Angular Frequency and In-Phase Motion: A Scientific Inquiry

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In a system oscillating in normal modes, the lowest angular frequency corresponds to particles oscillating in phase due to the nature of restoring forces and energy minimization. This in-phase motion allows for maximum energy efficiency, as all particles move synchronously, minimizing potential energy. The relationship between in-phase motion and the lowest frequency can be attributed to the system's tendency to stabilize at this frequency, leading to the least resistance and energy loss. While normal modes can exhibit various frequencies, the lowest one is uniquely tied to the coherent movement of all particles. Understanding this relationship is crucial for analyzing oscillatory systems in physics.
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My question is; why for a system oscillating in a normal mode does the lowest angular frequency relate to the particles in that system oscillating in phase. I understand that the signs of the corresponding vector are the same,but what relation is it to the lowest angular frequency.

Thanks for any suggestions
 
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Please refer to Einstein modes. Normal mode can have any frequency (not only lowest one).
 
But the question ask: why for in-phase motion is the associated angular frequency the lowest, intermediate or highest (obviously there are 3 normal modes). The answer is the lowest angular frequency, but the question then asks; is this to be expected? ie why is the lowest angular frequency associated with in phase motion
 

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