The 4 Principles of Dynamics

3. If one conceives that the motion of a body and the forces soliciting it are decomposed according to three straight lines perpendicular to each other, one can consider separately the motions and forces relative to each of these three directions.

For, because of the perpendicularity of the directions, it is evident that each of these partial motions can be regarded as independent of the other two and that it can receive alteration only from the part of the force which acts in the direction of this motion; from which one can conclude that these three motions must follow, each individually, the laws of rectilinear motions accelerated or retarded by given forces.

In rectilinear motion, the effect of the accelerating force consisting only in altering the velocity of the body, this force must be measured by the ratio between the increase or decrease of velocity during any instant and the duration of that instant, that is to say by the differential of velocity divided by that of time; and, as velocity itself is expressed, in varied motions, by the differential of space divided by that of time, it follows that the force in question will be measured by the second differential of space divided by the square of the first differential of time, supposed constant.

Therefore also the second differential of the space that the body traverses, or is supposed to traverse, according to each of the three perpendicular directions, divided by the square of the constant differential of time, will express the accelerating force with which the body must be animated according to that same direction and must, consequently, be equated to the actual force which is supposed to act in that direction.

This is the well-known principle of accelerating forces.

It is not necessary that the 3 directions to which one refers the instantaneous motion of the body be absolutely fixed.

It suffices that they be so during the duration of an instant. Thus, in curvilinear motions, one can take these directions at each instant, one in the tangent and the other two in the perpendiculars to the curve. Then the accelerating force which acts according to the tangent, and which is called the tangential force, will be entirely employed in altering the absolute velocity of the body and will be expressed by the element of this velocity divided by the element of time.

The normal forces, on the contrary, will only change the direction of the body and will depend on the curvature of the line it describes. By reducing the normal forces to a single one, this compound force must be found in the plane of curvature and be expressed by the square of the velocity divided by the osculating radius, since at each instant the body can be regarded as moved in the osculating circle.

It is thus that the known formulas of tangential forces and normal forces were found, which were long used to solve problems on the motion of bodies animated by given forces. Euler’s Mechanics, which appeared in 1736, and which must be regarded as the first great Work where Analysis was applied to the science of motion, is still entirely founded on these formulas; but they have been almost abandoned since, because a simpler manner of expressing the effect of accelerating forces on the motion of bodies has been found.

It consists in referring the motion of the body and the forces soliciting it to fixed directions in space. Then, employing, to determine the place of the body in space, three rectangular coordinates which have these same directions, the variations of these coordinates will evidently represent the spaces traversed by the body according to the directions of these coordinates;

Consequently, their second differentials, divided by the square of the constant differential of time, will express the accelerating forces which must act according to these same coordinates.

Thus, by equating these expressions to those of the forces given by the nature of the problem, one will have three similar equations which will serve to determine all the circumstances of motion.

This manner of establishing the equations of motion of a body animated by any forces by reducing it to rectilinear motions is, by its simplicity, preferable to all others; it should have presented itself first, but it appears that Maclaurin is the first who employed it in his Treatise on Fluxions, which appeared, in English, in 1742; it is now universally adopted.

4. These principles lets us:

• determine the laws of motion of a free body solicited by any forces, provided that the body is regarded as a point.
• determine the motion of several bodies which mutually attract on each other, according to a law which is a known function of distances.
• extend them to motions in resisting media, as well as to those which take place on given curved surfaces, for the resistance of the medium is nothing other than a force which acts in a direction opposite to that of the moving body

When a body is forced to move on a given surface, there is necessarily a force perpendicular to the surface which retains it there, and whose unknown value can be determined according to the conditions resulting from the very nature of the surface.

But the preceding principles are insufficient to solve the motion of several bodies which act on each other by impulse or pressure.

This needs a new principle which determines the force of bodies in motion, having regard to their mass and speed.

5. This principle consists in this: to impart to a given mass a certain velocity according to any direction, whether this mass is at rest or in motion, requires a force whose value[2] is proportional to the product of the mass by the velocity and whose direction is the same as that of this velocity.

This product of the mass of a body multiplied by its velocity is commonly called the quantity of motion of this body.

This is because it is the sum of the motions of all the material parts of the body.

