Poinsot’s criticism

Abstract

This chapter is devoted to Louis Poinsot’s criticisms toward Lagrange’s Méchanique analitique of 1788, assuming a reductionist approach. The first part introduces Poinsot’s mechanics which also takes account of the existence of constraints. Poinsot believes that VWLs have no interest for mechanics. The final part of the chapter is a report on the demonstration of a law of virtual work similar to that of Lagrange in which use is made of velocities and not infinitesimal displacements. Although Poinsot considers this demonstration only a trivial corollary of his mechanics, it had a remarkable success.

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Louis Poinsot was born in Paris in 1777 and died in Paris in 1859. He enrolled at the École polytechnique probably in 1794, without much mathematical background, and studied there for three years. The influence on him of Prony, who at that time was a professor at the École, was remarkable. Poinsot is generally considered as a minor figure in the history of mechanics, not comparable to the great ones: Euler, Lagrange, Laplace, Cauchy, etc. This may be true, but it does not mean that the attemption to understand the role of virtual work laws in mechanics, Poinsot’s position is not in the foreground. Among the scholars of some importance in the first half of the XIX century he was the only one who rather than enhancing the principle of virtual velocities, tried to prove that it was neither necessary nor useful for a coherent and efficient foundation of mechanics. According to him, once this mechanics was established, any demonstration of a virtual work law reduced to mere geometry. For this reason, and because his ‘demonstration’ has influenced most of the treatises of mechanics, I will dedicate a large space referring both to his work and to Bailhache’s interesting scientific biography [197]

.

Poinsot was not a prolific author; his main works reduce to:

• Éléments de statique 1803 (first edition) [195];

• Mémoire sur le compositions desmoments etla composition des aires, 1804 [193];

• Mémoire sur la théorie générale de l’équilibre et du mouvements des systèmes, 1806 [194];

• Mémoire sur la composition des moments en mécanique, 1804 [193];

• Remarque sur un point fondamentale de la Mécanique analytique de Lagrange, 1846 [198].¹

The most interesting of Poinsot’s contributions on virtual work principles is perhaps the Mémoire sur la théorie générale de l’équilibre, which was included in the latest editions of the Éléments de statiq ue. This work is derived from a review of a memoir, Sur la théorie générale de la mécanique, of the previous year which was read by Lagrange. The criticism was not completely in favor and demanded a radical revision. Poinsot accepted the request and sent a new draft to Lagrange. Lagrange sent it back to him, with a series of notes, but after it had already been published in the Journal de l’École polytechnique. Poinsot replied, even orally, point by point. The result of this discussion was that Lagrange realized the value of his interlocutor and had him appointed inspector general of the university. Poinsot was twenty nine years old.

In the following I will examine first an unpublished work entitled Considerations sur le principe des vitesses virtuelles of 1797, reported in full in [197], then the Mémoire sur la théorie générale de l’équilibre and its previous version, along with Lagrange’s annotations to it [197].

14.1 Considérations sur le principe des vitesses virtuelles

Poinsot wrote the Considérations sur le principe des vitesses virtuelles when he was twenty years old, and so when he had not yet fully developed his critical views on Lagrange’s virtual velocity principle. And, entering the École polytechnique cultural climate, he even provided a demonstration, largely following Prony’s approach (see Chapter 13). However, the critical insights that presage the development of Poinsot’s thought can already be seen.

In this regard, an interesting note was reported at the beginning of the work, when Poinsot introduced the virtual velocities:

Lines aa′, bb′, cc′, &c, are what scholars call the virtual velocities of the points a, b, c, & c., but if one wants to have only the value of the moment, one multiplies the force estimated for these lines in the direction of the force, i.e. projected onto them. To shorten, it is therefore convenient to call these projections themselves the virtual velocities [197].² (A.14.1)

This clarification indicates the attention Poinsot put on the virtual velocity concept, understood in the modern sense as a vector quantity.

Poinsot begins his demonstration of his version of the virtual work law taking for granted, as did Prony, the rule of composition of forces, but he does so in greater detail, as follows. Let P,Q and R be three forces in equilibrium on a plane, applied to a point a. Assume that a moves into a′ with an infinitesimal displacement. As demonstrated by Varignon, the static moments of the forces P and Q with respect to any point, for example a′, is equal to the static moment of the resulting R – or that of the balancing force with sign changed – evaluated for the same pole, and with reference to Fig. 14.1a , it is:

$$ Px+Qy+Rz=0, $$

(14.1)

Fig. 14.1

Reaction of a constraint

where x,y and z are the normal to the directions of P, Q and R conducted from a′. Indicating with p, q and r the components of aa′ on P, Q and R respectively, i.e. Bernoulli’s virtual velocities associated with aa′, it may be obtained easily:

$$ Pp+Qq+Rr=0 $$

(14.2)

In order to prove this, simply extend the lines x, y, z and build on them forces equal to P, Q, R, but rotated by a right angle and with origin in a′ (see Fig. 14.1b), for which now the normals are p, q, r. Because the equilibrium of P, Q, R persists also if they are rotated, from the balance of statics moments the relation (14.2) is obtained. This expresses the vanishing of the sum of the ‘moments’– Lagrange and Galileo terminology. The demonstration of Poinsot deserves attention because it shows the close analogy between Lagrange’s ‘moments’ and static ‘moments’. They derive from a different way to observe forces.

Poinsot then argues that the proof of the equation of moments is also valid in the case of any number of forces applied to a point and even for any number of free points, because a system of material points is in equilibrium if and only if all its points are in equilibrium

In the case of a system of constrained material points, the equation of moments is still valid, provided that in addition to the active forces P, Q, R, etc. also the reaction forces H, M, N, etc. and the corresponding virtual velocities h, m, n, etc., are considered, so it can be written:

$$ Pp+Qq+Rr+\mathrm{etc}.+Hh+Mm+Nn+\mathrm{etc}.=0. $$

(14.3)

Note the explicit introduction of the constraint reactions, perhaps according to the teachings of Prony. Poinsot in his later writings, will instead avoid the concept.

