Finding Normal Modes for Coupled Oscillators with Rotating Vectors

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I like coupled oscillators. Here are a couple of videos:

Finding normal modes with some tricks

Finding normal modes with the eigenvalue problem

Now I'm going to find the normal modes by varying the initial conditions.

Here is the python code I used
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Loved the live graph. Gonna have to play with that!

fizixx
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But, what is special about normal modes?

thejll