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To stay with finite (as opposed to infinite) degrees of freedom, Goldstein might have a complex discrete Fourier transform (DFT) $$ {\bf r}_j(t)~=~\sum_{k=1}^N e^{2\pi i jk/N }{\bf q}_j(t)$$ (or a real version thereof) in mind. Here the amplitudes ${\bf q}_j(t)$ play the role of generalized coordinates. This is for instance useful for finding normal modes of lattice models.