To prove the equation of moment in case of a body with ‘immutable distances’ (a rigid body) Poinsot bases his argument on the idea that there is equilibrium if (and only if) all the forces, as a result of translations, compositions and decompositions with the rule of the parallelogram can be reduced to only three forces – including constraint forces – balancing each other. These three forces should satisfy the law (14.2) which requires the vanishing of moments. Exploiting the fact that the total moment of forces does not change with the operations of translation and composition, Poinsot can then claim that the criterion of equilibrium of a rigid body is expressed by the annulment of the sum of all moments of forces acting on the body, including the constraint forces. But, when considering virtual displacements compatible with constraints, the moments of individual reactions, and hence their sum, are zero. The moments of the constraint forces applied to fixed points, because their virtual velocities are zero and “those of the forces normal to the resistance surface, are also zero because the projection of the infinitely small arc, described by the root of normal itself is zero”. Note that Poinsot, in his demonstration, in line with the scholars of the time, takes for granted the assumption of smooth constraints. Ultimately it can then be concluded that the criterion of equilibrium of a rigid body can be traced back to the annulment of the sum of the moments of the only active forces.

Instead of proceeding further, Poinsot warns: “We will also change the wording of the general principle of virtual velocities to avoid the idea of infinitely small motions and disturbance of the equilibrium, which are ideas foreign to the subject and leave something obscure in the spirit”. To clarify his position on the virtual displacement, I quote in full a note written on a separate sheet of the Considerations:

This will exclude the ideas of the infinitely small and disrupting the equilibrium; ideas that are alien to the subject, and the principle of virtual velocities appear as a simple theorem of geometry by ignoring those considerations that always leave something dark in the spirit. But it should be noted that this property of equilibrium that we study was discovered by means of these little velocity [motion], because those offer themselves naturally when you perturb a machine in equilibrium. It seems that through these movements the energies of the forces in motion of the machine are estimated. If a system is in equilibrium, you know the absolute value of each force, but not the effect it exerts on account of its position. Disturbing a bit the system to see what are the simultaneous movements that can take the points where forces are applied, some of these points are moving in the same direction of the forces, others are moving in the opposite direction, and the energy is evaluated as the product of forces by the velocity of the points of application, it is found that the energies that achieve their effect are the same as the energies overcome [197].³ (A.14.2)

Poinsot will succeed fully in order to eliminate the concept of virtual displacement only in the later works. Now, he limits himself to see under what conditions the equation of moments can be extended to the case of finite displacements. Besides the well-known examples of a straight lever and the inclined plane, Poinsot refers to results found by Fossombroni for parallel forces applied to the points of a line; cases that he generalizes by showing that the equation of moments is also true for finite displacements when the forces, parallel to each other, are applied to the points of a plane:

If a free system of invariable form is in equilibrium under all forces that are applied to it, assuming that all the forces acting at the junction of their directions with a plane situated at will, the equation of the moments will be valid whatever was the displacement of the system [197].⁴(A.14.3)

In addition to striving to eliminate the idea of infinitesimal virtual displacements, replacing these with the virtual velocities, Poinsot points to – or rather decides to follow in – Carnot’s footsteps, contending that the virtual velocities refer to changes of position occurring in a virtual time while the real time is frozen (i.e. that the virtual velocities and forces are not correlated, even when the forces depend on real motion). The balance is made with forces frozen at the instant in which the equilibrium should be studied:

It must be noted further that the system is supposed to move in any way, without reference to forces that tend to move it: the motion that you give is a simple change of position where the time has nothing to do at all [197].⁵ (A.14.4)

Toward the end of his text Poinsot writes: “It would therefore be futile to search for the metaphysics of the principle of virtual velocities and to endeavor to understand what they are in themselves the moments of the forces. Everything comes from the parallelogram of forces, where it is seen as the moments combine among them.”

Poinsot is not the only one who uses virtual velocity instead of virtual infinitesimal displacement. Fourier seems to put in the same plane the method of ‘fluxions’ (i.e. the velocity) and that of infinitesimal displacements. Poinsot, however, is the first to emphasize the need to use only virtual velocities, finally abandoning the infinitesimal displacements. In this he will be followed later by Ampère and Lagrange (in the second edition of the Théorie des fonctions analytique).

14.2 Théorie générale de l’équilibre et du mouvement des systèmes

The Théorie générale de l’équilibre et du mouvement des systèmes is much more mature than the previous text; it begins with historical considerations on virtual work laws, then develops a mechanical theory completely independent of it to finish by reducing the virtual velocity principle itself to a trivial theorem of ‘Geometry’. I will refer mainly to the edition of 1806, published in the XIII Chaier of the l’École polytechnique, but when it will be necessary I refer also to the text of 1805 and to the notes on the text reproduced in [197]. For the first part of the present section, which from certain points of view is the most interesting for what concerns the virtual velocity principle, I refer instead to the version of 1834 published in the Éléments de statique [195]. In it, historical references and comments to Lagrange’s work are much more extensive and interesting; the wide passage below clearly expresses Poinsot’s ideas with no need of comment:

The principle of virtual velocities was known for a long time as well as the majority of the other principles of mechanics. Galileo first noticed in the machines, the famous property of virtual velocities, that is the known relationship that exists between the applied forces and speeds that their points of application would take if the equilibrium of the machine should upset by an infinitely little amount. Johann Bernoulli saw in the full extent this principle that he enunciated with the great generality it has today. Varignon and the majority of the Geometers were careful to check it in virtually all matters of statics. And although there was no general proof, it was universally regarded as a fundamental law of the equilibrium of systems.

