I have begun reading Feynman & Hibbs Quantum Mechanics and Path Integrals. Knowing little about variational calculus or Lagrangians I found the following integration by parts opaque. I think if I saw it done once methodically it would clarify a lot. I have no problem with integration by parts in general.
On p. 27 he says that upon integration by parts the variation in S becomes
δS=[δx∂L∂˙x]tbta−∫tbtaδx[ddt(∂L∂˙x)−∂L∂x]dt.
Now
S=∫tbtaL(˙x,x,t)dt
in which L is the Lagrangian
L=m2˙x2−V(x,t)
and he says that to a first order
δS=S[ˉx+δx]−S[x]=0.
He shows S[x+δx] explicitly in (2-5) and from this derives (2-6).
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