Talk:D'Alembert's principle

couple of things

The last two sentences in the first par distract from the main narrative, which is very much overview at this point, and would best be pushed further down in the article.

The text describing the formula omits needs to mention the product with the virtual displacements. It sort of forgets to.

Will leave to others with more ownership of the article to consider. — Preceding unsigned comment added by 124.170.43.155 (talk) 00:37, 21 April 2014 (UTC)

I think that the last two sentences in the first section sum up the main contribution of d'Alembert's Principle, and the rest of the article fails to establish the main points of what he was doing. The whole concept of generalized coordinates is that when constraints are imposed on the analysis of a problem, the number of degrees of freedom for the analysis can be reduced by the number of imposed constraints. If there are N particles, there would in general to 3*N degrees of freedom. With k constraints, the number of degrees of freedom required to describe the problem are 3*N-k, and that is the number of generalized coordinates needed to describe the system.

Part of what makes d'Alembert's Principle work is that the constraint degrees of freedom do no work... because there is no motion in the direction of the constraint. That is the nature of a constraint, as assumed in this form of analysis. It is a beautiful simplification which means that the analyst does not need to solve for every reaction (constraint) force in a system in order to analyze the system dynamics.

Of course for statics analysis where the analyst is interested in the static loads in order to size members, reaction forces and moments are very important to be able to solve for. But for dynamics analysis, it is sufficient to assert that the reaction forces and moments for the constraints are simply 'as large as they need to be in order to impose the constraint'.

This article is very MUDDLED because it does not recognize the importance of the concept of generalized coordinates in dynamics analysis, and instead gets into using the principle of virtual work for solving statics problems. This article will leave people more confused after reading it rather than adding clarity.  :-(

Jacob Linder at the Norwegian University of Science and Technology has a series of lectures posted on YouTube that give a very thorough treatment of this material.

https://www.youtube.com/watch?v=Pw8tk3k9o7U

PoqVaUSA (talk) 15:04, 5 February 2015 (UTC)

First equation

What is r_i in the first equation? r_i are the components of position vector of the ith particle of the system. —Preceding unsigned comment added by 117.200.61.215 (talk) 13:39, 4 October 2009 (UTC)

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 $Fc_{i}$ 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)

There are two things going on here that are at cross purposes with each other. One is that d'Alembert's principle and Lagrange's equations, along with the concept of generalized coordinates, allow for analysis of motion without needing to solve for the forces and moments provided by each constraint on the system. For example, analysis of a cable and pulley system for operating the controls for a small aircraft can be done without calculating the reaction forces at each pulley. On the other hand, if you want to size the pulleys and the supporting structures for the pulleys, you may wind up with too many degrees of freedom to solve that problem without taking into account stiffness and deformation of the supports, hence the need for considering small deformations in directions besides the direction that the cable traverses. TWO DIFFERENT PROBLEMS. The first problem explicitly excludes consideration of any deformation in the direction of a constraint (that would be a violation of the constraint). The second one maybe can't be solved without imposing some small deformation 'in violation of the notional constraint'.

So, it's a bit like the beer commercial. Is it less filling, or does it taste great, or both? The simplification of the dynamics problem benefits from the assertion that no work is done in the direction of the constraints, and removes those degrees of freedom from the analysis. That won't let you size the structures for the constraints, but that's a different problem. (See link above for YouTube lectures by Jacob Linder.)

PoqVaUSA (talk) 15:44, 5 February 2015 (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.

If you take two particules P1 and P2, and consider a virtual displacement such that P1 moves (lets denote u1 that displacement) and not P2 (u2=0), if you denote Ai->j the force exerced by P1 on P2 you have W= u1*A2->1 + u2*A1->2= u1 *A2->1 + 0 != 0. like Thurth said the constraint vanishes _only_ if ones is considering a rigid body motion. By the way that principle seems to be very near of the principle of virtual power wich states :

