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### Normal modes of oscillation: how to find them? [duplicate]

Are normal modes the eigenvectors of the matrix $(\omega ^2 T- V)$ where $T$ is the matrix of kinetic energy and $V$ is the matrix of potential energy? Is it the only way to express them? How can I ...
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### Interpreting physical meaning of normal modes

What really is a normal mode? Maybe it's because of my teachers but I find it really abstract. I know that "numerically" corresponds to the eigenvectors of the equation $\ddot{X}= -M^{-1}KX$ ...
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### How to calculate the potential energy of coupled oscillators?

The equations of motion that describe the above situation is given by: $$m \ddot{x_1} = -2kx_1 + kx_2$$ $$m \ddot{x_2} = -2kx_2 + kx_1$$ Now I want to work out the potential energy of this system. ...
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### Why do two masses connected to each other by a spring have the same frequency of oscillation?

Why do two masses, connected to each other by a spring, and each connected to a wall by a spring, have the same frequencies of oscillation when perturbed? In solving for the motion of the masses, ...
If you take a system like the one in the image, and you do the $y=x'$ trick to turn it into a first-order system of equations ($x_{1}$ or $x_{2}$ being the displacement of the mass $m_{1}$ or $m_{2}$ ...