There's an integral ${\int\limits_{t_1}^{t_2}}(\frac{\partial{L}}{\partial{q}}{\delta}q+\frac{\partial{L}}{\partial{v}}{\delta}v)dt=0$. [1.]

$ {\delta}v={\frac{d{\delta}q}{dt}}$ [2.]

I should get $ [\frac{\partial{L}}{\partial{v}}{\delta}q]_{t_1}^{t_2}+{\int\limits_{t_1}^{t_2}}(\frac{\partial{L}}{\partial{q}}-\frac{d}{dt}\frac{\partial{L}}{\partial{v}}){\delta}q dt = 0$ [3.] from [1.] using integration by parts and [2.], but I don't know how exactly should I calculate it. This is taken from Landau's and Lifshitz's "Mechanics" more precisely Chapter I, §2.


Given $$ \int_{t_1}^{t_2}\textrm{d}t\,\left(\frac{\partial L}{\partial q}\,\delta q + \frac{\partial L}{\partial v}\,\delta v \right)= \int_{t_1}^{t_2}\textrm{d}t\,\left(\frac{\partial L}{\partial q}\,\delta q\right) + \int_{t_1}^{t_2}\textrm{d}t\,\left(\frac{\partial L}{\partial v}\,\frac{d}{dt}\delta q \right) $$ then the second contribution on the right hand side can be re-written as $$ \int_{t_1}^{t_2}\textrm{d}t\,\left(\frac{\partial L}{\partial v}\,\frac{d}{dt}\delta q \right) = \int_{t_1}^{t_2}\textrm{d}t\,\left(\frac{d}{dt}\left(\frac{\partial L}{\partial v} \delta q\right)-\delta q\frac{d}{dt}\left(\frac{\partial L}{\partial v} \right) \right). $$ The first piece can be integrated out, being a total derivative, and the second one can be grouped together with the first overall contribution.

P. S. My personal experience is to never read Landau & Lifshitz's to learn from.

  • $\begingroup$ I agree with the last sentence! $\endgroup$ – Soba noodles Sep 22 '15 at 21:34
  • 4
    $\begingroup$ In my opinion, Landau & Lifshitz's start being useful after you have learnt from another book. $\endgroup$ – gented Sep 22 '15 at 22:24
  • $\begingroup$ Yes, the material is thoroughly presented, but not in a pedagogical manner. I didn't really clarify what I agreed with, but it seems we're both thinking the same. $\endgroup$ – Soba noodles Sep 22 '15 at 22:27
  • $\begingroup$ @gented Hello, I'm a 12th grader, I know Newtonian and Picked Landau and Lifshitz to learn Lagrangian Mechanics. So, I'm having the same bit of confusion every time. So, can you make me understand How $$\int \limits_{t_1}^{t_2} \left ( \frac{\partial L}{\partial v} \frac d {dt} \delta q \right)dt = \int \left ( \frac d {dt} \left ( \frac{\partial L}{\partial v} \delta q \right)- \delta q \frac d {dt} \left ( \frac{\partial L}{\partial v} \right) \right)dt$$ As far I know, integration with parts is $$\int a db = ab - \int b da$$ isnt it? Any help is appericiated! $\endgroup$ – user208739 Apr 7 '19 at 15:05

Partial integration is employed only for the second term in (1): $$\underset{{t}_{1}}{\overset{{t}_{2}}{\int }}\frac{\partial L}{\partial v}\delta v=\underset{{t}_{1}}{\overset{{t}_{2}}{\int }}\frac{d}{dt}\left(\frac{\partial L}{\partial v}\delta q\right)dt-\underset{{t}_{1}}{\overset{{t}_{2}}{\int }}\left(\frac{d}{dt}\frac{\partial L}{\partial v}\right)\delta qdt$$ The first term comes out as $$\frac{\partial L}{\partial v}\delta {q|}_{{t}_{1}}^{{t}_{2}},$$ the second one is left under the integral.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.