$\left\{{\begin{matrix}\forall v^{\star }\in \varepsilon ,\\{\mathcal {P}}_{a}(v^{\star })={\mathcal {P}}_{e}(v^{\star })+{\mathcal {P}}_{i}(v^{\star })\\\forall v_{R}^{\star }\in \varepsilon _{R},\\{\mathcal {P}}_{i}(v_{R}^{\star })=0\end{matrix}}\right.$

where $\varepsilon _{R}$ is the set of the virtual rigid movements and $\varepsilon$ the one of the virtual movements -- Drébon (talk) 17:55, 13 July 2008 (UTC)

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)

It seems that the principles are valid for non rigid body constraints. BUT nobody has shown that in the non rigid body case the sum of the virtual work done by constraints vanishes. This is the main simplification this principle allows! So the given derivation in the article ist just a special case. It can be shown that for those special cases, D'Alembert is equivalent to Newtonen, due tho orthogonal virtual displacments. Snake707-de.wikipedia --87.145.69.208 (talk) 21:03, 3 June 2010 (UTC)

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. United States of America: CBS College Publishing. ISBN 0-03-063366-4.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)

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)

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)

I'm not sure the form of the principle stated in the article there is conform, but if you take it as the principle of virtual power (see my above remark or http://fr.wikipedia.org/wiki/Principe_des_puissances_virtuelles) you can derive Newton's law and much more from that principle Drébon (talk) 18:00, 13 July 2008 (UTC)

Sorry guys, but its just wrong to state, that D'Alembert's principle is equivalent to Newtons Second Law. This misses important simplifications, such that the sum of the virtual work, vanishes. Snake707 -- de.Wikipedia --87.145.69.208 (talk) 21:00, 3 June 2010 (UTC)

Usually the virtual displacements in d'Alemberts principle $(F-ma)\delta r=0$ are along constrained lines. However, for an unconstrained (=free) particle, there are no restrictions on the $\delta r$. For the equation to be valid for all $\delta r$, the equation $F-ma=0$ must hold, which is just Newton's second law. I would not really call this an "equivalence" but sure the two are related. --Dogbert66 (talk) 15:28, 14 April 2012 (UTC)

What is the "it" in the sentence "And it is true only for some very special cases e.g. rigid body constraints." The grammar implies that d'Alembert's principle is true only for some very special cases. If the intent is to say that the equivalency to Newton's second law is true only for some very special cases, then the intent has failed. Either way, this ambiguity should be clarified and resolved. (Given the arguments above, this entire section could use some editing.) — Preceding unsigned comment added by IAmGaryAmI (talk • contribs) 11:10, 24 August 2018 (UTC)

Well, d'Alembert's principle of virtual work is equivalent to Newton's laws, as can be seen from the fact that they both lead to the exact same equations of motion. In fact, one learns in a course on advanced classical mechanics that d'Alembert's principle leads to (a version of) the Euler-Lagrange equations. Nerd271 (talk) 03:29, 24 June 2020 (UTC)

Danger ... Derivation

Hello,

my textbooks says, that the principle of D'Alembert can only be derived in very easy cases e.g. a mass particle, or a rigid body. The problem in deriving the principle for not rigidly bound masses is the anistropy of the configuration space. The problem can be "solved" by approximation with rational masses.

Source: Eckhard Rebhan: Theoretische Physik: Mechanik, section 5.4. "Exkurs: Ableitung des D'Alembertsche Prinzips"

mfg —Preceding unsigned comment added by 87.145.74.41 (talk) 20:26, 13 March 2010 (UTC)

Vanishing forces of constraint

The introduction states

D'Alembert's contribution was to demonstrate that in the totality of a dynamic

system the forces of constraint vanish. That is to say that the generalized forces :{\mathbf Q}_{j} need not include constraint forces.

