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The confusion for me comes from page 46 of Goldstein, where he says

"However, it has been proven that no such varied path can be constructed unless [the differential equations of constraint] are integrable, in which case the constraints are actually holonomic."

In the second edition, he gives a citation in a footnote. He continues:

"A variational principle leading to the correct equations of motion can nonetheless be obtained when the varied paths are constructed from the actual motion by virtual displacments."

I have consulted other numerous texts (Fetter and Walecka, Lanczos, Corben and Stehle, Marion and Thornton, Symon, etc., etc.) as well as lecture notes from the web in an effort to get this straight in my head. If anyone can help clarify this issue for me I would greatly appreciate it.

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The type of non-holonomic constraints, that Ref. 1 is discussing at this point, are so-called semi-holonomic constraints $$ a_{\ell}(q,\dot{q},t)~\equiv~ \sum_{j=1}^n a_{\ell j}(q,t)\dot{q}^j+a_{\ell t}(q,t)~=~0, \qquad \ell~\in \{1,\ldots, m\}. \tag{1}$$ The upshot is the following:

  1. On one hand, it is possible to use d'Alembert's variational principle to incorporate semi-holonomic constraints (1) into the Lagrange equations with the use of Lagrange multipliers $\lambda^1,\ldots ,\lambda^m$, cf. this Phys.SE post. Note in particular that there is no stationary action principle associated with this first case.

  2. On the other hand, if one tries to consider the extended stationary action principle for the action $$\widetilde{S}[q,\lambda]~=~\int\! dt ~\widetilde{L}(q,\lambda,\dot{q},t),$$ $$ \widetilde{L}(q,\lambda,\dot{q},t)~\equiv~ L(q,\dot{q},t)~+~\sum_{\ell=1}^m\lambda^{\ell}a_{\ell}(q,\dot{q},t), \tag{2}$$ with Lagrange multipliers $\lambda^1,\ldots ,\lambda^m$ for the semi-holonomic constraints (1), the corresponding Euler-Lagrange (EL) equations are inconsistent with the underlying Newton's laws, cf. e.g. my Phys.SE answer here.

Note that the Lagrange multipliers play slightly different roles in the two cases.

References:

  1. Herbert Goldstein, Classical Mechanics, Chapter 1 and 2.
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