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From my understanding, my instructor told me that in order to use the Lagrangian, defined as $$L \equiv T - V,$$ to find the equations of motion via the Euler-Lagrange equations, the generalized coordinates of the system must be derived from the positions of the particles in an inertial reference frame. Is this correct? Also, is this requirement built into the definition of generalized coordinates?

What if we use the most general definition of the Lagrangian, i.e., the one that says the Lagrangian is the function such that the Euler-Lagrange equations give the correct equations of motion?

  • Lagrangian mechanics is essentially equivalent to Newtonian mechanics (I mean this in the logical sense as in "Lagrangian mechanics if and only if Newtonian mechanics", of course Lagrangian mechanics in practice is not equivalent to Newtonian mechanics). So any assumptions needed for Newtonian mechanics must be carried over into Lagrangian mechanics – BioPhysicist Apr 07 '19 at 23:24
  • @AaronStevens Perhaps the reason you commented as opposed to answered is that you're aware that your comment doesn't constitute a proof, it just appeals to intuition. Consider, for example, the equation $$\frac{d}{dt} \sum_\alpha \mathbf{r}\alpha' \times \mathbf{p}\alpha' = \sum_\alpha \mathbf{r}\alpha' \times \mathbf{F}\alpha^\textrm{ext},$$ in deriving it one assumes Newton's laws, but it's independent of whether or not the coordinates $\mathbf{r}_\alpha$ (not primed) are those pertaining to inertial reference frame. –  Apr 07 '19 at 23:40
  • I wasn't telling you my intuition. Any classical mechanics text I have seen shows a proof of "Lagrangian mechanics if and only if Newtonian mechanics". I commented because I am not going to just put a proof as an answer, and I am not at a place to write out a full answer. – BioPhysicist Apr 07 '19 at 23:48
  • With that being said, you can still describe non-inertial frames using forces, etc. similar to Newtonian mechanics, so I am sure you can modify the Lagrangian in a way to use Lagrangian mechanics in non-inertial frames (for example, maybe an effective potential) – BioPhysicist Apr 07 '19 at 23:55
  • @AaronStevens I'm not questioning your statements, what I'm claiming is that what you said doesn't prove that the coordinates must be in an inertial reference frame. The statements themselves are just fine, they just don't constitute a proof. –  Apr 08 '19 at 00:00
  • I never claimed that my comment was a proof... I was saying proofs exist and telling you the conclusions one would draw from such proofs – BioPhysicist Apr 08 '19 at 00:09
  • @AaronStevens You're right, I was just making sure we're on the same page. –  Apr 08 '19 at 00:10
  • Doesn't "the generalized coordinates of the system must be derived from the positions of the particles in an inertial reference frame" in this context really mean that the Lagrangian should not be dependent on derivatives of higher order than first order? It's certainly possible to express a Lagrangian in terms that are frame independent. – S. McGrew Apr 08 '19 at 00:46
  • @S.McGrew I think it does mean that. But then is the Lagrangian still $T - V$? –  Apr 08 '19 at 02:05
  • Possible duplicate: https://physics.stackexchange.com/q/99923/2451 – Qmechanic Apr 08 '19 at 04:45

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