Action Definition: Integral of Lagrangian Over Time

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Discussion Overview

The discussion revolves around the definition of action in physics, specifically why it is expressed as the integral of the Lagrangian over time. Participants explore the implications and interpretations of this definition, touching on concepts of energy and motion.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions the meaning of defining action as the integral of the Lagrangian, particularly regarding the interpretation of the integral of kinetic minus potential energy.
  • Another participant suggests that while the integral itself may not have inherent meaning, its variation leading to the equations of motion is significant.
  • A different viewpoint proposes that the integral can be seen as a measure of energy usage, indicating that the path taken by a particle minimizes this energy, although it is noted that it actually takes a 'stationary' path.
  • A later reply mentions finding a detailed explanation in the Feynman lectures, indicating a search for clarity on the topic.

Areas of Agreement / Disagreement

Participants express differing interpretations of the integral's meaning and its implications for understanding motion, indicating that the discussion remains unresolved with multiple competing views.

Contextual Notes

Some assumptions about the nature of the integral and its physical interpretation are not fully articulated, and there is a lack of consensus on the implications of the action's definition.

o_neg
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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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