Daniel Bernoulli; Inertia

10. d’Alembert’s Treatise on Dynamics of 1743, put an end to these challenges by offering a direct method for solving, or at least for putting into equations, all the problems of Dynamics.

It reduces all the laws of motion of bodies to those of their equilibrium and thus brings Dynamics back to Statics.

Jacques Bernoulli used this principle to explain the center of oscillation.

But d’Alembert envisaged this principle in a general manner as to give it all the simplicity and fruitfulness.

If one impresses on several bodies motions which they are forced to change because of their mutual action, it is clear that one can regard these motions as composed of those which the bodies will actually take, and of other motions which are destroyed; from which it follows that these latter must be such that the bodies animated by these only motions would be in equilibrium.

Such is the principle that d’Alembert gave in his Treatise on Dynamics and of which he made happy use in several problems, and especially in that of the precession of the equinoxes.

This principle does not immediately furnish the equations necessary for the solution of problems in Dynamics, but it teaches how to deduce them from the conditions of equilibrium.

Thus, by combining this principle with the ordinary principles of the equilibrium of the lever or of the composition of forces, one can always find the equations of each problem;

But the difficulty of determining the forces which must be destroyed, as well as the laws of equilibrium between these forces, often makes the application of this principle embarrassing and painful.

The solutions which result from it are almost always more complicated than if they were deduced from less simple and less direct principles, as one can convince oneself by the second Part of the same Treatise on Dynamics[8].

11. If one wished to avoid the decompositions of motions that this principle requires, one would only have to establish at once the equilibrium between the forces and the motions generated, but taken in opposite directions. For, if one imagines that one impresses on each body, in the opposite direction, the motion it must take, it is clear that the system will be reduced to rest; consequently, these motions must destroy those which the bodies had received and which they would have followed without their mutual action; thus there must be equilibrium between all these motions, or between the forces which can produce them.

This manner of recalling the laws of Dynamics to those of Statics is in truth less direct than that which results from d’Alembert’s principle, but it offers more simplicity in applications; it comes back to that of Herman and Euler, who employed it in the solution of many problems of Mechanics, and it is found in some Treatises on Mechanics under the name of d’Alembert’s Principle.

12. I have reduced all Statics to a single general formula which gives the laws of equilibrium of any system of bodies drawn by as many forces as one wishes.

One could therefore also reduce all Dynamics to a general formula; for, to apply to the motion of a system of bodies the formula of its equilibrium, it will suffice to introduce into it the forces which proceed from the variations of the motion of each body, and which must be destroyed.

The development of this formula, having regard to the conditions depending on the nature of the system, will give all the equations necessary for the determination of the motion of each body, and one will only have to integrate these equations, which is the business of Analysis.

13. This formula has the general equations which contain the following principles:

• Conservation of living forces
• Conservation of the motion of the center of gravity
• Conservation of moments of rotation or Principle of areas
• Principle of least action

These principles are the general results of the laws of Dynamics rather than primitive principles,

These are often employed in the solution of problems.

Conservation of living forces

14. The Conservation of living forces was found by Huygens but in a form slightly different from my version.

It deals with the problem of centers of oscillation.

The principle, as it was used in the solution of this problem, consists in the equality between the descent and the rise of the center of gravity of several heavy bodies which descend conjointly, and which then rise separately, being reflected upwards each with the velocity it had acquired.

Now, by the known properties of the center of gravity, the path traversed by this center, in any direction, is expressed by the sum of the products of the mass of each body by the path it has traversed according to the same direction, divided by the sum of the masses.

On the other hand, by Galileo’s theorems, the vertical path traversed by a heavy body is proportional to the square of the velocity it has acquired in descending freely, and with which it could rise again to the same height.

Thus Huygens’ principle reduces to this: in the motion of heavy bodies, the sum of the products of the masses by the squares of the velocities at each instant is the same, whether the bodies move conjointly in any manner, or whether they freely traverse the same vertical heights.

This is also what Huygens himself remarked in a few words, in a short writing relating to the methods of Jacques Bernoulli and L’Hôpital for centers of oscillation.

Up to that point, this principle had only been regarded as a simple theorem of Mechanics.

Leibniz established a difference between:

• dead forces or pressures which act without actual motion
• living forces which accompany this motion
  • The measure of this is the products of masses and squares of velocities

Jean Bernoulli adopted this and saw in this principle that the sum of the living forces of several bodies:

• remains the same, while these bodies act on each other by simple pressures
• is constantly equal to the simple living force which results from the action of the actual forces moving the bodies.

He thus:

• called this conservation of living forces
• used it successfully to solve some problems which had not yet been solved

Daniel Bernoulli later:

• extended this principle
• deduced from it the laws of motion of fluids in vessels

Finally, he rendered it very general, in the Mémoires de Berlin in 1748, by showing how one can apply it to the motion of bodies animated by any mutual attractions or attracted towards fixed centers by forces proportional to any functions of the distances.

This furnishes a finite equation between the velocities of the bodies and the variables which determine their position in space; so that, when by the nature of the problem all these variables reduce to a single one, this equation suffices to solve it completely.

This is the case of that of centers of oscillation. In general, the conservation of living forces always gives a first integral of the different differential equations of each problem, which is of great utility on many occasions.

