6.11.

It is also clear to me that if 6.11 is a solution to (1), then its real and imaginary parts will also be solutions, and also that the solution of (1) must be real, as it is a real homogenous linear differential equation.

What I don't understand is why we say that (only) the real part corresponds to the actual motion of the system.

In addition, wouldn't it be better if we denoted the solution to the equation (1) as $\eta_{real}$,and the solution in eq. 6.11 as $\eta_{complex}$?

(I read this and this before posting, but I don't get it yet…)

Best Answer

I think the reason for the sentence

"It is understood of course that it is the real part of (6.11) that is to correspond to the actual motion"

Is that $\eta_i$ represents (likely) the displacement of a particle from a certain position. Something for example measured in meters and therefore, obviously, a real quantity. This is the reason for the "of course" in the sentence as a complex value wouldn't make sense.

This however is a bit cheating. We have a mathematical description of reality. Said description allowed to be augmented to complex values. But if the description is correct it must give back real values for proper (real) data.

And this is indeed what happens. First you should notice that there are actually two solutions

$$ \eta_i = C_\pm e^{\pm i \omega t} $$

Any linear combination of the above is still a solution. The point is that, if we did things correctly, for proper initial conditions (i.e. initial conditions that are compatible with meaningful data) the solution should automatically turns out real.

To summarize, the solution of the equation, being a second order ODE, depends on two initial conditions (for example initial position and velocity). If these values correspond to a meaningful motion, the end solution turns out automatically real.

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