Dynamics (Encyclopedic order. Understanding. Reason. Philosophy or Science. Science of Nature) has its proper meaning as the science of forces or motor causes, that is to say the forces which place bodies in motion.

This word is constructed from the Greek word δύναμιζ, force which comes from the verb δύναμαι, I can .

M. Leibniz was the first to have used this term to designate the most transcendental part of mechanics, which is involved with the motion of bodies that are due to actual and continuously acting motor forces. The general principle of Dynamics when taken in this sense is that the product of the accelerating or decelerating force by time is equal to the element of speed and the reasoning given is that speed increases or decreases at every instant in virtue of the small repeated increments that the moving force provides to the body during that instant; to which one should see the article Accelerator and the article Cause.

The word Dynamics has been in wide use for a number of years among the Geometers, in particular to signify the science of the motion of bodies which act one against the other, irrespective of the way in which it can occur, either by pushing, either by pulling through the intermediary of a random body between, and to which they are attached as with a string, an rigid lever, a plane etc.

Following this definition, the problems from which one determines the laws surrounding the percussion of bodies are problems of Dynamics. See Percussion

In regards to the problems which it is necessary to determine the motion of several bodies which are held together one and another with some flexible or rigid rod and by which they mutually change their movements; the first which has been resolved for this type is that which is known today as the problem of the centers of oscillation .

It is a matter to determine the motion of a number of weights attached to the same pendulum rod; in order to understand where the difficulty lays, one must first of all observe that if any of these weights were attached alone to the rod, would it describe in the first instant of its motion a small arc of whose length would be the same at any place on the rod where it was attached. Since the rod being pulled from its vertical position from any location on the rod where the weight is placed, the action of gravity on to it is the same and must produce the same effect at the first instant. That is why each one of these weights which are attached to the rod tends to describe a small line which is equal for all these weights. Since the rod is supposedly rigid, it is impossible that all this weight travels through all of the lines at the first instant; however those that are nearest to the center of suspension must evidently travel a lesser distance than those which are further must cover a greater distance. It is therefore necessarily due to the rigidness of the rod, that the speed with which each weights tends to move be altered and that instead of each being the same, it increases in the lesser weights and diminishes in the greater ones. However, according to which law must it increase or diminish? The problem consists of this and one will find the solution in the article Oscillation .

M. Huyghens and a number of others after him solved this problem by different methods. Since that time and especially since about twenty years, geometers have applied themselves to diverse questions regarding this type. The Mémoires of the Petersburg Academy offer us several of these questions solved by MM. Johann and Daniel Bernoulli, father and son; and by M. Euler whose names today are so famous. MM. Clairaut, de Montigny and d’Arcy have also published in the Mémoires of the Academy of Science some solutions to some problems in Dynamics and the first of these three geometers has provided the Mémoires académique 1742 some methods that facilitate the solution of a large number of questions which are related to this science. In 1743, I published a treatise of Dynamics where I give a general principal to resolve all of these types of problem. This can be read in the preface concerning this subject:

“As this part of Mechanics is no less curious than it is difficult and that the problems which are involved are composed of a class which is very extensive, and the greatest geometers have applied themselves assiduously for number of years, but they have only resolved a small number of these of this type and only under particular conditions. The greater part of the solutions that they have given to us, are applicable besides to the one here to principles that no one as yet proved in the general way; those for example as the conservation of kinetic energy (see conservation of kinetic energy under the word Force). Therefore I thought it necessary to extend myself principally on this subject and to show how to resolve all the questions of Dynamics using the a same and simple and direct method which and which is nothing more that the combination of the principles of equilibrium and composed motion; I have demonstrated their use in selected number of chosen problems, of which some are already known, others entirely new, and others that were poorly solved, even by some of the great geometers”.

Here I will explain in a few words in what my principle consist of in order to solve these types of problems. Let us imagine that we impress onto several bodies, motion that they are unable to conserve due to their interaction and that they are forces to alter and to change into other. It is certain that the motion that each body possessed prior can be seen as composed of two other motions at will ( see Decomposition and Composition of motion ), and that we may understand as one of these composed motions as the one that each body must take in virtue of the action of other bodies. Furthermore if each body, instead of having the primary motion which was impressed on to it had received instead the composed motion, it is certain that each of these bodies would have conserved this motion without changing anything, since by the supposition that the motion of each body takes onto itself. Thus the other composed motion must be such that it disturbs nothing in the primary composed motion, that is to say that the second motion must be such for each body as though it had been impressed alone and without any other the system would have remained at rest.

From there it follows that to find the motion of several bodies which act one on the other, one must decompose the motion that each body has received and the tendency it has to motion into two other motions of which one is destroyed and the other is such and so directed that the action of the surrounding bodies cannot alter it nor make it change. One will find in the articles Oscillation, Percussion and elsewhere the applications of this principle which will demonstrate the ease with which it are used.

By this it is easy to see that all of the laws of motion of bodies are reduced to the laws of equilibrium; since to solve any problem in Dynamics , one must decompose the motion of each body into two, of which since one is supposedly known the other will also necessarily. . However one of these motions must be such that the bodies which follow do not bother each other, that is to say that if they are attached to a rigid rod that this rod should not suffer neither breakage nor extension and that the bodies always remain at the same distance one from the other and the second motion must be such that if it were impressed alone, the rod or the system in general would remain in equilibrium. This condition of rigidness of the rod and the condition of equilibrium will always provide all the requisite equations to find the direction and value in each body of the composed motion and consequently the direction and the value of the other.

I believe that I can assure that there are no problems in Dynamics that cannot be easily solved even saying that it is practically like playing with this principle or at least that it can be easily reduce into an equation, since that is all that can be demanded from Dynamics and the solution or the integration of the equation is following this an affair of pure analysis. One will be convinced of what I am saying here by reading through the different problems in my treatise on Dynamics ; I selected the most difficult ones that I could and I believe to have solved them in a simple and direct way that the questions permitted. Since the publication of my treatise on Dynamics in 1743, I have had frequent need to apply the principle, either in the research of the motion of fluids in containers of different shapes ( see my treatise on the equilibrium of the motion of fluids, 1744), either the oscillations in a fluid that cover a spherical surface ( see my research on Winds, 1746) or concerning the theory of the precession of the equinoxes and the nutation of the Earth’s axis in 1749, or to the resistance of fluids in 1752 and finally other problems of this nature. I have always found this principle of an extreme ease and fecundity, I dare to say and without hesitation and I would of the discoveries of another, and I could produce for this subject very serious and authentic witnesses. It appears to me that this principle is reduces all of the problems of a body’s motion to the most simple consideration to that of equilibrium. See Equilibrium. It does not based on any poorly conceived or obscure metaphysics, and it does not consider within motion that which is not really there and that is to say in the distance travelled and the time taken to cover that distance. It does not take use either of the actions or the forces, neither in one word any of the secondary principles which sometimes can be useful in themselves and sometimes useful to shorten or facilitate the solutions, but which will never be primitive principles, since metaphysics will never be clear.