# Hamilton's principle with nonholonomic constraints in Goldstein

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?

• I am pretty sure that the derivation is not as straightforward as Goldstein might be suggesting. Note that if you follow your approach carefully you get (just one constraint for simplicity) $\frac{\partial L}{\partial q_i}-\frac{d}{dt}\frac{\partial L}{\partial \dot q_i}+\mu\frac{\partial f}{\partial q_i}-\frac{d}{dt}\left(\mu\frac{\partial f}{\partial\dot q_i}\right)=0$, since the multiplier depends on time. This means you should know the time derivative of $\mu$ but this does not make much sense (at least for me). Jul 8, 2017 at 0:35
• As far as I can tell, your phrasing of the constraints is as in the third edition, but the result you quote seems to be as in the second edition (it seems to be equation 2.27 in the second edition rather than the third). The treatments (in particular the kinds of constraints considered) in the two cases seem to me to be actually very different, and that the confusion comes from mixing the two. In particular, in the third edition, where they use the constraints you use, they give the result that @Diracology has suggested. Jul 8, 2017 at 12:30

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)  \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 .

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.

• Idea for later: Consider Hamiltonian Lagrangian $L_H=\sum_{j=1}^np_j\dot{q}^j -H(q,p,t) +\sum_{\ell=1}^m\lambda^{\ell} f_{\ell}(q,p,t)$. Sep 13, 2022 at 14:29
• Notes for later: Non-holonomic constraints $f_{\ell}(q,\dot{q},t)=0$ with pertinent rank condition can be brought on normal form $\dot{q}^{\ell}-g^{\ell}(q,\text{other }\dot{q},t)=0$. It is not clear how to apply d'Alembert's principle. Normal form $\dot{q}^{\ell}-g^{\ell}(q,\text{other }\dot{q},t)=0$ is not equivalent to one-form $\mathrm{d}q^{\ell}-g^{\ell}(q,\text{other }\dot{q},t)\mathrm{d}t=0$. If they were, it would also violate Chetaev eqs. Sep 14, 2022 at 13:52