General Formula Of Dynamics For The Motion Of A System Of Bodies Animated By Any Forces

1. When the forces acting on a system of bodies are arranged in accordance with the laws set forth in the first Part of this Treatise, these forces destroy each other mutually and the system remains in equilibrium.

But, when equilibrium does not take place, the bodies must necessarily move, obeying wholly or in part the action of the forces soliciting them. The determination of the motions produced by given forces is the object of this second Part.

We will mainly consider here accelerating and retarding forces whose action is continuous, like that of gravity, and which tend to impress at each instant an infinitely small and equal velocity upon all particles of matter.

When these forces act freely and uniformly, they necessarily produce velocities which increase as time; and one can regard the velocities thus generated in a given time as the simplest effects of these sorts of forces and, consequently, as the most suitable to serve as their measure.

In Mechanics, the simple effects of forces must be taken as known, and the art of this science consists solely in deducing from them the compound effects which must result from the combined and modified action of the same forces.

2. I will therefore suppose that, for each accelerating force, one knows the velocity it is capable of impressing on a moving body by always acting in the same manner, during a certain time which we will take as the unit of times.

We will measure the accelerating force by this same velocity, which must be estimated by the space that the moving body would traverse in the same time if it were continued uniformly; now it is known, by Galileo’s theorems, that this space is always double that which the body has actually traversed by the constant action of the accelerating force.

Moreover, one can take a known accelerating force for the unit and refer all others to it. Then it will be necessary to take as the unit of spaces double the space that the same force, continued equally, would make the body traverse in the time one wishes to take as the unit of times, and the velocity acquired in this time by the continuous action of the same force will be the unit of velocities. In this manner, forces, spaces, times and velocities will be only simple ratios, ordinary mathematical quantities.

For example, if one takes gravity at the latitude of Paris as the unit of accelerating forces, and one counts time in seconds, one should then take (30{,}196) Paris feet as the unit of spaces traversed, because (15{,}098) feet is the height from which a body abandoned to itself falls in one second under this latitude; and the unit of velocities will be that which a heavy body acquires in falling from this height.

3. These preliminary notions assumed, let us consider a system of bodies, arranged with respect to each other in any way and animated by any accelerating forces.

Let (m) be the mass of any one of these bodies, regarded as a point; for the greatest simplicity, let us refer the absolute position of the same body at the end of any time (t) to three rectangular coordinates (x, y, z). These coordinates are supposed always parallel to three fixed axes in space, which intersect perpendicularly at a point called the origin of coordinates; they consequently express the rectilinear distances of the body from three planes passing through the same axes.

Thus, because of the perpendicularity of these planes, the coordinates (x, y, z) represent the spaces by which the moving body recedes from the same planes; consequently,

[ \frac{dx}{dt},\qquad \frac{dy}{dt},\qquad \frac{dz}{dt} ]

will represent the velocities which this body has at any instant to recede from each of these planes and to move along the extensions of the coordinates (x, y, z); and these velocities, if the body were afterwards abandoned to itself, would remain constant in subsequent instants, by the fundamental principles of the theory of motion.

But, by the connection of the bodies and by the action of the accelerating forces which solicit them, these velocities take, during the instant (dt), the increments

[ d\frac{dx}{dt},\qquad d\frac{dy}{dt},\qquad d\frac{dz}{dt} ]

which it is required to determine. One can regard these increments as new velocities impressed on each body, and, dividing them by (dt), one will have the measure of the accelerating forces employed immediately to produce them; for, however variable the action of a force may be, one can always, by the nature of the Differential Calculus, regard it as constant during an infinitely small time, and the velocity generated by this force is then proportional to the force multiplied by the time; consequently, the force itself will be expressed by the velocity divided by the time.

Taking the element (dt) of time as constant, the accelerating forces in question will be expressed by

[ \frac{d^{2}x}{dt^{2}},\qquad \frac{d^{2}y}{dt^{2}},\qquad \frac{d^{2}z}{dt^{2}}, ]

and, multiplying these forces by the mass (m) of the body on which they act, one will have

[ m\frac{d^{2}x}{dt^{2}},\qquad m\frac{d^{2}y}{dt^{2}},\qquad m\frac{d^{2}z}{dt^{2}} ]

for the forces employed immediately to move the body (m) during the time (dt), parallel to the axes of the coordinates (x, y, z). One will therefore regard each body (m) of the system as impelled by such forces; consequently, all these forces must be equivalent to those by which the system is supposed to be solicited, and whose action is modified by the very nature of the system; and the sum of their moments must always be equal to the sum of the moments of these latter forces, by the theorem given in the first Part (sect. II, art. 15).

4. We will in the sequel employ the ordinary characteristic (d) to represent differentials relative to time, and we will denote the variations which express virtual velocities by the characteristic (\delta), as we have already done in some problems of the first Part.

Thus one will have

[ m\frac{d^{2}x}{dt^{2}}\delta x,\qquad m\frac{d^{2}y}{dt^{2}}\delta y,\qquad m\frac{d^{2}z}{dt^{2}}\delta z ]

for the moments of the forces

[ m\frac{d^{2}x}{dt^{2}},\qquad m\frac{d^{2}y}{dt^{2}},\qquad m\frac{d^{2}z}{dt^{2}} ]

which act according to the coordinates (x, y, z) and tend to increase them; the sum of their moments can therefore be represented by the formula

[ \sum m\left(\frac{d^{2}x}{dt^{2}}\delta x+\frac{d^{2}y}{dt^{2}}\delta y+\frac{d^{2}z}{dt^{2}}\delta z\right), ]

assuming that the summation sign (\sum) extends to all the bodies of the system.