But up to Lagrange, the Geometers were oriented more to prove or to extend the general principles of science, than to obtain a general rule for problem solving, or rather they had not yet put this great problem, which alone represents all the mechanics. It was then a happy idea by relying on the principle of virtual velocities as an axiom and, without stopping to consider it in itself, to be concerned only to get a uniform method of calculation to derive the equations of motion and balance in all possible systems. Thus overcoming all the difficulties of mechanics, avoiding, so to speak, to address the Science itself, one transforms it into a matter of calculation, and this transformation, the objective of Mécanique analytique [Méchanique analitique], appeared as a striking example of the power of analysis.

Nevertheless, since, in this work one was at first careful only to consider this beautiful development of Mechanics, which seemed to derive everything from a single formula, it was believed that natural science was made and one just has to try to prove the principle of virtual velocities. But this research has highlighted the difficulties of the principle itself. This law is so general, where vague and strange ideas on infinitely small movements and the disturbance of the balance mix, does nothing but to become dark in his examination and Lagrange’s book is not giving anything more clear than the course of calculations, one sees well that the mists were not avoided in the way of mechanics, because they were, so to speak together, at the very origin of this science.

A general proof of the principle of virtual velocities has basically to put the entire mechanics on another basis, because the demonstration of a law that encompasses a science cannot be other than the reduction of this science to another law so general, but obvious, or at least easier than before, thus making it useless. So for the reason that the principle of virtual velocities contains all the mechanics, and that needs a thorough demonstration, it cannot serve as a primary basis. Trying to prove it on the basis of such a happy use has made is to try to go through this use; either finding some other law just as fruitful, but more clear, or founding on the principles of an ordinary general equilibrium theory, from which then the virtual velocities becomes just a corollary. So the state in which Lagrange had brought the science was not a demonstration of the principle of virtual velocities, which might be sought immediately. The Mécanique analytique [Méchanique analitique], as the author conceived it, is basically what it should be, and the demonstration of the principle of virtual velocities is not lacking at all, because if one tried to put it at the beginning of this book in a general and well developed way, the work would be made, i.e. this demonstration would include already all the mechanics.

It should therefore be considered that Lagrange placed himself with a single shot on one of the high points of science in order to discover some general rule to solve, or at least to put in the form of equations all problems of mechanics, and this objective has been fully achieved. But to form the science itself, one has to produce a theory that dominates equally all points of view. One needs to go straight, not to the obscure principle of virtual velocities, but to the clear rule that can be extracted from the solution of problems. And this natural and direct search, which alone can satisfy our spirit, is the main purpose of the memoir that is going to be read [195].⁶ (A.14.5)

The above introduction to the Théorie générale, contains an attack on the Méchanique analitique much stronger than that of the introduction to the same work in 1806, read by Lagrange. Poinsot argues that the principle of virtual velocities is obscure and unintuitive. The darkness comes from two factors: the first is contested by all opponents of the principle of virtual work, Stevin above all, that in the study of equilibrium it does not make sense to consider a perturbation, a virtual movement. The second factor concerns the nature of infinitesimals, the notion of infinitesimal was not so clear to Poinsot, or at least it was less clear than the concept of velocity that Poinsot will choose. For this ‘darkness’, according to Poinsot, the virtual velocity principle cannot be assumed as a principle of mechanics, and any attempt to prove it does not make sense because it means replacing this principle with another principle, equally general but more clear, which makes the virtual velocity principle unnecessary. Moreover, there is no practical advantage to introduce it.

Poinsot’s position is incomprehensible to a modern reader, accustomed to handbooks of mechanics based on a highly formalized approach, in which axioms fall from nothing and there is no need for justification other than success in explaining mechanical phenomena. At the beginning of the discussion the equations of Lagrange themselves are assumed [148]⁷– or those of Hamilton – often as axioms, and these equations are anything but intuitive. Poinsot’s position becomes clear when examined in the perspective of the epistemology of the XIX century, essentially in Aristotelian style. A principle must be self-evident, and even if it is not required with Aristotle, to be evident to the pure intellect, it must at least reflect the more immediate experience. According to Poinsot who embraces Aristotle’s opinion, a principle cannot be proved, otherwise it is not a principle; according to other scholars of his time, a principle is not necessarily obvious, but it can be proved and this can be done starting from metaphysical arguments – that is, with topics outside the science of which the ‘principle’ is a principle – or from within the discipline, but with very elementary arguments.

Poinsot’s criticism of the possibility to regard Lagrange’s virtual velocity law as a principle, therefore, appears to be partly reduced to a linguistic fact, all depending on what it is understood by ‘principle’. Of different values is instead the criticism for which the demonstrations reported until now are unsatisfactory, or that the principle of virtual work is neither simple nor fundamental. This seems unfair.

The proofs of Lagrange, Fourier and Carnot, who do not depart from the ‘usual’ principles of mechanics, are certainly very interesting. Lagrange connects the principle of virtual velocities to the pulley, Fourier to the lever. Carnot starts instead by the impact regarded as a phenomenon that can be characterized in a simple and obvious way. Even the demonstrations that originate from the usual principles of mechanics, such as Prony’s, do not seem less interesting than the demonstration reported by Poinsot himself. Then, in ease of use, if not in enunciation, the superiority of the virtual velocity principle compared to other seemingly simple principles is proved by its diffusion in treatises of mechanics. As to whether it is fundamental there seems to be no doubt, with some difficulty in dealing with friction forces, but, however, the criticism of Poinsot certainly was not referring to them.

