What is an physical interpretation of these harmonic oscillators: $$\ddot{x}+i\cdot x=0$$ and $$\ddot{x}-1\cdot x=0.$$

I assume that the system satisfies this second order DE $$\ddot{x}+\omega^2\cdot x=0.$$ I.e. when $\omega$ is imaginary. Here $i$ is the imaginary unit. Which properties do these oscillator posses and what are examples of them ? Do they have a name ?


1 Answer 1


Assume we look for solutions in the canonical form $x(t) = X e^{s t}$, and then sum all the solutions we get exploiting the linearity of the problem. At the very end, once we get the most general expression for the solutions, we could find the multiplicative constants using initial conditions (not given here).

For the first equation $\ddot {x} + i x = 0$, we get $(s^2 + i ) X e^{s t}= 0$ and thus $s^2 = -i = e^{-i \pi/2+n2\pi}$ ($n \in \mathbb{Z})$, whose two solutions are

$s_1 = e^{i 3\pi/4} = \dfrac{1}{\sqrt{2}}(-1 +i)$
$s_2 = e^{-i \pi/4} = \dfrac{1}{\sqrt{2}}(1 - i)$

and putting these two solutions together,

$x(t) = X_1 e^{-\frac{1}{\sqrt{2}}t} \left(\cos\left(\frac{1}{\sqrt{2}}t\right) + i \sin\left(\frac{1}{\sqrt{2}}t\right)\right) + X_2 e^\frac{1}{\sqrt{2}}t \left(\cos\left(\frac{1}{\sqrt{2}}t\right) - i \sin\left(\frac{1}{\sqrt{2}}t\right)\right)$.

For the second equation $\ddot {x} - x = 0$, we get $(s^2 - 1 ) X e^{s t}= 0$ and thus $s^2 = 1$, whose two solutions are

$s_1 = -1$
$s_2 = 1$

and putting these two solutions together,

$x(t) = X_1 e^{-t} + X_2 e^t $.

  • $\begingroup$ Your answer is nice so I up-voted, but I cannot mark as solved since I was asking about the physical interpretation of these 2 harmonic oscillators with imaginary frequency. Do you know something along these lines ? I think at least that the physical system will be weird one! $\endgroup$ Sep 4, 2022 at 13:58
  • $\begingroup$ Where did you meet that expressions? O guess that these expressions (especially the one with the $i$ factor) comes as a convenient way to write a system of two ODEs. Defining $x(t) \in \mathbb{C}$, you can write it as the sum of its real and imaginary parts $x(t) = u(t)+iv(t)$, with $u(t), v(t) \in \mathbb{R}$. The equation $\ddot{x} + i x = 0$ on the complex field, is equivalent to the system of real equations $\ddot{u} - v = 0$, $\ddot{v} + u= 0$. The solution of this system of equations corresponds to the real and imaginary parts of the expression of $x(t)$ provided in the answer. $\endgroup$
    – basics
    Sep 4, 2022 at 14:52
  • $\begingroup$ Right then. Can this system of two real equations be (or is it assumed to be) the description of a single oscillator ? How ? $\endgroup$ Sep 4, 2022 at 18:28

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