How Are Definite Integrals Related to the Principle of Least Action?

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Discussion Overview

The discussion revolves around the relationship between definite integrals and the principle of least action, specifically in the context of a derivation presented in Goldstein's work. Participants explore mathematical theorems related to definite integrals and their implications for variations in the endpoints of integrals.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the validity of a specific mathematical theorem regarding definite integrals, suggesting that the expression ∫a+cb+df(x)dx = f(a)c-f(b)d may not hold universally.
  • Another participant notes that the theorem only applies when variations in the endpoints are small, indicating that the integral over a small interval approximates the function value at a point multiplied by the interval's width.
  • There is a repeated inquiry about why the variation is expressed as L(t2)Δt2 - L(t1)Δt1 instead of L(t2 +Δt2) - L(t1 +Δt1), with the latter seeming more logical to some participants.
  • One participant argues that if the deltas are small, it should not matter where the function is evaluated within the small interval, suggesting the use of a Taylor expansion to illustrate this point.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate formulation of variations in the context of the principle of least action, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

The discussion highlights potential limitations in the assumptions regarding the size of variations and the evaluation points within integrals, which remain unresolved.

Ben Geoffrey
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This is with regard to my doubt in the derivation of the principle of least of action in Goldstein

Is there any theorem in math about definite integrals like this ∫a+cb+df(x)dx = f(a)c-f(b)d

The relevant portion of the derivation is given in the image.
 

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That only works when the variations in the endpoints are small. The integral over a small interval is approx. the function value at a point in the interval times the width of the interval.
 
But why is the variation due to ends points L(t2)Δt2 - L(t1)Δt1 rather than L(t2 +Δt2) - L(t1 +Δt1) . Makes more sense if it is L(t2 +Δt2) - L(t1 +Δt1)
 
Ben Geoffrey said:
But why is the variation due to ends points L(t2)Δt2 - L(t1)Δt1 rather than L(t2 +Δt2) - L(t1 +Δt1) . Makes more sense if it is L(t2 +Δt2) - L(t1 +Δt1)

If the deltas are small it makes no difference where you evaluate the function within the small interval.

To see this use a Taylor expansion.
 
thank you
 

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