14.2.1 Poinsot’s principles of mechanics

I now pass to the exposition of the Théorie générale de l’équilibre de systèmes, according to the text of 1806, beginning by enumerating the principles assumed. From this exposition it soon becomes clear that it would just turn against Poinsot the same criticism of vagueness that he ascribes to his colleague scientists, because he does not set clearly and definitively the principles he uses. Although one can give the excuse that some of them were already submitted in the Elements de statique of 1803, also the others do not seem so obvious.

The first principle, which Poinsot gives as well known, an axiom, is generally called the solidification principle. For this principle, if constraints, both internal and external, are added to a system of bodies in equilibrium, the equilibrium is not altered [194].⁸ The principle was used by Stevin, Clairaut and Euler for the study of fluids (by Lagrange, Laplace and Ampère in previous chapters) and will be used later by Piola and Duhem (see Chapter 17) to obtain the indefinite equilibrium equations of a three-dimensional continuum.

The second principle is presented as the fundamental property of equilibrium, it asserts that a necessary condition for the equilibrium of a system of bodies free from external constraints is that all the forces applied at various points can be reduced to any number of pairs of collinear forces equal and opposite to each other. The condition becomes sufficient for a body with invariant distances, i.e. a rigid body [194].⁹

The third principle is required by the second, even if not explicitly, and concerns the possibility to decompose a force into other forces with the rule of the parallelogram and to move a force along its line of action. It is embarrassing that Poinsot does not state explicitly this principle, which, perhaps, is the most important and complex. He evidently takes it for granted even though it is difficult to argue that it is inherently more intuitive than the virtual velocity principle. Just as it is not very intuitive to accept the second principle, for which the reduction of forces to a number of pairs of opposing forces to each other is a necessary condition for the equilibrium [194].¹⁰

The fourth principle concerns constrained material points moving on a surface and asserts the need of the orthogonality of the active forces to the surface for equilibrium:

In the equilibrium of systems, any force must be perpendicular to the surface of the curve on which its point of application would move if all the other points were considered as fixed [194].¹¹ (A.14.6)

A fifth principle concerns the mechanical superposition for constraints, that is if in a system of bodies or points there are more constraints, they are able to absorb the sum of the forces that each constraint is capable of absorbing separately [194].¹²

Before considering in detail the constrained systems of material points, Poinsot explores the consequences of the second and third principles, i.e. some necessary conditions for equilibrium. Without going into detail he imposes the equivalence between the applied active forces Pi and a system of a pair of forces equal and opposite, an R _ij agent along the line joining the material points ij. Notice that Poinsot is not using a principle of action and reaction, as for example Prony and Laplace did, but only takes an algebraic position because the lines joining the material points are in general only imaginary and do not represent rods, for example. For a system of n material points there are 3n – 6 possible connections and then 3n – 6 components R _i j to be considered as unknowns and 3n equations that express the equivalence between the active forces P _i and their decomposition R _ij . So there is a surplus of six equations which must be verified by the assigned active forces. Poinsot does not exhibit these equations of equilibrium, perhaps considering them irrelevant, perhaps because they are well exemplified in current treatises on mechanics as the cardinal equations of statics.

Fig. 14.2 illustrates the above for the case of four material points, where there are 3 × 4 – 6 = 6 connections. The four forces P ₁,P ₂ ,P ₃,P₄ are decomposed in the six – couples of – forces R ₁₂,R ₁₃,… ,R ₃₄, a priori unknowns. Among P _i and R _ij there are the twelve equivalency equations of the kind:

$$ \begin{array}{l}{X}_1={R}_{12}\cos {\alpha}_{12}+{R}_{13}\cos {\alpha}_{13}+{R}_{14}\cos {\alpha}_{14}\\ {Y}_1={R}_{12}\cos {\beta}_{12}+{R}_{13}\cos {\beta}_{13}+{R}_{14}\cos {\beta}_{14}\\ {Z}_1={R}_{12}\cos {\gamma}_{12}+{R}_{13}\cos {\gamma}_{13}+{R}_{14}\cos {\gamma}_{14}\\ \kern9em \cdot \cdot \cdot \\ {X}_4=-{R}_{14}\cos {\alpha}_{14}-{R}_{24}\cos {\alpha}_{24}-{R}_{34}\cos {\alpha}_{34}\\ \kern8.88em \cdot \cdot \cdot \end{array} $$

(14.4)

where α _ij , β _ij , γ _ij are the angles that the forces R _ij form with axes x, y, z respectively, and X _i , Y _i ,Z _i are the components of forces P _i along the same lines. By eliminating

Fig. 14.2

Decomposition of forces

R _ij the six cardinal equations of statics are achieved. Notice that the existence of solutions for R _ij is only a necessary condition for equilibrium, as stated by the second principle of Poinsot’s mechanics.

14.2.1.1 System of material points constrained by a unique equation

Then Poinsot passes to a more in-depth analysis of systems of constrained material points with the use also of his first and fourth principles. For the sake of simplicity he considers the system of four points shown in Fig. 14.3, the six mutual distances of which, indicated by m, n, p, q, r, s, are subject to the equation of constraint:

$$ L\left(m,n,p,q,r,s\right)=0. $$

Applying the principle of solidification, the equilibrium conditions on the external forces of a point, such as x ₁, do not change if the other three points are assumed as fixed. If m, n, p are the distances of x ₁ from the other three points, the condition of constraint takes the form, depending only on m, n, p:

$$ L\left(m,n,p,\overline{q},\overline{r},\overline{s}\right)=0, $$

(14.5)

in which the values of $\overline{q},\overline{r},\overline{s}$ are assigned.

Relation (14.5) defines a surface, whose normal at x ₁ has as components in the directions m,n,p the quantities L′(m),L′(n),L′(p), where the apex denotes the partial derivative with respect to the variable in parentheses. Applying the fourth principle, i.e. the hypothesis of smooth constraints, when the point x ₁ is in equilibrium, it is necessary that the external force applied to it has the direction defined by the components L′(m),L′(n),L′(p). The same holds true for the other points.

