I Physical meaning of Lagrangian?

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The Lagrangian in classical mechanics, defined as the difference between kinetic and potential energy (L = T - V), lacks a unique physical interpretation due to its nonuniqueness; it can be altered by adding total time derivatives without affecting the equations of motion. This implies that while the Lagrangian is crucial for calculations, it may not represent a measurable quantity with intrinsic physical meaning. The discussion highlights that a genuine Lagrangian leads to differential equations governing motion, whereas functions that satisfy the Euler-Lagrange equations identically do not provide dynamical information. Despite its abstract nature, the Lagrangian contains all necessary information to describe a physical system. Ultimately, its utility in physics remains significant, even if it lacks intuitive physical representation.

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