Lagrangian: kinetic matrix Z_ij and mass matrix k_ij

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SUMMARY

The discussion centers on the Lagrangian mechanics, specifically addressing the kinetic matrix \( Z_{ij} \) and the mass matrix \( k_{ij} \). Participants clarify that the kinetic term for fluctuations is diagonal due to the orthogonality of the modes involved. The normalization factor \( \sqrt{m} \) is introduced to ensure proper scaling of the kinetic term. Additionally, \( Z_{ij} = \delta_{ij} \) indicates that the kinetic matrix is identity-like, simplifying the analysis of the system's dynamics.

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  • Understanding of Lagrangian mechanics
  • Familiarity with matrix algebra
  • Knowledge of normal modes in physics
  • Experience with classical mechanics textbooks or papers
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  • Review classical mechanics textbooks focusing on Lagrangian formulations
  • Study the concept of normal modes and their implications in mechanical systems
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Students of physics, researchers in classical mechanics, and anyone studying Lagrangian dynamics will benefit from this discussion.

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Screen Shot 2023-10-04 at 08.43.45.png

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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