Consider now two points x ₁ and x ₃ of the system joined by the line m of Fig. 14.3. As mentioned above, for the equilibrium, the components of the forces applied to the two points x ₁and x ₃ in the directions that connect them to other points must have components in the form:

$$ \alpha {L}^{\prime }(m),\alpha {L}^{\prime }(n),\alpha {L}^{\prime }(p)\mathrm{and}\;\beta {L}^{\prime }(m),\beta {L}^{\prime }(q),\beta {L}^{\prime }(r) $$

(14.6)

respectively, where α and β are constants of proportionality. But for the second principle of Poinsot’s mechanics, it is necessary that the components of the forces in the direction m which joins x ₁ and x ₃ are equal and opposite, so it should be α = β. It can be concluded that in order to have equilibrium, the components of external forces in six directions should be proportional to the partial derivatives of L′(m),L′(n),L′(p),L′(q),L′(r),L′(s). This result extends to any number of points connected by a single condition of constraint.

Fig. 14.3

Constrained material points

After these considerations Poinsot encounters some difficulty in the use of the components of forces in the global reference frame. His difficulties come from having delivered the conditions of constraints by means of the distances between points rather than by means of their coordinates with respect to the coordinate system, as it would seem more natural, at least to a modern reader. The reasons for this are quite complex although may be not so interesting [197]. A few years later Cauchy [66] will resume the reasoning of Poinsot using constraint equations expressed by means of the coordinates of the points.

I avoid referring to those aspects that do not have a very important conceptual value and, without giving the proof I pass to exposing the first conclusion of Poinsot which ensures that to have equilibrium in a system of any number of particles, subject to a constraint of type L = 0, the components of the forces applied to each point x _i should be proportional to the quantities:

$$ \frac{\partial L}{\partial {x}_i},\frac{\partial L}{\partial {y}_i},\frac{\partial L}{\partial {z}_i}, $$

(14.7)

with (x _i ,y _i ,z _i ) the Cartesian coordinates of the i-th point.

This result was already obtained by Lagrange with a different approach. Moreover according to Poinsot, relations (14.7) provide the directions the external forces need to have so they are equilibrated, instead of according to Lagrange, being directions of constraint forces that rise for the equilibrium.

Developing derivatives of (14.7) it is then:

$$ \begin{array}{l}\frac{\partial L}{\partial {x}_i}=\frac{\partial L}{\partial p}\frac{\partial p}{\partial {x}_i}+\frac{\partial L}{\partial q}\frac{\partial q}{\partial {x}_i}+\frac{\partial L}{\partial r}\frac{\partial r}{\partial {x}_i}+\mathrm{etc}.\\ \frac{\partial L}{\partial {y}_i}=\frac{\partial L}{\partial p}\frac{\partial p}{\partial {y}_i}+\frac{\partial L}{\partial q}\frac{\partial q}{\partial {y}_i}+\frac{\partial L}{\partial r}\frac{\partial r}{\partial {y}_i}+\mathrm{etc}.\\ \frac{\partial L}{\partial {y}_i}=\frac{\partial L}{\partial p}\frac{\partial p}{\partial {z}_i}+\frac{\partial L}{\partial q}\frac{\partial q}{\partial {z}_i}+\frac{\partial L}{\partial r}\frac{\partial r}{\partial {z}_i}+\mathrm{etc}.\hbox{,}\end{array} $$

(14.8)

where p,q, r which represent the distances of the various points, should be considered as functions of their Cartesian coordinates x ₁, y ₁ ,z ₁ ,x ₂, etc. in a fixed frame of reference.

14.2.1.2 System of material points constrained by more equations

Poinsot can then turn to the case with more than one constraint condition:

$$ \begin{array}{l}L(m,n,p,q,r,s)=0\\ M(m,n,p,q,r,s)=0\\ N(m,n,p,q,r,s)=0\\ \kern4.56em \mathrm{etc}.\end{array} $$

(14.9)

In his words:

First, the mere fact that the points of the system are linked together by the first equation L = 0 the forces:

$$ \begin{array}{l}\lambda \sqrt{{\left(\frac{\partial L}{\partial x}\right)}^2+{\left(\frac{\partial L}{\partial y}\right)}^2+{\left(\frac{\partial L}{\partial z}\right)}^2}\\ \lambda \sqrt{{\left(\frac{\partial L}{\partial {x}^{\prime }}\right)}^2+{\left(\frac{\partial L}{\partial {y}^{\prime }}\right)}^2+{\left(\frac{\partial L}{\partial {z}^{\prime }}\right)}^2}\\ \lambda \sqrt{{\left(\frac{\partial L}{\partial {x}^{''}}\right)}^2+{\left(\frac{\partial L}{\partial {y}^{''}}\right)}^2+{\left(\frac{\partial L}{\partial {z}^{''}}\right)}^2}\\ \kern10em \&\mathrm{c}.\kern1em \\ \kern5.76em \end{array} $$

can be applied to them, where λ designates any undetermined coefficient and being each force perpendicular to the surface L = 0, when one considers the three coordinates of the point of application as the only variables.

Second, because the points of the system are linked together by the second equation M = 0, it is still possible to apply the respective forces:

$$ \begin{array}{l}\mu \sqrt{{\left(\frac{\partial M}{\partial x}\right)}^2+{\left(\frac{\partial M}{\partial y}\right)}^2+{\left(\frac{\partial M}{\partial z}\right)}^2}\\ \mu \sqrt{{\left(\frac{\partial M}{\partial {x}^{\prime }}\right)}^2+{\left(\frac{\partial M}{\partial {y}^{\prime }}\right)}^2+{\left(\frac{\partial M}{\partial {z}^{\prime }}\right)}^2}\\ \mu \sqrt{{\left(\frac{\partial M}{\partial {x}^{''}}\right)}^2+{\left(\frac{\partial M}{\partial {y}^{''}}\right)}^2+{\left(\frac{\partial M}{\partial {z}^{''}}\right)}^2}\\ \kern10em \&\mathrm{c}.\kern1em \\ \kern5.76em \end{array} $$

with μ a new indeterminate coefficient, and each of these forces being perpendicular to the surface represented by the equation M = 0, when the three coordinates of the point of application are considered as the only variables.

