Dynamics

Dynamics is the science of accelerating or retarding forces and of the varied motions they must produce.

This science is entirely due to the moderns.

Galileo laid its first foundations.

Before him, forces acting on bodies were considered only in a state of equilibrium.

Although the acceleration of heavy bodies and the curvilinear motion of projectiles could only be attributed to the constant action of gravity, no one had determined the laws of these everyday phenomena from such a simple cause.

Galileo was the first to take this important step and thereby opened a new and immense career for the advancement of Mechanics.

This discovery is explained and developed in the work entitled Discorsi e dimostrazioni matematiche intorno a due nuove scienze, which appeared for the first time in Leiden, in 1638.

The discoveries of the satellites of Jupiter, the phases of Venus, the sunspots, etc., required only telescopes and assiduity.

But it took an extraordinary genius to unravel the laws of nature in phenomena that had always been before one’s eyes, but whose explanation had nevertheless always escaped the researches of philosophers.

Huygens added to the theory of the acceleration of heavy bodies that of the motion of pendulums and of centrifugal forces.

He thus prepared the way for the great discovery of universal gravitation.

Mechanics became a new science with Newton.

Finally, the invention of the Infinitesimal Calculus enabled geometers to reduce to analytic equations the laws of motion of bodies.

The investigation of forces and of the resulting motions has since become the principal object of their labors.

I have proposed here to offer them a new means of facilitating this investigation.

1. The theory of varied motions and of the accelerating forces which produce them is founded on these general laws.

Every motion impressed on a body is, by its nature, uniform and rectilinear.

Different motions impressed at the same time or successively on the same body combine in such a way that the body is found at each instant at the same point in space where it ought to be, in effect, by the combination of these motions, if they each existed really and separately in the body.

The force of inertia and of compound motion consist of these 2 laws.

Galileo first perceived these 2 principles. He deduced from them the laws of motion of projectiles, by composing the oblique motion, the effect of the impulse communicated to the body, with its perpendicular fall due to the action of gravity.

With regard to the laws of the acceleration of heavy bodies, they are naturally deduced from the consideration of the constant and uniform action of gravity, by virtue of which, bodies receiving in equal instants equal degrees of velocity according to the same direction, the total velocity acquired after any time must be proportional to that time.

This constant ratio of velocities to time must itself be proportional to the intensity of the force that gravity exerts to move the body.

So that, in motion on inclined planes, this ratio must not be proportional to the absolute force of gravity, as in vertical motion, but to its relative force, which depends on the inclination of the plane and is determined by the rules of Statics.

This provides an easy means of comparing with each other the motions of bodies descending on planes differently inclined.

However, Galileo did not discover in this manner the laws of the fall of heavy bodies.*

He began, on the contrary, by supposing the notion of a uniformly accelerated motion, in which velocities increase as the times.

He deduced from it geometrically the principal properties of this kind of motion and especially the law of the increase of spaces in the ratio of the squares of the times.

Then he assured himself, by experiments, that this law actually takes place in the motion of bodies falling vertically or on any inclined planes.

But, to be able to compare with each other motions on different inclined planes, he was first obliged to admit this precarious principle, that the velocities acquired in descending from equal vertical heights are always equal.

It was only a short time before his death, and after the publication of his Dialogues, that he found the demonstration of this principle by the consideration of the relative action of gravity on inclined planes, a demonstration which was afterwards inserted in other editions of this Work.

2. The constant ratio which, in uniformly accelerated motions, must subsist between velocities and times, or between spaces and the squares of times, can therefore be taken as the measure of the accelerating force which acts continually on the moving body; because, in fact, this force can only be estimated by the effect it produces in the body, which consists in the velocities generated or in the spaces traversed in given times.

Thus it suffices, for this estimation of forces, to consider the motion produced in any time, finite or infinitely small, provided that the force is regarded as constant during that time.

Consequently, whatever the motion of the body and the law of its acceleration may be, since, by the nature of the Differential Calculus, the action of any accelerating force can be regarded as constant during an infinitely small time, one can always determine the value of the force acting on the body at each instant, by comparing the velocity generated in that instant with the duration of the same instant, or the space which it causes to be traversed during the same instant with the square of the duration of that instant.

It is not even necessary that this space has been actually traversed by the body.

It suffices that it can be traversed by a compound motion, since the effect of the force is the same in either case, by the principles of motion set forth above.

Huygens:

• found that the centrifugal forces of bodies moving in circles with constant velocities are as the squares of the velocities divided by the radii of the circles
• was able to compare these forces with the force of gravity at the surface of the Earth

This can be seen by his theorems on centrifugal force, published in 1673 at the end of the Treatise entitled Horologium oscillatorium.

By combining this theory of centrifugal forces with that of evolutes, of which Huygens is also the author, and which reduces to circular arcs each infinitely small portion of any curve, it was easy for him to extend it to all curves.

But it was reserved for Newton to take this new step and to complete the science of varied motions and of the accelerating forces which can generate them.

This science now consists only in a few very simple differential formulas.

But Newton consistently used the geometric method simplified by the consideration of first and last ratios, and, if he sometimes used analytic calculus, it is solely the method of series that he employed, which must be distinguished from the differential method, although it is easy to bring them together and recall them to the same principle.

The geometers who treated, after Newton, the theory of accelerating forces almost all contented themselves with generalizing his theorems and translating them into differential expressions.

Hence the different formulas of central forces found in several Works on Mechanics, but which are no longer much used, because they apply only to curves supposed to be described by virtue of a single force tending towards a center, and we now have general formulas for determining motions produced by any forces.