7
$\begingroup$

I am studying from Goldstein's Classical Mechanics, 3rd edition. In section 2.4, he discussed Hamiltion's principle with semiholonomic constraints. The constraints can be written in the form $f_\alpha(q_1,...,q_n;\dot{q_1},...,\dot{q_n};t)=0$ 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?

$\endgroup$
  • 1
    $\begingroup$ 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). $\endgroup$ – Diracology Jul 8 '17 at 0:35
  • $\begingroup$ 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. $\endgroup$ – diracula Jul 8 '17 at 12:30
3
$\begingroup$

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the treatment of Lagrange equations for non-holonomic constraints in Ref. 1 is inconsistent with Newton's laws, and has been retracted on the errata homepage for Ref. 1.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

References:

  1. H. Goldstein, Classical Mechanics; 3rd ed; 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.)
  2. M.R. Flannery, The enigma of nonholonomic constraints, Am. J. Phys. 73 (2005) 265.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.