Discussion
In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?
Answer: B
The statement that does NOT apply is the normal coordinates are orthogonal to one another.
Normal coordinates are the combinations of the original coordinates that diagonalise both the kinetic and potential energies at once. In terms of them,
\[ \begin{aligned} T &= \tfrac{1}{2}\sum_{k}\dot{\eta}_{k}^{2} \\ V &= \tfrac{1}{2}\sum_{k}\omega_{k}^{2}\eta_{k}^{2} \end{aligned} \]
so the potential energy is a sum of squares with constant coefficients, each coordinate oscillates independently, and each has its own eigen-frequency \(\omega_{k}\). Those three statements are the defining properties.
Orthogonality is not a property of the coordinates themselves. The mode (eigen)vectors from which they are built are orthogonal only with respect to the kinetic-energy (mass) matrix, not in the ordinary sense, so the key treats this as the statement that does not apply.
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