The general law of equilibrium

1. The general law of equilibrium in machines is that forces or powers are reciprocally proportional to the velocities of the points where they are applied, estimated according to the direction of these powers:

This law constitutes what is commonly called the principle of virtual velocities, a principle long recognized as the fundamental principle of equilibrium, as we have shown in the previous Section, and which can, consequently, be regarded as a kind of axiom of Mechanics.

To reduce this principle to a formula, let us suppose that powers

… directed along given lines, are in equilibrium. Let us conceive that, from the points where these powers are applied, straight lines equal to

… are drawn and placed in the directions of these powers; and let us denote, in general, by

… the variations or differences of these lines, insofar as they can result from any infinitely small change in the position of the different bodies or points of the system.

It is clear that these differences will express the spaces traversed in the same instant by the powers

… according to their own directions, supposing that these powers tend to increase the respective lines

…

The differences

… will thus be proportional to the virtual velocities of the powers

…

and may, for greater simplicity, be taken for these velocities.

That established, let us first consider only two powers

… in equilibrium. By the law of equilibrium between two powers, it will be necessary that the quantities

… be to each other in inverse ratio to the differentials

…

but it is easy to conceive that there could be no equilibrium between two powers, unless they are disposed in such a way that, when one of them moves according to its own direction, the other is constrained to move in a direction opposite to its own; from which it follows that the values of the differences

…

must be of opposite signs; therefore, the values of the forces

…

being supposed both positive, one will have, for equilibrium,

2. Thus we have, in general, for the equilibrium of any number of powers ( \mathrm{P, Q, R, \dots} ) directed along the lines ( p, q, r, \dots ) and applied to any system of bodies or points arranged among themselves in any way whatsoever, an equation of this form:

[ \mathrm{P} , dp + \mathrm{Q} , dq + \mathrm{R} , dr + \dots = 0. ]

This is the general formula of Statics for the equilibrium of any system of powers.

We shall call each term of this formula, such as ( \mathrm{P} , dp ), the moment of the force ( \mathrm{P} ), taking the word moment in the sense that Galileo gave it, that is to say, for the product of the force multiplied by its virtual velocity; so that the general formula of Statics consists in the sum of the moments of all the forces being equal to zero.

To make use of this formula, the difficulty reduces to determining, according to the nature of the given system, the values of the differentials ( dp, dq, dr, \dots ).

We will therefore consider the system in two different and infinitely close positions, and we will seek the most general expressions of the differences in question, by introducing into these expressions as many undetermined quantities as there are arbitrary elements in the variation of the position of the system.

We will then substitute these expressions of ( dp, dq, dr, \dots ) into the proposed equation, and this equation must hold, independently of all the indeterminates, in order for the equilibrium of the system to subsist in general and in all directions.

We will therefore set separately equal to zero the sum of the terms affected by each of the same indeterminates, and we will thus obtain as many particular equations as there are indeterminates.

Their number must always be equal to that of the unknown quantities in the position of the system; therefore, by this method, we will obtain as many equations as are needed to determine the state of equilibrium of the system.

This is how all authors who have so far applied the principle of virtual velocities to the solution of problems in Statics have proceeded.

But this manner of employing this principle often requires constructions and geometric considerations which make the solutions as long as if they were deduced from the ordinary principles of Statics.

This is perhaps the reason that has prevented this principle from being valued and used as much as it seems it should have been, given its simplicity and generality.

3. The object of this Work being to reduce Mechanics to purely analytical operations, the formula we have just derived is very suitable for fulfilling it.

It is only a matter of analytically expressing, in the most general way, the values of the lines ( p, q, r, \dots ) taken in the directions of the forces ( \mathrm{P, Q, R, \dots} ), and we will obtain, by simple differentiation, the values of the virtual velocities ( dp, dq, dr, \dots )

One must only be careful that, in Differential Calculus, when several quantities vary together, it is supposed that they all increase at the same time by their differentials; and, if by the nature of the problem some of them must decrease while others increase, then the minus sign is given to the differentials of those which must decrease.

The differentials ( dp, dq, dr, \dots ), which represent the virtual velocities of the forces ( \mathrm{P, Q, R, \dots} ), must therefore be taken positively or negatively, according as these forces tend to increase or decrease the lines ( p, q, r, \dots ) which determine their direction; but, since the general formula of equilibrium does not change by changing the signs of all its terms, it will be permissible to regard indifferently as positive the differentials of lines which increase or decrease together, and as negative the differentials of those which vary in the opposite direction.

Thus, regarding the forces as positive, their moments ( \mathrm{P}dp, \mathrm{Q}dq, \dots ) will be positive or negative, according as the virtual velocities ( dp, dq, \dots ) are positive or negative, and when one wishes to make the forces act in opposite directions, one need only give the minus sign to the quantities representing these forces, or change the signs of their moments.

