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

[Ben Geoffrey]

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+c_b+df(x)dx = f(a)c-f(b)d

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

 

[PeroK]

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.

 

[Ben Geoffrey]

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)

 

[PeroK]

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.

 

[Ben Geoffrey]

thank you