Hamilton's principle with nonholonomic constraints in Goldstein

Question

I am studying from Goldstein's Classical Mechanics, 3rd intl' edition, 2013. In section 2.4, he discussed Hamilton's principle with nonholonomic constraints. The constraints can be written in the form $$f_\alpha(q_1,...,q_n;\dot{q_1},...,\dot{q_n};t)~=~0\tag{2.24}$$ where $\alpha=1,...,m$. Using variational priciple, we get

$$\delta\int_{t_1}^{t_2}\left(L+\sum_{\alpha=1}^m \mu_\alpha f_\alpha\right)\mathrm dt=0 \tag{2.26}$$

where $\mu_\alpha=\mu_\alpha(t)$.

But how can he get the formula

$$\frac{d}{dt}\frac{\partial L}{\partial\dot{q_k}}-\frac{\partial L}{\partial q_k}=-\sum_{\alpha=1}^m \mu_\alpha \frac{\partial f_\alpha}{\partial\dot{q_k}} \tag{2.27}$$

for $k=1,...,n$ from the previous formula?

When I go through the steps as in section 2.3, I get $$\frac{dI}{d\beta}=\int_{t_1}^{t_2}\sum_{k=1}^n\left(\frac{\partial L}{\partial q_k}-\frac{d}{dt}\frac{\partial L}{\partial\dot{q_k}}+\sum_{\alpha=1}^m \mu_\alpha\left(\frac{\partial f_\alpha}{\partial q_k}-\frac{d}{dt}\frac{\partial f_\alpha}{\partial\dot{q_k}}\right)\right)\frac{\partial q_k}{\partial\beta}\mathrm dt$$ where $\beta$ denotes the parameter of small change of path: \begin{align} q_1(t,\beta)&=q_1(t,0)+\beta\eta_1(t)\\ q_2(t,\beta)&=q_2(t,0)+\beta\eta_2(t)\\ &\ \,\,\vdots \end{align} Using the same argument as in the part of holonomic constraint in section 2.4, I get $$\frac{\partial L}{\partial q_k}-\frac{d}{dt}\frac{\partial L}{\partial\dot{q_k}}+\sum_{\alpha=1}^m \mu_\alpha \left(\frac{\partial f_\alpha}{\partial q_k}-\frac{d}{dt}\frac{\partial f_\alpha}{\partial\dot{q_k}}\right)=0$$ for $k=1,...,n$.

What am I missing?

Answer 1

TL;DR: Note that the treatment of Lagrange equations for non-holonomic constraints in Refs. 1 & 2 is inconsistent with Newton's laws, and has been retracted on the errata homepage for Ref. 2. See Ref. 3 for details.

Longer explanation:

1. The main point of Goldstein's section 1.4 was to start from d'Alembert's principle (DAP) and derive Lagrange equations (LE) for holonomic constraints$^1$.

2. Therefore (although Goldstein does admittedly not state this clearly$^2$), the main point of section 2.4 should be to start from DAP and derive LE for affine non-holonomic constraints (=semi-holonomic constraints).

3. In fact, more generally, for independent non-holonomic one-form constraints $$\begin{align}\omega_{\ell}~\equiv~& \sum_{j=1}^na_{\ell j}(q,\dot{q},t)\mathrm{d}q^j+ a_{\ell 0}(q,\dot{q},t)\mathrm{d}t~=~0, \cr \ell~\in~&\{1,\ldots, m\}, \end{align} \tag{NH1C}$$ one may show that DAP leads to LE $$\begin{align} \frac{d}{dt}\frac{\partial T}{\partial \dot{q}^j}-\frac{\partial T}{\partial q^j}~=~&Q_j+\sum_{\ell=1}^m\lambda^{\ell} a_{\ell j}, \cr j~\in ~&\{1,\ldots, n\}.\end{align} \tag{LE}$$

4. Now Refs. 1 & 2 use instead independent non-holonomic constraints $$ f_{\ell}(q,\dot{q},t)~=~0, \qquad \ell~\in~ \{1,\ldots, m\}. \tag{NHC}$$ Eqs. (NHC) and (NH1C) are equivalent for affine non-holonomic constraints, but not in general.

5. Eq. (2.27) in Ref. 1 is essentially Chetaev's equations (CE) [5] $$ \begin{align}\frac{d}{dt}\frac{\partial T}{\partial \dot{q}^j}-\frac{\partial T}{\partial q^j}~=~&Q_j+\sum_{\ell=1}^m\lambda^{\ell}\frac{\partial f_{\ell}}{\partial \dot{q}^j}, \cr j~\in~& \{1,\ldots, n\}. \end{align}\tag{CE}$$ DAP plus affine non-holonomic constraints (where $\frac{\partial f_{\ell}}{\partial \dot{q}^j}$ has maximal rank) imply CE, but not for general non-holonomic constraints [6].

References:

1. H. Goldstein, Classical Mechanics, 3rd intl' ed, 2013; Section 2.4. Eq. (2.26) is wrong/misleading at best.

2. H. Goldstein, Classical Mechanics, 3rd ed, 2001; Section 2.4. Errata homepage. (Note that this criticism only concerns the treatment in the 3rd edition; the results in the 2nd edition are correct.)

3. M.R. Flannery, The enigma of nonholonomic constraints, Am. J. Phys. 73 (2005) 265.

4. E.J. Saletan & A.H. Cromer, A Variational Principle for Nonholonomic Systems, Am. J. Phys. 38 (1970) 892. Ref. 1 cites Ref. 4.

5. N.G. Chetaev, Izv. Fiz.-Mat. Obsc. Kaz. Univ. 6 (1933) 68. The Chetaev term $\sum_{\ell=1}^m\lambda^{\ell} \frac{\partial f_{\ell}}{\partial \dot{q}^j}$ is invariant under reparametrizations of the constraints $f^{\prime}_k= f_{\ell} M^{\ell}{}_k$ and $\lambda^{\ell}=M^{\ell}{}_k\lambda^{\prime k}$.

6. M.R. Flannery, D'Alembert-Lagrange analytical dynamics for nonholonomic systems, J. Math. Phys. 52 (2011) 032705; p. 22.

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$^1$ In this answer we will assume the commutativity rule $$\frac{d}{dt}\delta q^j=\delta \frac{d}{dt}q^j,\tag{CR}$$ cf. e.g. this related Phys.SE post.

$^2$ Goldstein confusingly refers to Hamilton's principle, which goes against the main paradigm of using Newton's laws as a first principle.