[…]

It is clear that there will be equilibrium on the basis of all these forces, because there would be equilibrium in particular in each group of each equation [194].¹³ (A.14.7)

Poinsot has implicitly accepted that more constraints working at the same time do not interact with each other and that the overall effect is the sum of individual effects (it is the fifth principle of his mechanics). He realizes that this fact is not very evident and tries to overcome a little below, in a passage that I do not refer for lack of space. He himself did not deem it good enough and then will return to this point in the Elements de statique since the eighth edition of 1841 (for further clarification on the issue see the work of Ampère on the virtual velocity principle examined in Chapter 13). Even Lagrange in the Théorie des fonctions analytique, made the same assumption of Poinsot on the superposition of constraints but he did not feel the need to justify the fact.

The Théorie générale ends with the following theorem:

Whatever the equations governing the coordinates of various points of the system are, for equilibrium, each of them requires that to these points are applied forces, along their coordinates, proportional to the derivatives of these equations with respect to these coordinates, respectively.

Thus, representing with L = 0, M = 0, etc. any equation between the coordinates x, y, z, x′, y′, z′, etc. of the different points, and with λ, μ, etc. any of the undetermined coefficients, the components of the forces that must be applied to these points should satisfy:

$$ \begin{array}{l}X=\lambda \left(\frac{dL}{dx}\right)+\mu \left(\frac{dM}{dx}\right)+\&\mathrm{c}.\\ Y=\lambda \left(\frac{dL}{dy}\right)+\mu \left(\frac{dM}{dy}\right)+\&\mathrm{c}.\\ Z=\lambda \left(\frac{dl}{dz}\right)+\mu \left(\frac{dM}{dz}\right)+\&\mathrm{c}.\\ {X}^{\prime }=\lambda \left(\frac{dL}{d{x}^{\prime }}\right)+\mu \left(\frac{dM}{d{x}^{\prime }}\right)+\&\mathrm{c}.\\ {Y}^{\prime }=\lambda \left(\frac{dL}{d{y}^{\prime }}\right)+\mu \left(\frac{dM}{d{y}^{\prime }}\right)+\&\mathrm{c}.\\ {Z}^{\prime }=\lambda \left(\frac{dL}{d{z}^{\prime }}\right)+\mu \left(\frac{dM}{d{z}^{\prime }}\right)+\&\mathrm{c}.\end{array} $$

If the indeterminate λ,μ, etc. are eliminated from these equations, there will remain the equilibrium equations themselves, i.e. the relationships that must take place between the applied forces and the coordinates of their points of application [194].¹⁴ (A.14.8)

The equations above allow the solution of the static problem. Given a set of forces X, Y, Z, X′, Y′, etc. verify whether the system of material points is in equilibrium in a given configuration x, y, z, x′, y′, etc. This can be made as Poinsot suggests, by solving the equations obtained by eliminating the indeterminate multipliers, or more simply, by considering that previous equations define a linear system of algebraic equations in λ, μ, etc. with known coefficients. The linear system may be determined, undetermined or overdetermined; if it admits at least a solution then the system of material points is in equilibrium.

Note that Poinsot is establishing the mechanics of constrained bodies without reference to the concept of constraint reaction though it was there accepted at the École polytechnique. He is even more rigorous than Lagrange and leaves no physical meaning to the ‘indeterminate’ ‘coefficients’ λ, μ, etc. that once known are generally interpreted as constraint forces.

The memoir ends with an interesting conclusion and two notes: the first regards the comments on the role of constraints, which is not particularly illuminating. The second note concerns the demonstration of the virtual work principle and is of great interest, which is why I quote it in full.

14.3 Demonstration of the virtual velocity principle

In the demonstration which follows, Poinsot replaces, for the first time unequivocally, the virtual displacements (infinitesimal) with virtual velocities, that he also calls ‘actual’ to emphasize that there are no assumptions of smallness. Poinsot declares he wants to prove Lagrange’s virtual velocity principle; actually, however, because of his use of velocity instead of infinitesimal displacements he is going to prove a slightly different principle, which however is still a virtual work principle. According to Poinsot, because this is almost an immediate consequence of the mechanics he has developed, which takes into account the role of constraints, and follows nearly immediately when the equations of equilibrium and constraints are written side by side, its proof has not a great value and also the principle in itself is of little interest. Personally I do not share Poinsot’s opinion and consider Lagrange’s proof very interesting and among the most cogent ever given.

Note II

Demonstration of the principle of virtual velocities. Identity of this principle with the general theorem object ofthe previous Memoir.