Can someone provide (a link to) this demonstration? If the forces of constraint vanish in the totality of the dynamic system, one could express this as $\mathbf {C} _{i}=0$ and enter this in the section Derivation_for_special_cases. 82.92.241.131 (talk) 07:03, 8 September 2010 (UTC)

History

My professor (who will rename nameless, but is a venerable scholar in physics) said that Newton already derived this principle, and was referred to as his "4th law". (more specifically, he claimed that Newton lumped what are usually called the 2nd and 3rd law together, and this was the orginal 3rd law, and later became a "4th"). This makes sense since it is an additional principle for dealing with constraints, therefore it must be considered a fundamental law (or axiom) in Newtonian mechanics. He said he read about this in a book by Lord Kelvin, and that later authors brushed it aside as being "not as fundamental" as the other 3 laws. I did a quick search and couldn't find any information, but I figured I'd mention it. Perhaps someone could look into this. Danski14^(talk) 20:39, 8 September 2010 (UTC)

Geometry

D'Alembert's principle of virtual displacements opens a way to find orbits of mechanical systems under constraint qualifications. If there are no constraints then it says only that the zero vector is the only vector perpendicular to all other vectors. Suppose that the original mechanical systems has n degrees of freedom and that there are m < n functionally independent holonomic-scleronomic constraints in implicit form, then these constraints have in every point m linear independent normals and n - m linear independent tangents which are perpendicular to the normals. These tangents are the virtual displacements of the principle. The vector of Newtons equations of motion must be perpendicular to all tangents. In Lagrange's multiplier method, this vector must be a linear combination of the normals (constraint forces) and the factors of the linear combination are the multipliers. In the first case, we obtain a system of n-m differential equations and m "algebraic" eqations (the constraints) for n variables, and a system of n differential equations and m "algebraic" equations for n+m variables in the second case. Therefore, the result is a differential-algebraic system in any case. Both methods are dual to each other in geometrical sense by a simple result on the range of a matrix. --E.W.Gekeler (talk) 17:16, 17 June 2013 (UTC)

• Eckart W.Gekeler: "Mathematical Methods for Mechanics." Springer-Verlag, Heidelberg 2008, ISBN 978-3-540-69278-2, e-ISBN 978-3-540-69279-9

In my opinion this analysis is correct, and shows that the following statement is wrong: "To date nobody has shown that D'Alembert's principle is equivalent to Newton's Second Law. This is true only for some very special cases e.g. rigid body constraints. However, an approximate solution to this problem does exist."Prof McCarthy (talk) 23:40, 6 July 2013 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:D'Alembert's principle/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Geometry guy 20:36, 1 June 2007 (UTC) It seems the word "work" is missing. It is not clearly explained why Lagrange should be credited. What means variational? Is it about small variations?

Last edited at 20:36, 1 June 2007 (UTC). Substituted at 12:39, 29 April 2016 (UTC)

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General or special?

"D'Alembert's principle is a more general case . And it is true only for some very special cases..." General, but only works for special cases. I think this could be expressed in a less confusing way. --Inc (talk) 21:12, 18 March 2019 (UTC)

Confusing formulation of the principle

The statement of the principle in the beginning is unclear and confusing in several ways. First, the equation after the statement is different from what the statement claims it to be: the momenta and the constraints of the system which are in the text are missing in the equation. Second, inertial forces, the essence of d'Alembert's principle, are missing altogether. Third, the statement speaks about projection, which is a geometrical term. There is a whole discipline Projective Geometry. Here, nobody can tell how momenta are projected on virtual displacements. Fourth, virtual displacements require much explanation and sending the reader to Virtual displacement with heavy analytical mechanics most of which doesn't have anything in common with d'Alembert's principle is a big distraction and obstacle to understand the principle itself. If virtual displacement is needed in further discussion it should be explained briefly and in relation to the topic.

I'll wait for some time for someone to convince me that this text can stay as it is, and then shall replace it with a clearer and more exact statement. Lantonov (talk) 13:18, 31 October 2020 (UTC)

Using the Lagrangian?

We sometimes encounter a formulation of d'Alembert principle that uses the Lagrangian. Instead of writing $$\sum _{i}(\mathbf {F} _{i}-m_{i}{\dot {\mathbf {v} }}_{i}-{\dot {m}}_{i}\mathbf {v} _{i})\cdot \delta \mathbf {r} _{i}=0,$$ it is written as Hamilton's principle, but for virtual changes $\delta \mathbf {r} _{i}$ of the $i$-th particle that are consistent with the constraints. It is more general than the form used above. Is it a form that was also proposed by d'Alembert or is it a misnomer? Discussion is welcome. AntonyKers (talk) 14:29, 2 March 2023 (UTC)