Why is force equal to the rate of change of momentum?

[John_Doe]

What's the lagrangian you're using?

 

[Son Goku]

This factor is important because it allows you to set the rest of the integrand to zero (in general, just because the integral of a function is zero doesn't mean the function is zero). Hope this helps.
By the way, you wouldn't happen to go to Georgia Tech would you?

Thanks for the part in bold, I was actually looking at what I'd written after wondering why it wasn't $$\delta{S} = constant$$.
As for your question, I go to NUI Maynooth in Ireland.

What's the lagrangian you're using?

A general one for a conservative force.