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So I'm doing some work from Taylor's mechanics book. He says for the problems in the book, we require the system to be holonomic - that is the number of generalized coordinates = number of Deg. of freedom. Why does this have to be the case?

I have been looking through his proof for a single particle, where he proves the Lagrangian, for the correct path taken by the particle, minimizes the action integral, but he doesn't say 'for this step in the proof to be true, we require the system to be holonomic'.

So why does this feature have to be true?

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  • $\begingroup$ I don't have this book, but you'll be more likely to get an answer if you give a page and equation number $\endgroup$ – zeldredge Sep 4 '15 at 17:05
  • $\begingroup$ How could the number of degrees of freedom be more than the number of generalized coordinates? Now, you can set up the problem so that the number of generalized coordinates is greater than the degrees of freedom, but to solve it you have to add constraints (which, ultimately reduce the number of coordinates to the number of degrees of freedom). $\endgroup$ – Jon Custer Sep 4 '15 at 17:18
  • $\begingroup$ Thanks for the replies.The proof is on pages 252/253 for anyone interested. I understand there are nonholonomic systems, but I don't understand where in his proof he uses the fact the system is holonomic $\endgroup$ – milanios Sep 4 '15 at 17:31
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    $\begingroup$ @JonCuster: A system can easily have $n$ degrees of freedom but require more than $n$ coordinates to describe. The standard example is a sphere rolling without slipping on a table in two dimensions; it requires five coordinates to fully describe (2 for position on the table, 3 for orientation of the ball), but it only has three degrees of freedom, and one cannot solve for any two of the coordinates in terms of the other three. $\endgroup$ – Michael Seifert Sep 4 '15 at 20:19
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  1. Actually, that the constraints are holonomic is not always sufficient. E.g. there could still be sliding friction.

  2. What is needed in the derivation of Lagrange equation's from Newton's laws is D'Alembert's principle, which we will write in the form$^1$ $$\sum_{i=1}^N {\bf F}^{(c)}_i\cdot \delta {\bf r}_i~=~0, \tag{1} $$ cf. Ref. 1, i.e. that the total virtual work of the constraint forces ${\bf F}^{(c)}_i$ on $N$ point particles at positions ${\bf r}_1,\ldots,{\bf r}_N, $ is zero.

  3. It is possible to show that broad classes of constraint forces of holonomic types satisfy D'Alembert's principle, see e.g. this Phys.SE post and links therein.

References:

  1. J.R. Taylor, Classical Mechanics, 2005; eq. (7.49).

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$^1$It is tempting to call eq. (1) the Principle of virtual work, but strictly speaking, the principle of virtual work is just D'Alembert's principle for a static system.

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