Using Complex Numbers to find the solutions (simple Q.)

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Discussion Overview

The discussion revolves around the use of complex numbers in solving the equations of an un-damped harmonic oscillator, particularly focusing on how to select the appropriate part of the solution (real or imaginary) based on the nature of the forcing function (sine or cosine). The scope includes conceptual understanding rather than advanced mathematical proofs.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests using Euler's equation to solve for both sine and cosine simultaneously and selecting the real or imaginary part based on the forcing function.
  • Another participant argues that the observable must satisfy a differential equation and is real, thus the real part of the solution should be chosen, regardless of the forcing function.
  • A different participant expresses that they believe the imaginary part should be chosen when the forcing function is sine, emphasizing their focus on the nature of the particular solution.
  • One participant requests clarification or visual representation to better understand the argument regarding the selection of the imaginary part for sine functions.

Areas of Agreement / Disagreement

Participants express disagreement regarding which part of the solution (real or imaginary) should be chosen based on the type of forcing function. There is no consensus on the justification for the method of using Euler's formula in this context.

Contextual Notes

Participants acknowledge the relevance of boundary conditions and the potential clarity offered by considering damped cases, but these aspects are not the focus of the current discussion.

Who May Find This Useful

This discussion may be useful for students or individuals interested in the application of complex numbers in differential equations, particularly in the context of harmonic oscillators and the justification of solution methods.

JTC
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Say you have an un-damped harmonic oscillator (keep it simple) with a sine or cosine for the forcing function.

We can exploit Euler's equation and solve for both possibilities (sine or cosine) at the same time.

Then, once done, if the forcing function was cosine, we choose the real part as the particular solution. If it was sine, we choose the imaginary solution.

Can someone say, in words, how we know this works?

I know it works because I can do it. How do we justify it? Most books just do it, but never explain why it works.

I am NOT looking for an advanced mathematical proof. I just want to know, in simple words, how we know this works.
 
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JTC said:
Then, once done, if the forcing function was cosine, we choose the real part as the particular solution. If it was sine, we choose the imaginary solution.
No we don't. The observable satifies a differential equation and the observable is real. We 'choose' the real part of the solution.

A sine is a cosine with a phase difference. That way we have an amplitude and a phase as constants so that we can satisfy the boundary conditions.

Check out the damped case as well, maybe it will be a bit clearer then
 
BvU said:
No we don't. The observable satifies a differential equation and the observable is real. We 'choose' the real part of the solution.

A sine is a cosine with a phase difference. That way we have an amplitude and a phase as constants so that we can satisfy the boundary conditions.

Check out the damped case as well, maybe it will be a bit clearer then

As I understood, and worked out, if the forcing function was sine, we WOULD choose the imaginary.

And I am ONLY discussing the nature of the particular solution. I am not interested in (in the case of damping--yes, bringing that back in), the transient solution.

I am aware that I CAN work this out by sticking with the complex solution and consolidating it with a phase shift. But I am not interested in that discussion either.

I would just like to know WHY this method of using Euler's formula to extract the real or imaginary (again, NOT interested in the phase right here), works.
 
JTC said:
As I understood, and worked out, if the forcing function was sine, we WOULD choose the imaginary
Hard for me to guess how it looks -- my telepathic capabilities have proven to be quasi non-existent --, so can you post something to show that ?
 
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