In the Memoir we have been content to observe that from the theorem on the expression of the general equilibrium of forces, one could easily switch to the principle of virtual velocities. But this principle is so famous in the history of mechanics that one cannot fail to point out a few words with these steps. I am very happy to do this, since the principle of virtual velocities is not only a corollary of the general proposition stated above, but I think even identical to it when one looks at it from his own point of view, and sets it out in a comprehensive manner. Let the system be defined by the following equations between the coordinates of the bodies:

$$ \begin{array}{l}f(x,y,z,{x}^{\prime },{y}^{\prime },{z}^{\prime },\&\mathrm{c}.)=0.\\ \phi (x,y,z,{x}^{\prime },{y}^{\prime },{z}^{\prime },\&\mathrm{c}.)=0.\\ \kern9em \&\mathrm{c}.\end{array} $$

Suppose to impress to all bodies any of the velocity that can actually occur without violating the terms of the constraints. The coordinates x,y,z,x′,y′,z′, &c., will vary with time t, of which they must be considered functions, and because the impressed velocities:

$$ \frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt},\frac{d{x}^{\prime }}{dt},\&\mathrm{c}. $$

(A)

could be admissible by the constraints, as supposed, it will be necessary that they satisfy the equations:

$$ \begin{array}{l}{f}^{\prime }(x)\frac{dx}{dt}+{f}^{\prime }(y)\frac{dy}{dt}+{f}^{\prime }(z)\frac{dz}{dt}+{f}^{\prime}\left({x}^{\prime}\right)\frac{d{x}^{\prime }}{dt}+{f}^{\prime}\left({y}^{\prime}\right)\frac{d{y}^{\prime }}{dt}+\&\mathrm{c}.=0\\ {\phi}^{\prime }(x)\frac{dx}{dt}+{\phi}^{\prime }(y)\frac{dy}{dt}+{\phi}^{\prime }(z)\frac{dz}{dt}+{\phi}^{\prime}\left({x}^{\prime}\right)\frac{d{x}^{\prime }}{dt}+{\phi}^{\prime}\left({y}^{\prime}\right)\frac{d{y}^{\prime }}{dt}+\&\mathrm{c}.=0\\ \kern22em \&\mathrm{c}.\end{array} $$

(B)

obtained from the previous (A) and it will be sufficient to ensure that they meet them so that the constraint conditions are met.

Now if one multiplies these equations for the undetermined coefficients λ, μ, &c. and adds, it follows that the velocities satisfy the sole following condition, no matter λ, μ, &c.

$$ \begin{array}{l}\left[\uplambda{f}^{\prime}\right(x)+\mu {\phi}^{\prime }(x)+\&c.]\frac{dx}{dt}+\left[\uplambda{f}^{\prime}\right(y)+\mu {\phi}^{\prime }(y)+\&c.]\frac{dy}{dt}+\\ \left[\uplambda{f}^{\prime}\right(z)+\mu {\phi}^{\prime }(z)+\&c.]\frac{dz}{dt}+\left[\uplambda{f}^{\prime}\right({x}^{\prime })+\mu {\phi}^{\prime }({x}^{\prime })+\&c.]\frac{d{x}^{\prime }}{dt}+\\ \kern6em \left[\uplambda{f}^{\prime}\right({y}^{\prime })+\mu {\phi}^{\prime }({y}^{\prime })+\&c.]\frac{d{y}^{\prime }}{dt}+\&\mathrm{c}.=0\\ \kern9em \end{array} $$

(C)

But the functions which multiply the velocities,

$$ \frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt},\frac{d{x}^{\prime }}{dt},\&\mathrm{c}. $$

are nothing but (after what has been proven) the general expressions of the forces which can be balanced on the system. Assuming therefore that the forces X,Y,Z,X′,Y′,Z′, &c., are effectively balanced, it is:

$$ X\frac{dx}{dt}+Y\frac{dy}{dt}+Z\frac{dz}{dt}+{X}^{\prime}\frac{d{x}^{\prime }}{dt}+{Y}^{\prime}\frac{d{y}^{\prime }}{dt}+{Z}^{\prime}\frac{d{z}^{\prime }}{dt}+\&\mathrm{c}.=0 $$

(D)

Instead of the three components X, Y, Z, multiplied for the corresponding velocities:

$$ \frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt} $$

it can be considered the resultant P, multiplied by the resulting velocity dx/dt, dy/dt, dz/dt, projected into the direction of P, which I will call ds/dt; the same can be done for the other forces, and it will be:

$$ P\frac{ds}{dt}+{P}^{\prime}\frac{d{s}^{\prime }}{dt}+{P}^{''}\frac{d{s}^{''}}{dt}+\&\mathrm{c}.=0. $$

That is, if the forces are in equilibrium on any system, the sum of their products for the velocities, one wants to give their bodies, whatever they may be, but allowed by their constraints, will always be zero, by estimating these velocities along the directions of forces. One can see from this, it is possible to take any velocities of finite value, which are measured by any straight lines that would be described simultaneously by the body if their links are suddenly broken and each of them run away freely toward their part.

Because of constraints among the bodies the velocities vary in each moment, when one wants to measure these velocities using the spaces themselves that the bodies actually describe, these spaces should be taken infinitely small, otherwise they no longer would measure the impressed velocities, and in this way one falls into the virtual velocities themselves, where the principle is to lose some of its clarity.

In fact it follows from what we have said, that this beautiful property of equilibrium can be stated as follows:

When the different bodies of a system run any of the movements which do not violate in any way the link established between them, i.e. the system is continuously in one of those configurations allowed by the constraint equations, it can be sure that the forces that will be capable of being balanced in these configurations, when the system passes in them, are such that multiplied by the velocity of the bodies projected onto their directions, the sum of all these products is necessarily equal to zero.

In this way, the principle no longer maintains any trace of the ideas of the infinitely small movements and disturbance of the equilibrium, which seem extraneous to the issue and leave some darkness in the spirit.

