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While deriving the Lagrange equations from d'Alembert's principle, we get from $$\displaystyle\sum_i(m\ddot x_i-F_i)\delta x_i=0\tag{1}$$ to $$\displaystyle\sum_k (\frac {\partial\mathcal L}{\partial\ q_k}-(\frac d {dt}\frac {\partial\mathcal L}{\partial\dot q_k}))\delta q_k=0\tag{2}$$

However, from the above step, we get to the below step only after assuming holonomic constraints: $$(\frac {\partial\mathcal L}{\partial\ q_k}-(\frac d {dt}\frac {\partial\mathcal L}{\partial\dot q_k})=0.\tag{3}$$

Why is it that we have to assume holonomic constraints for that transition? My guess is that it has something to do with that if the constraints are not holonomic, then the virtual displacement are not always perpendicular to the trajectory of the body, but I can't see the mathematical connection between these.

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  • $\begingroup$ Is this from some reference? Which page? Is it online? $\endgroup$
    – Qmechanic
    Commented Sep 1, 2023 at 18:02
  • $\begingroup$ @Qmechanic it is from the lecture notes published by my professor... $\endgroup$
    – gluon
    Commented Sep 1, 2023 at 18:23
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    $\begingroup$ Judging by your follow-up questions: I get the impression that what you are looking for is a conversational forum. However, the stackexchange environment is not a conversational environment. The stackexchange concept is that you take the time to formulate a focused question, such as to allow a focused answer. The concept is that if there is a new question then that new question should be submitted as a separate question. If you prefer a conversational forum: an example of that is: physicsforums $\endgroup$
    – Cleonis
    Commented Sep 1, 2023 at 19:19

1 Answer 1

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  1. To go from (2) to (3) we only need that the generalized coordinates are independent variables.

  2. The holonomic constraints were used at a previous stage to define the generalized coordinates in the first place, see e.g. Ref. 1.

References:

  1. H. Goldstein, Classical Mechanics; Section 1.3.
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  • $\begingroup$ 1. why do we need the generalized coordinates to be independent? 2. what happens if they are not? $\endgroup$
    – gluon
    Commented Sep 1, 2023 at 18:55
  • $\begingroup$ 1. E.g. to deduce eq. (3). 2. Then there are further constraints. $\endgroup$
    – Qmechanic
    Commented Sep 1, 2023 at 18:59
  • $\begingroup$ yes, but why is it important? why can't we deduce eq 3 when the generalized coordinates are dependent on each other? $\endgroup$
    – gluon
    Commented Sep 1, 2023 at 19:08
  • $\begingroup$ Because eq. (3) no longer holds. Eq. (3) will be modified by the extra constraints. $\endgroup$
    – Qmechanic
    Commented Sep 1, 2023 at 19:13

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