Action Definition: Integral of Lagrangian Over Time

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The action is defined as the integral of the Lagrangian over time because setting its variation to zero yields the equations of motion. This integral represents the measurement of energy used, specifically the difference between kinetic and potential energy. While the interpretation of the integral can be complex, it can be visualized as the path a particle takes that minimizes energy usage. A reference to the Feynman Lectures on Physics provides further clarity on this concept. Understanding this framework is essential for grasping the principles of classical mechanics.
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Hi,

why is the action defined as the integral of the Lagrangian over time?
i don't see the meaning of integral on the (kinetic - potential) energy

thanks,
ori.
 
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The integral has no meaning by itself, but setting its variation equal to zero leads to the equations of motion.
 
You can see the integral as a way of measuring how much of something is used. So in this case, the integral measures how much (K-U) energy is being used up, and the path that a particle takes minimizes this (this isn't technically correct... it takes a 'stationary' path, but looking at it this way could be beneficial to the student visually).
 
thanks for the replay.

I found a detailed explanation in the Feynman lectures on physics vol 2
 

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