Thus, forces are measured by the quantities of motion they are capable of producing, and conversely the quantity of motion of a body is the measure of the force that the body is capable of exerting against an obstacle, and which is called percussion.

It follows that if 2 non-elastic* bodies come to collide directly in opposite directions with equal quantities of motion, their forces must counterbalance and destroy each other.

Consequently, the bodies must stop and remain at rest.

But, if the impact took place by means of a lever, it would be necessary, for the destruction of the motion of the bodies, that their forces follow the known law of equilibrium of the lever.

Descartes first perceived this principle. But he was mistaken in its application to the impact of bodies since he believed that the same quantity of absolute motion should always be conserved[3].

Wallis was the first who had a clear idea of this principle.

He used it successfully to discover the laws of the communication of motion in the impact of hard or elastic bodies, as can be seen in the Philosophical Transactions of 1669 and in the third Part of his Treatise on Motion, printed in 1671.

The product of mass and velocity expresses the finite force of a body in motion.

• Likewise, the product of mass and the accelerating force, which we have seen to be represented by the element of velocity divided by the element of time, will express the elementary or nascent force.

This quantity, if one considers it as the measure of the effort that the body can make by virtue of the elementary velocity it has taken or tends to take, constitutes what is called pressure.

But, if one regards it as the measure of the force or power necessary to impart this same velocity, it is then what is called motive force.

Thus, pressures or motive forces will destroy each other or be in equilibrium if they are equal and directly opposed, or if, being applied to any machine, they follow the laws of equilibrium of that machine.

6. When bodies are joined together, in such a way that they cannot freely obey the impulses received and the accelerating forces by which they are animated, these bodies exert on each other continual pressures which alter their motions and render their determination difficult.

The simplest problem of this kind is that of the center of oscillation.

The Letters of Descartes give the first traces of research on this.

Mersenne had proposed to geometers to determine the size that a body of any figure must have, so that, being suspended from a point, it makes its oscillations in the same time as a thread of given length loaded with a single weight at its end.

Descartes observes that this question has some relation to that of the center of gravity and that, just as in a heavy body falling freely there is a center of gravity around which the efforts of gravity of all the parts of the body are in equilibrium, so that this center descends in the same manner as if the rest of the body were annihilated or concentrated in the same center.

Thus, in heavy bodies which turn around a fixed axis, there must be a center, which he calls the center of agitation, around which the agitation forces of all the parts of the body counterbalance each other, so that this center, being free from the action of these forces, can be moved as it would be if the other parts of the body were annihilated or concentrated in this same center; that, consequently, all bodies in which this center will be equally distant from the axis of rotation will make their vibration in the same time.

Descartes gives a general method for determining this center of agitation in bodies of any shape by:

1. Seeking the center of gravity of the agitation forces of all the parts of the body
2. Estimating these forces by the products of the masses multiplied by the velocities, which are here proportional to the distances from the axis of rotation
3. Supposing that the parts of the body are projected onto the plane which passes through its center of gravity and through the axis of rotation, so that they preserve their distances from this axis

Descartes’ solution became a subject of contention between him and Roberval who claimed that:

• it was only good when all the parts of the body are actually or can be supposed to be placed in the same plane passing through the axis of rotation
• in all other cases one should consider only the motions perpendicular to the plane passing through the axis of rotation and through the center of gravity of the body
• one should refer each particle to the point where this plane is met by the direction of the motion of this particle, a direction which is always perpendicular to the plane drawn through this particle and through the axis of rotation.

But it is easy to prove that, with respect to the axis of rotation, the moments of the forces estimated in this manner are always equal to those of the forces estimated according to Descartes’ method[4].

Roberval claimed, with more foundation, that:

• Descartes had only sought the center of percussion, around which the shocks or moments of percussion are equal
• to find the true center of oscillation of a heavy pendulum, it was also necessary to have regard to the action of gravity, by virtue of which the pendulum moves.

But this research was superior to the Mechanics of those times.

Geometers continued to assume tacitly that the center of percussion was the same as that of oscillation.

Huygens was the first who considered this latter center from its true point of view.

He therefore thought this was an entirely new problem.

• Not being able to solve it by the known laws of motion, he invented a new, but indirect, principle as the conservation of living forces.