From this results this general property of equilibrium: that any system of forces in equilibrium remains so even if each of the forces comes to act in the opposite direction, provided that the constitution of the system suffers no change from a change in direction of all the forces.

4. Whatever the forces that act upon a given system of bodies or points, they can always be regarded as tending towards points situated on the lines of their direction.

We shall name these points the centers of the forces, and we can take for the lines ( p, q, r, \dots ) the respective distances from these centers to the points of the system to which the forces ( \mathrm{P, Q, R, \dots} ) are applied. In this case, it is clear that these forces will tend to decrease the lines ( p, q, r, \dots ); consequently, the minus sign should be given to their differentials; but, by changing all the signs, the general formula becomes equally:

[ \mathrm{P} , dp + \mathrm{Q} , dq + \mathrm{R} , dr + \dots = 0. ]

Now the centers of the forces can be outside the system, or within the system and form part of it, which distinguishes forces into external and internal.

In the first case, it is evident that the differences ( dp, dq, dr, \dots ) express the entire variations of the lines ( p, q, r, \dots ) due to the change in the system’s situation; they are consequently the complete differentials of the quantities ( p, q, r, \dots ), considering as variable all quantities relative to the system’s situation, and as constant those relating to the position of the different centers of the forces.

In the second case, some of the bodies of the system will themselves be the centers of the forces acting upon other bodies of the same system, and, because of the equality between action and reaction, these latter bodies will be at the same time the centers of the forces acting upon the former.

Let us therefore consider two bodies[2] which act upon each other with any force ( \mathrm{P} ), whether this force comes from the attraction or repulsion of these bodies, or from a spring placed between them, or from any other manner.

Let ( p ) be the distance between these two bodies, and ( dp’ ) the variation of this distance insofar as it depends on the change of situation of one of the bodies; it is clear that, relative to this body, we will have ( \mathrm{P} dp’ ) for the virtual moment of the force ( \mathrm{P} ).

Similarly, if we denote by ( dp’’ ) the variation of the same distance ( p ), resulting from the change of situation of the other body, we will have, relative to this second body, the moment ( \mathrm{P} dp’’ ) of the same force ( \mathrm{P} ); therefore the total moment due to this force will be represented by ( \mathrm{P}(dp’ + dp’’) ).

But ( dp’ + dp’’ ) is the complete differential of ( p ), which we will designate by ( dp ), since the distance ( p ) can only vary by the displacement of the two bodies: therefore the moment in question will be expressed simply by ( \mathrm{P} dp ). This reasoning can be extended to any number of bodies.

5. It follows that, to obtain the sum of the moments of all the forces of a given system, whether these forces are external or internal, one need only consider in particular each of the forces acting on the different bodies or points of the system, and take the sum of the products of these different forces multiplied each by the differential of the respective distance between the two termini of each force, i.e., between the point on which this force acts and the point towards which it tends, considering, in these differentials, as variable all quantities which depend on the situation of the system, and as constant those which relate to the external points or centers, i.e., considering these points as fixed, while one varies the situation of the system.

This sum, being set equal to zero, will give the general formula of Statics.

6. To give the analytic expression of this formula all the generality as well as the simplicity of which it is susceptible, we will refer the position of all the bodies or points of the given system, as well as that of the centers, to rectangular coordinates parallel to three fixed axes in space.

We will denote, in general, by ( x, y, z ) the coordinates of the points to which the forces are applied, and we will then distinguish them by one or more accents, relative to the different points of the system.

We will similarly designate by ( a, b, c ), the coordinates for the centers of the forces.

It is evident that the distances ( p, q, r, \dots ) between the points of application and the centers of the forces will be expressed, in general, by the formula:

…

in which the quantities ( a, b, c ), will be constant or at least must be regarded as such, while ( x, y, z ), vary, in the case where they relate to points situated outside the system and where the forces are external; but, in the case where the forces are internal and emanate from some of the bodies of the system itself, these quantities ( a, b, c ), will become ( x’’, y’’, z’’ ), and will consequently be variable.

Having thus the expressions of the finite quantities ( p, q, r, \dots ) as known functions of the coordinates of the different bodies of the system, one need only differentiate in the usual manner, considering these coordinates as the only variables, to obtain the desired values of the differences ( dp, dq, dr, \dots ) which enter into the general formula of equilibrium.

7. But, although one can always regard the forces ( \mathrm{P, Q, R, \dots} ) as tending towards given centers, however, since the consideration of these centers is foreign to the question, in which ordinarily only the quantity and direction of each force are considered as given, here are more general ways of expressing the differences ( dp, dq, dr, \dots ).

And first, by supposing, which is always permissible, that the force ( \mathrm{P} ) tends towards a fixed center, we have