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We restrict that $$\delta q\mid _{t_{1}}= \delta q\mid _{t_{2}}=0$$ while applying Hamilton Principle ($\delta\int_{t_{1}}^{t_{2}}Ldt=0$) to get Euler-Lagrange’s Equations. Hence adding a $$\frac{d}{dt}\Lambda(q,t)$$ doesn’t change the principle as $$\delta\int_{t_{1}}^{t_{2}}\frac{d}{dt}\Lambda(q,t)dt=\delta\Lambda\mid_{t_{1}}^{t_{2}}=0 \ .$$

My question is, why don’t we restrict $$\delta\dot{q}\mid_{t_{1}}=\delta\dot{q}\mid_{t_{2}}=0$$ as well, so that adding a $$\frac{d}{dt}\Lambda(q,\dot{q},t)$$ won’t change the principle? Why is it no longer feasible just because $$\dot{q}=\frac{dq}{dt} \ ?$$

Qmechanic
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Aimikan
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1 Answers1

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Typically we assume that the EL equations are of 2nd order (and that the Lagrangian is of 1st order). Then 2 boundary conditions are needed per DOF. Imposing both Dirichlet and Neumann boundary conditions would overconstrain the system.

However, OP's proposal makes sense for higher-order systems, cf. e.g. this Phys.SE post.

Qmechanic
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  • Thanks a lot, the answer and the links are really helpful. – Aimikan Jul 18 '23 at 15:05
  • Thanks again, I’ve been reading many SE posts concerning the BCs these days, and I am impressed that you stated $\delta p=0$ is incorrectly used to give canonical transformation in many textbooks in this post.But almost all the textbooks I’ve read made this mistake, so it will be of great help if you can recommend textbooks or articles that deal with the BCs correctly, or textbooks focusing on the principle of variation especially when it comes to the BCs? – Aimikan Jul 21 '23 at 08:07
  • Hi @Aimikan. Thanks for the feedback. I don't have any relevant references off the top of my head. – Qmechanic Jul 21 '23 at 09:19