I Virtual displacement is not consistent with constraints

[Kashmir]

Goldstein 3rd ed says

"First consider holonomic constraints. When we derive Lagrange's equation from either Hamilton's or D'Alembert's principle, the holonomic constraint appear in the last step when the variations in the $q_i$ were considered independent of each other. However, the virtual displacements in the $\delta q_{I}$'s may not be consistent with constraints. If there are $n$ variables and $m$ constraint equations $f_\alpha$ of the form Eq. (1.37), the extra virtual displacements are eliminated by the method of Lagrange undetermined multipliers.

I do not understand the parts that the virtual displacements may be inconsistent with constraints because earlier on in the book he defines virtual displacement as the infinitesimal change of the coordinates *consistent with the forces and constraints imposed on the system at the given instant $t$* (pg 16).

 

[vanhees71]

Goldstein's 3rd edition is utterly flawed in the issue of nonholonomic constraints. Rather read the 2nd edition, which is correct! It's an example for a very good textbook ruined by people (obviously not by the original autor) who think they have to modernize it ;-)).

You precisely nailed the error in your last paragraph. Indeed the nonholonomic constraints are constraints on the virtual displacements (no matter whether you use the d'Alembert or the action principle, and it's at the given instant of time, and time is not to be varied).

For more details clarifying the issues with Goldstein's 3rd edition, see

https://doi.org/10.1119/1.1830501

As you can read there, in the meantime the authors of Goldstein's 3rd edition have retracted their errorneous treatment on a webpage:

http://astro.physics.sc.edu/Goldstein/

 

[Kashmir]

Goldstein's 3rd edition is utterly flawed in the issue of nonholonomic constraints. Rather read the 2nd edition, which is correct! It's an example for a very good textbook ruined by people (obviously not by the original autor) who think they have to modernize it ;-)).

You precisely nailed the error in your last paragraph. Indeed the nonholonomic constraints are constraints on the virtual displacements (no matter whether you use the d'Alembert or the action principle, and it's at the given instant of time, and time is not to be varied).

For more details clarifying the issues with Goldstein's 3rd edition, see

https://doi.org/10.1119/1.1830501

As you can read there, in the meantime the authors of Goldstein's 3rd edition have retracted their errorneous treatment on a webpage:

http://astro.physics.sc.edu/Goldstein/

Thank you so much. I couldn't read the article in the journal and saw the errata. Could you please tell me are the virtual displacements consistent with the non holonomic constraints?
I found yet another article on arxiv that says that the virtual displacements can be in contrast with the constraints in non holonomic case. https://www.google.com/url?sa=t&sou...gQFnoECAMQAQ&usg=AOvVaw0RLIZuNZW0fswKBeqKr0o7

 

[vanhees71]

As far as I can see from glancing over the paper, there the statement about anholonomic constraints is correct (Fig. 1). It's a constraint on the "allowed" virtual displacements, which must be implemented by using Lagrange multipliers for the constraints in the form in the lower right corner of Fig. 1 to get the same correct equations of motion from Hamilton's principle in Lagrangian form as you get from d'Alembert's principle.