When there is equilibrium, it is clear that the principle holds for all the systems of velocities that the points could have, passing through the configuration that is considered. But, when one wants to start from the principle enunciated in such a way that it ensures the equilibrium one should require that it holds for this infinite number of velocity systems. There is a plethora of conditions, and it is possible to show that it is enough to say that the equation (D) must be verified for all systems of velocity allowed by the constraint equations (B) or (bringing together, as we did above, all these equations in one (C) by means of indeterminate λ, μ, &c.), it suffices to say that the equation (D) of the moments must be verified for all systems of velocities:

$$ \frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt},\frac{d{x}^{\prime }}{dt},\&\mathrm{c}. $$

But since, by definition, each of these systems of velocities must satisfy the equation (C), which amounts to say that all forces X ,Y, Z, X′,Y′,Z′, &c. that multiply the velocities:

$$ \frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt},\frac{d{x}^{\prime }}{dt},\frac{d{y}^{\prime }}{dt},\frac{d{z}^{\prime }}{dt},\&\mathrm{c}. $$

in the equation (D), have to be proportional to the functions:

$$ \begin{array}{l}\left[\uplambda{f}^{\prime}\right(x)+\mu {\phi}^{\prime }(x)+\&c.]\frac{dx}{dt},\left[\uplambda{f}^{\prime}\right(y)+\mu {\phi}^{\prime }(y)+\&c.]\frac{dy}{dt},\left[\uplambda{f}^{\prime}\right(z)+\mu {\phi}^{\prime }(z)+\&c.]\frac{dz}{dt},\\ \left[\uplambda{f}^{\prime}\right({x}^{\prime })+\mu {\phi}^{\prime }({x}^{\prime })+\&c.]\frac{d{x}^{\prime }}{dt},\left[\uplambda{f}^{\prime}\right({y}^{\prime })+\mu {\phi}^{\prime }({y}^{\prime })+\&c.]\frac{d{y}^{\prime }}{dt},\&\mathrm{c}.\\ \kern9em \end{array} $$

that multiply the same velocities as in the general equation (C), which requires for them the only conditions of constraints. So the principle of virtual velocities well set out, i.e. where all the ideas that one can make are clear: it is perfectly identical to the general theorem which is the subject of this memoir. I say exactly the same thing, namely that for the equilibrium, the components of the forces applied to bodies, by virtue of each equation must be proportional to the derivatives of these [constraint] equations with respect to these coordinates, which was to be proved.¹⁵

Moreover, it would have been taken to recognize this identity by a description of the ordinary principle of virtual velocities, by making well aware of the true meaning that it needs to be given. In fact, the general problem of statics is not only to seek the relationship between the forces which are in equilibrium, on the system, but rather [to seek] the general expression of the forces that may be continually equilibrated in any configurations where it can go under the constraint equations. The general equation given by the principle of virtual velocities is not, if I may speak so, the relation of an instant; it in no way should consider simply the equilibrium of the system in the configuration where it is, but also throughout the sequence of configurations where it can be, for it is this sequence of configurations that characterizes its definition [emphasis added] So the equation of moments does not say that one has to take the forces of a magnitude sufficient so that it is satisfied, but (since these forces must vary with the configuration) [it says] how one must choose these functions of the coordinates, so that the equation of moments remain continually satisfied. Now, under the constraint conditions themselves, one knows that between the velocities that the bodies can simultaneously have, it must apply the linear equation (C), the coefficients of which are the derivatives of functions given with respect to the coordinates by which this velocity is estimated. The equation of moments says that the forces of equilibrium must be represented by the derivative of these functions, therefore, to prove it, it is necessary to show how these forces are actually equilibrated or it must look directly for what functions of the coordinates can represent the forces of equilibrium, as we did from the beginning. This is why most of the demonstrations which trace back the principle of virtual velocities either to other principles or to the known law of some simple machine as the lever, &c., seem to us more justifications than real demonstrations. All in fact, even the happiest, that of Mr. Carnot, do not refer at all to the general definition of the system, as if the machine was, so to speak, voila, and one does not see anything but the ropes where the powers are applied. It may well be proved or made clear through some construction more or less simple that if one perturbs a bit the equilibrium, these powers must be in a relationship with the extensions allowed to its ropes, but this cannot provide that the current ratios forces considered as numeric values, and does not show at all its forms of expressions that are peculiar to them.

This disturbance of the balance would not know, in no event with which machine one has to do, and the same relationship between the applied forces, could occur even if the machines were of quite different constitution, and each of them, however, imposes to the expression of the forces that are generated, a different form that one should always see and find, if the difficulty of the theorem were fully resolved. So the property of virtual velocities remains not less mysterious, and there is no real demonstration. I mean an open and clear explanation, where one sees not only that it works well but that it is a consequence of the general definition of the considered system.

It is perhaps in a similar way, and to get the equation of moments as an equation identical that Mr. Laplace considered only the equations representing the link of the various parts of the system, and has, moreover, used other principles besides the composition of forces and the equality of action and reaction, which can be considered as elements of the equilibrium theory. As it is, after all, either one wants to start from the principle of the virtual velocities to follow its significance up to the end, or he directly attacks the problem of mechanics, which is simpler, one is conducted to look for the functions of the coordinates that give the forces of equilibrium in all the configurations that can be obtained in the system, in obedience to the relationships between the coordinates of the different bodies. This is exactly the problem we set ourselves, and our goal clear and distinct was to resolve it through the first principles of statics and geometry [194].¹⁶ (A.14.9)

I do not see that the text of Poinsot needs comment. On the basis of his mechanical theory, in which the role of constraints is clearly explained, and on the basis of his definition of virtual velocity, he can easily demonstrate a virtual work law which is a variant of Lagrange’s virtual velocity principle, both for its necessary and sufficient parts.

Poinsot maintains that the virtual velocity principle allows the study of the equilibrium not only in a given configuration: “but also in the entire sequence of configurations where it can be, for it is this sequence of configurations which characterizes its definition”. In such a way it can also lead to solution of another interesting problem of statics. Assigned a given configuration x,y,z,x′,y′, etc. find a set of forces X ,Y, Z, X′ ,Y′, etc. for with the equilibrium is satisfied. This can be made by solving the equations obtained from the virtual work law by eliminating the indeterminate multipliers.