My Graduate Mechanics text (Walecka and Fetter) says in the chapter on Small Oscillations, that the Modal Matrix Diagonalizes the Lagrangian(adsbygoogle = window.adsbygoogle || []).push({});

[tex] L = \sum_{\sigma=1}^\infty (\dot{\zeta_\sigma}^2 - \omega_\sigma \zeta_\sigma^2) [/tex]

where [tex]\zeta[/tex] are the normal coordinates related to the original coordinates [tex]\eta[/tex] via the modal matrix.

But as far as I understand, the Lagrangian is simply a number, not a matrix. So what does the diagonalization of the Lagrangian mean.

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# Diagonalized Lagrangian

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