Virtual displacement is not consistent with constraints

In summary, Goldstein's 3rd edition discusses the issue of nonholonomic constraints and the elimination of extra virtual displacements through the use of Lagrange undetermined multipliers. However, this treatment is flawed and has been acknowledged by the authors themselves. Virtual displacements can be inconsistent with nonholonomic constraints, and proper implementation of Lagrange multipliers is necessary to obtain correct equations of motion.
  • #1
Kashmir
465
74
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).
 
Physics news on Phys.org
  • #2
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/
 
Last edited:
  • Like
Likes Kashmir
  • #3
vanhees71 said:
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
 
  • #4
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.
 

1. What is virtual displacement?

Virtual displacement is a concept used in mechanics to describe a hypothetical, infinitesimal change in the position of a system or object. It is used to analyze the behavior of a system under different conditions and to determine the equilibrium state of the system.

2. How is virtual displacement different from actual displacement?

Virtual displacement is a theoretical concept and does not involve any physical movement of the system or object. It is an imaginary change that is used to analyze the behavior of a system, while actual displacement refers to a real change in the position of the system or object.

3. What does it mean for virtual displacement to be consistent with constraints?

A system is said to be constrained when it is subject to certain limitations or restrictions, such as a fixed point of rotation or a restricted range of motion. Virtual displacement is considered consistent with constraints when it satisfies these limitations and does not violate any of the constraints placed on the system.

4. Why is virtual displacement not consistent with constraints?

In some cases, virtual displacement may not satisfy the constraints placed on a system. This can happen when the constraints are too restrictive or when the virtual displacement is not small enough to be considered infinitesimal. In such cases, the concept of virtual displacement may not accurately describe the behavior of the system.

5. How does the inconsistency of virtual displacement with constraints affect the analysis of a system?

If virtual displacement is not consistent with constraints, it means that the theoretical analysis of the system may not accurately reflect its behavior in reality. This can lead to incorrect predictions and conclusions about the system's equilibrium state and overall behavior. It is important to carefully consider the constraints and the validity of virtual displacement when using it in analysis.

Similar threads

Replies
17
Views
2K
Replies
7
Views
697
Replies
2
Views
731
Replies
3
Views
1K
Replies
1
Views
843
Replies
22
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
2K
Replies
4
Views
12K
Replies
13
Views
2K
Replies
1
Views
2K
Back
Top