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I am confused by this remark in the derivation of Lagrange equations from d'Alembert's principle in Goldstein:

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I am not comfortable that I understand why, at this late stage of the derivation, they are saying we can now freely choose to specialize to the case where the virtual displacements are independent, which implies the independence of the $δq_j$ has not already been firmly established.

I feel this result was already implicit in the derivation. Specifically, in order to derive eq. (1.51):

$$\frac{∂\textbf{v}_i}{∂\dot{q_j}} = \frac{∂\textbf{r}_i}{∂{q_j}}\tag{1.51}$$

we have to assume:

$$\frac{∂\dot{q_j}}{∂\dot{q_k}} = \delta_{jk}.$$

I think this already implies the $q_j$ and $δq_j$ must also be independent, and so this is already the only case where (1.52) is valid, rather than a special case as they suggest. What am I misunderstanding? Can the generalized velocities be independent even if the generalized coordinates and virtual displacements are not?

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heranias
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1 Answers1

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  1. OP has a point. Already above eq. (1.38) Ref. 1 writes that the generalized coordinates generalized coordinates $q^1,\ldots, q^n,$ are independent variables, and this fact is then used repeatedly e.g. when defining partial derivatives wrt. $q^j$ and $\dot{q}^k$.

  2. To try to justify/make sense of the paragraph above eq. (1.53), perhaps Goldstein has in mind that there are additional constraints beyond the holonomic constraints that defines the generalized coordinates $q^1,\ldots, q^n,$ from the positions ${\bf r}_1, \ldots, {\bf r}_N,$ in the first place, cf. eq. (1-38)? In fact, Goldstein considers this very situation later in Section 2.4. (NB: Be aware of erratum to Section 2.4.)

References:

  1. H. Goldstein, Classical Mechanics; Section 1.4.
Qmechanic
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  • Thanks for at least agreeing I'm not making an obvious oversight. I will add that I actually first noticed this same issue in the derivation in Nolting's Analytical Mechanics book, and I went to Goldstein to try to clear it up to no avail. So at least two authors are saying something similarly unclear to me on this point. Greiner's and Scheck's books explicitly make the holonomic assumption at the start of the derivation so this issue cannot arise for them. Taylor and Morin don't cover d'Alembert's principle at all. – heranias Sep 25 '23 at 07:18