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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

$$ \delta S = \left[\delta x \frac{\partial L}{\partial \dot{x}} \right]_{t_a}^{t_b} - \int_{t_a}^{t_b}\delta x\left[\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}\right] dt. \tag{2-6} $$

Now

$$ S = \int_{t_a}^{t_b}L(\dot{x},x,t) dt \tag{2-1}$$

in which $L$ is the Lagrangian

$$ L = \frac{m}{2}\dot{x}^2 - V(x,t)\tag{2-2}$$

and he says that to a first order

$$\delta S = S[\bar{x}+ \delta x] - S[x] = 0. \tag{2-4}$$

He shows $S[x+\delta x]$ explicitly in (2-5) and from this derives (2-6).

Qmechanic
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daniel
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    You should not try to learn quantum mechanics, either in the path integral or in the canoncial quantiztation approach without having learned the Lagrangian and Hamiltonian formalisms of classical mechanics. You will only grow more and more confused if you try it. – ACuriousMind Jul 05 '14 at 15:52
  • I do agree with ACuriousMind above and strongly encourage you to learn about Lagrangian and Hamiltonian formalism. However, (2-6) must be thougt as the variation of S if you integrate L along a trajectory x(t) + dx(t), expanding L in Taylor series with respect to such dx(t) is the way to obtain (2-6). Tell me if you need more detail and I'll post this more accurately – giulio bullsaver Jul 05 '14 at 16:15
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    Do you know what $\delta L$ is, given that $L= L[x,\dot x] $? Hint: Think of differentials in ordinary calculus. – Danu Jul 05 '14 at 16:22
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    I believe it's $\delta \dot{x}\frac{\partial L}{\partial \dot{x}}+ \delta x\frac{\partial L}{\partial x}~ ?$ – daniel Jul 05 '14 at 16:25
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    Right. Now, apply partial integration to the first term. This should yield the desired result – Danu Jul 05 '14 at 16:34
  • Comment to the question (v2): It seems that OP is asking for the standard integration-by-parts argument in calculus of variations that leads to Euler-Lagrange equations. This is shown in many textbooks on the subject (including the above two Wikipedia links $\uparrow$), and e.g. this Phys.SE post and links therein. – Qmechanic Jul 08 '14 at 07:24
  • Don't listen to @ACuriousMind. I never took a course on classical mechanics (still haven't) or learned the Lagrangian or Hamiltonian formalisms before I did QM. It's not as necessary as others might have you believe. Just find a good textbook and work through all the problems. Eventually it will come to you – Jim Jul 08 '14 at 13:05
  • I will strongly protest @Jim 's advice - just because one learns to solve QM problems (which are very easy, really) doesn't mean one truly understands QM, and the added perspectives with Lagrangians and especially Hamiltonians can only help with this understanding. Being able to solve textbook problems is a means to an end, never an end in itself in physics. –  Jul 09 '14 at 13:19
  • @ChrisWhite: Hah! I didn't see Jim's note until just now. As a tactical matter since books are always in my budget I won't lose anything by getting a feel for Lagrangians, etc. So I will be taking ACuriousMind's advice. But I also suspect some folks can make the necessary inferences with sufficient exposure. And we need contrarians! – daniel Jul 09 '14 at 13:42

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