Discussion
In the case of two coupled identical pendulums, in general :
Answer: C
The potential energy is a homogeneous quadratic function in terms of normal coordinates.
For two coupled pendulums, in terms of the actual displacements \(x_{1}, x_{2}\), the potential energy (for small oscillations) is \(\frac{1}{2}k(x_{1}^{2} + x_{2}^{2}) + \frac{1}{2}k'(x_{1} - x_{2})^{2}\), which contains a cross term \(x_{1}x_{2}\). Normal coordinates \(q_{1} = x_{1} + x_{2}\) and \(q_{2} = x_{1} - x_{2}\) remove the cross term, so the potential becomes a sum of squares, each mode oscillating independently.
The point of the statement is this separation into squares. The other options put the special, diagonal form in the actual coordinates, or deny it in normal coordinates, which is the reverse of what normal coordinates achieve.
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