I What is the significance of the T - V Lagrangian of a system?

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The quantity ∫x T - V represents the action in a physical system, which is crucial because the actual trajectory minimizes this action. In the context of a factory, it signifies the optimization of resource allocation between different production processes. For a system's motion, it highlights how energy is exchanged between kinetic and potential forms, leading to the derivation of equations of motion through the Euler-Lagrange equations. The significance of the T - V Lagrangian extends to understanding symmetries and conservation laws in physics, linking to fundamental principles like Hamilton's stationary action. Overall, the action principle is a foundational concept that governs the dynamics of systems across various contexts.
  • #31
PS in this case, you can see that we have a common average velocity between all potential paths. And, any attempt to vary the velocity produces a greater increase in the action when ##v > \frac X T## than the reduction when ##v < \frac X T##. In other words, the ##v^2## term ensures that uniform velocity minimises the integral. Although, it's a bit more effort to prove that formally.
 

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