Talk:D'Alembert's principle

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[edit] First equation

What is ri in the first equation?

[edit] constraint forces

The article says about constraint forces:

D'Alembert should be credited with demonstrating that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces {\mathbf F}_{i} need not consider constraint forces.

But that would only be the case e.g. the constraint being a fixed rod between two mass points (completely insided the system). But analytical mechanics also considers constraints like a mass point connected by a rod to "the laboratory". In that case the sum of constraint forces don't vanish but for the constraint forces Fci it still holds

(1) \sum_{i}{\mathbf {Fc}}_{i} \cdot \delta{\mathbf r}_{i} = 0.

And therefore they can be still left out from the equation

(2) \sum_{i}\left({ {\mathbf F}_{i} - \dot {\mathbf p}_{i} }\right) \cdot \delta{\mathbf r}_{i} = 0.

In Landau/Lifschitz and some German textbooks , (1) is called the d'Alembert's principle.

Pjacobi 18:28, 27 May 2007 (UTC)

It seems to me that the article, in its derivation of d'Alembert's Princlple, needs to be more explicite in mentioning that the constraint forces involve internal forces between particles which are of action and reaction type, and are not necessarily perpendicular to the virtual displacements. Otherwise, the reader may falsely assume that only the constraint forces perpendicular to the virtual displacement are involved. This is important. Because, after the removal of constraint forces, the difference between applied forces and inertial force may not be equal to 0 for each particle. Thurth 06:01, 30 September 2007 (UTC) —Preceding unsigned comment added by Thurth (talk • contribs)
I could be wrong, but I think that any force acting in the direction of displacement contributes work, even if it is an internal reaction force. However I also think this work would be canceled out by an equal and opposite reaction on another particle. Also, this discussion doesn't make much sense to me in the discussion of the movement of particles, whose collisions (ignoring interacting fields attached to particles) take place over an infinitesimal distance (and also instantaneously).ChrisChiasson 02:53, 1 October 2007 (UTC)

Consider any two particles with positions \mathbf{r}_i,\mathbf{r}_j in the system, the rigid body internal force constraint is

(r_i-r_j)^2=c_{ij}^2.

Therefore, the virtual displacement should satisfy the constraint

(\mathbf{r}_i-\mathbf{r}_j)(\delta\mathbf{r}_i-\delta\mathbf{r}_j)=0.

So there are 2 possibilities:

  1. \delta\mathbf{r}_i=\delta\mathbf{r}_j: In this case, \delta\mathbf{r}_i\cdot \mathbf{F}_{ij}=-\delta\mathbf{r}_j\cdot \mathbf{F}_{ji}. The total work done by the two forces is 0.
  2. (\mathbf{r}_i-\mathbf{r}_j)\perp(\delta\mathbf{r}_i-\delta\mathbf{r}_j)=0.: Since \mathbf{F}_{ij}\ \|\ \mathbf{F}_{ji}\ \|\ (\mathbf{r}_i-\mathbf{r}_j), the total work done by the two forces is again 0.

So the total work done by the two opposing forces is 0.

The principle of virtual work and d'Alembert's principle are valid when under rigid body constraint, which was the condition imposed in Goldstein's book "Classical Mechanics". When rigid body constraint is not imposed, are the two principles still valid?Thurth 06:11, 1 October 2007 (UTC)

The virtual diaplacement can not be assumed to be orthogonal to the constraint forces. If there is any possible virtual displacement not orthogonal to any constraint force, there is virtual work done by the constraint force. Then, the constraint force term can not be eliminated Thurth 07:01, 27 October 2007 (UTC) —Preceding unsigned comment added by Thurth (talk • contribs)

[edit] name

The name of this person is written as d'Alembert in this article. Can anyone please confirm if this the appropriate way to write the name. I have seen several text books where the capital D is used. Ref : A first course in continuum mechanics Y.C.Fung . Pjacobi could you please take a look at Ch.1 page 8 Mathematical Formulation of Physical Problems in the book mentioned earlier. I believe you are stating correct.

—Preceding unsigned comment added by 129.240.21.182 (talk) 13:30, 3 September 2007 (UTC)

It is also spelled with a capitol D in my text: Torby, Bruce (1984). "Energy Methods", Advanced Dynamics for Engineers, HRW Series in Mechanical Engineering (in English). United States of America: CBS College Publishing. ISBN 0-03-063366-4. :269 ChrisChiasson 16:05, 27 September 2007 (UTC)

In Goldstein's book "Classical Mechanics", the person's name is D'Alembert, and the principle is called D'Alembert's Principle. However, in Beer & Johnston's book "Vector Mechanics for Engineers", the name is d'Alembert. I guess, to make sure the usage is correct, you need to consult with a French specialist. Thurth 05:34, 1 October 2007 (UTC) —Preceding unsigned comment added by Thurth (talk • contribs)
Is there some kind of process that can be requested for this?ChrisChiasson 15:03, 2 October 2007 (UTC)

[edit] WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as stub, and the rating on other projects was brought up to Stub class. BetacommandBot 09:47, 10 November 2007 (UTC)

[edit] Equivalence between D'Alembert's priciple and Newton's 2nd Law

Having been derived from Newton's second law, D'Alembert's principle is a refinement of Newton's second law. How can it be equivalent to Newton's second law?--Thurth 04:13, 17 November 2007 (UTC)