15. The second principle is from Newton’s inertia.

d’Alembert later:

• extended this principle
• showed that, if each body is solicited by a constant accelerating force which acts along parallel lines, or which is directed towards a fixed point and acts in proportion to the distance, the center of gravity must describe the same curve as if the bodies were free; to which one can add that the motion of this center is, in general, the same as if all the forces of the bodies, whatever they may be, were applied there, each according to its own direction.

This principle serves to determine the motion of the center of gravity independently of the respective motions of the bodies, and that thus it can always furnish three finite equations between the coordinates of the bodies and time, which will be integrals of the differential equations of the problem[9].

16. The third principle is was discovered at the same time by Euler, Daniel Bernoulli and d’Arcy, but under different forms.

According to Euler, Daniel Bernoulli, this principle is: in the motion of several bodies around a fixed center, the sum of the products of the mass of each body by its velocity of circulation around the center and by its distance from the same center is always independent of the mutual action that the bodies can exert upon each other, and remains the same as long as there is no external action or obstacle.

Daniel Bernoulli gave this principle in Volume 1 of the Mémoires de l’Académie de Berlin in 1746.

Euler gave it the same year in volume I of his Opuscules

d’Arcy’s principle in the Mémoires of 1747 only appeared in 1752, is that the sum of the products of the mass of each body by the area which its radius vector describes around a fixed center on the same projection plane is always proportional to time.

One sees that this principle is a generalization of the beautiful theorem of Newton on areas described by virtue of any centripetal forces; and to perceive the analogy or rather the identity with that of Euler and Daniel Bernoulli, one need only consider that the velocity of circulation is expressed by the element of the circular arc divided by the element of time, and that the first of these elements, multiplied by the distance from the center, gives the element of area described around this center; from which one sees that this latter principle is only the differential expression of that of d’Arcy.

This author afterwards presented his principle under another form which brings it closer to the preceding, and which consists in this: the sum of the products of the masses by the velocities and by the perpendiculars drawn from the center to the directions of the bodies is a constant quantity. From this point of view, he even made it a kind of metaphysical principle which he calls the conservation of action, to oppose it or rather to substitute it for that of least action; as if vague and arbitrary denominations constituted the essence of the laws of nature and could, by some secret virtue, erect into final causes simple results of the known laws of Mechanics.

However that may be, the principle in question holds generally for all systems of bodies which act upon each other in any way, either by threads, inflexible lines, laws of attraction, etc., and which are moreover solicited by any forces directed to a fixed center, whether the system is otherwise entirely free, or whether it is constrained to move around this same center.

The sum of the products of the masses by the areas described around this center and projected onto any plane is always proportional to time; so that, by referring these areas to three mutually perpendicular planes, one has three first-order differential equations between time and the coordinates of the curves described by the bodies; and it is properly in these equations that the nature of the principle we have just spoken of consists.

Principle of Least Action

17. I call the fourth principle as least action as analogy with what Maupertuis had given.

This principle says that in the motion of bodies acting on each other, the sum of the products of the masses by the velocities and by the spaces traversed is a minimum.

Maupertuis deduced from it:

• the laws of reflection and refraction of light
• those of the impact of bodies, in two Memoirs in 1744 and 1746 in Berlin.

But these applications are too particular to establish a general principle.

They have moreover something vague and arbitrary, which can only render uncertain the consequences one might draw from them for the very accuracy of the principle.

I think it is wrong to place this principle on the same line as those I have just explained.

But I have another way of envisaging it more generally and rigorously.

Euler gave the first idea of it at the end of his Treatise on Isoperimetrical Problems in 1744.

• He showed that in trajectories described by central forces, the integral of velocity multiplied by the element of the curve always makes a maximum or a minimum.

Euler had found this property in the motion of isolated bodies. It seemed limited to these bodies.

I have extended, through the conservation of living forces, to the motion of any system of bodies which act on each other in any manner.

This gave this new general principle: the sum of the products of the masses by the integrals of the velocities multiplied by the elements of the spaces traversed is constantly a maximum or a minimum.

I call it least action.

I regard it not as a metaphysical principle but as a simple and general result of the laws of Mechanics.

I used it to solve several difficult problems of Dynamics.

This principle, combined with that of living forces according to the rules of the calculus of variations, directly gives all the equations to solve each problem.

From this is born a method equally simple and general for treating questions concerning the motion of bodies.

But this method is itself only a corollary of that which forms the object of the second Part of this Work and which has, at the same time, the advantage of being drawn from the first principles of Mechanics.

1 no void - commonality 2 action - density 3 conservation - bal 4 inertia - spread

1 - Conservation of living forces 2 - Conservation of moments of rotation or Principle of areas 3 - Conservation of the motion of the center of gravity 4 Therefore, if we assume the system to be entirely free, that is, the bodies simply connected to each other in some way, but without being restrained or hindered by fixed supports or any external obstacles, it is easy to understand that the conditions resulting from the nature of the system can only concern the quantities ξ, η, ζ, ξ′, η′, ζ′, … and not at all the quantities x, y, z, x, y, z, whose differentials will consequently remain independent and indeterminate.- Principle of least action