Graduate Lagrangian: kinetic matrix Z_ij and mass matrix k_ij

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The discussion revolves around the diagonal nature of the kinetic term for fluctuations in Lagrangian mechanics and the addition of the square root of mass for normalization. Participants express confusion about the significance of Z_ij being equal to delta_ij and seek clarification on these concepts without specific context from the example referenced. The importance of citing the relevant textbook or paper for clarity is emphasized. Understanding these terms is crucial for grasping the underlying principles of the Lagrangian framework. Overall, the conversation highlights the need for clearer explanations in theoretical physics discussions.
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Can somebody explain why the kinetic term for the fluctuations was already diagonal and why to normalize it, the sqrt(m) is added? Any why here Z_ij = delta_ij?
Quite confused about understanding this paragraph, can anybody explain it more easily?
 
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How can we answer this question, without knowing, what the example actually is? It's also mandatory to quote the textbook/paper you picture is taken from.
 
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