5. Let now (\mathrm{P, Q, R, \ldots}) be the given accelerating forces, which solicit each body (m) of the system towards the centers to which these forces are supposed to tend; and let (p, q, r, \ldots) be the rectilinear distances of each of these bodies from the same centers. The differentials (\delta p, \delta q, \delta r, \ldots) will represent the variations of the lines (p, q, r, \ldots) proceeding from the variations (\delta x, \delta y, \delta z) of the coordinates (x, y, z) of the body (m); but, as the forces (\mathrm{P, Q, R, \ldots}) are supposed to tend to diminish these lines, their virtual velocities must be represented by (-\delta p, -\delta q, -\delta r, \ldots) (Part I, sect. II, art. 3); therefore the moments of the forces (m\mathrm{P}, m\mathrm{Q}, m\mathrm{R}, \ldots) will be expressed by (-m\mathrm{P}\delta p, -m\mathrm{Q}\delta q, -m\mathrm{R}\delta r, \ldots), and the sum of the moments of all these forces will be represented by

[ -\sum m(\mathrm{P}\delta p + \mathrm{Q}\delta q + \mathrm{R}\delta r + \ldots). ]

Equating this sum to that of the preceding article, one will have

[ \sum m\left(\frac{d^{2}x}{dt^{2}}\delta x+\frac{d^{2}y}{dt^{2}}\delta y+\frac{d^{2}z}{dt^{2}}\delta z\right) = -\sum m(\mathrm{P}\delta p + \mathrm{Q}\delta q + \mathrm{R}\delta r + \ldots) ]

and transposing the second member,

[ \sum m\left(\frac{d^{2}x}{dt^{2}}\delta x+\frac{d^{2}y}{dt^{2}}\delta y+\frac{d^{2}z}{dt^{2}}\delta z\right) + \sum m(\mathrm{P}\delta p + \mathrm{Q}\delta q + \mathrm{R}\delta r + \ldots) = 0. ]

This is the general formula of Dynamics for the motion of any system of bodies.

6. It is evident that this formula differs from the general formula of Statics, given in the first Part (sect. II), only by the terms due to the forces (m\frac{d^{2}x}{dt^{2}}, m\frac{d^{2}y}{dt^{2}}, m\frac{d^{2}z}{dt^{2}}) which produce the acceleration of the body (m) along the extensions of the three coordinates (x, y, z). Indeed, we saw in the preceding Section (art. 11) that these forces, being taken in the opposite direction, i.e., being regarded as tending to diminish the lines (x, y, z), must equilibrate the actual forces (\mathrm{P, Q, R, \ldots}), which are supposed to act to diminish the lines (p, q, r, \ldots); so that one need only add to the moments of these latter forces the moments of the forces (m\frac{d^{2}x}{dt^{2}}, m\frac{d^{2}y}{dt^{2}}, m\frac{d^{2}z}{dt^{2}}), for each body (m), to pass at once from the conditions of equilibrium to the properties of motion (Part I, sect. II, art. 4).

7. The same rules that we gave in the first Part (sect. II) for the development of the general formula of Statics will therefore apply also to the general formula of Dynamics.

One need only observe:

1º That the differences which we had marked by the ordinary characteristic (d), to represent variations, will henceforth always be marked by the characteristic (\delta);

2º That the characteristic (d) will always be relative to time (t), as well as the corresponding characteristic (\int) for integrations, except in partial differences, where it is indifferent which characteristic is employed;

3º That, to represent the elements of a curve or of a surface, or, in general, of a system composed of an infinity of particles, one will employ the characteristic (\mathrm{D}), which corresponds to the integral characteristic (\sum). Thus, when one wishes to extend to motion the formulas we gave for equilibrium, in the first Part (sect. V, Chap. III and IV), it will be necessary everywhere to change the characteristic (d) to (\mathrm{D}), to obtain the expression for the sum of the moments of all the forces.

8. When the motion takes place in a resisting medium, one can regard the resistance of the medium as a force acting in the opposite direction to that of the body and which can, consequently, be supposed to tend towards a point on the tangent.

Suppose the resistance is (\mathrm{R}); to obtain its moment (-\mathrm{R}\delta r), one need only consider that one has, in general,

[ r = \sqrt{(x-l)^{2} + (y-m)^{2} + (z-n)^{2}}, ]

(l, m, n) being the coordinates of the center of the force (\mathrm{R}); therefore

[ \delta r = \frac{x-l}{r}\delta x + \frac{y-m}{r}\delta y + \frac{z-n}{r}\delta z. ]

Take the center of the force (\mathrm{R}) on the tangent of the curve described by the body and very close to it; one will set, for this,

[ x-l = dx,\qquad y-m = dy,\qquad z-n = dz, ]

which will give, taking (ds) as the element of the curve,

[ \frac{x-l}{r} = \frac{dx}{ds},\qquad \frac{y-m}{r} = \frac{dy}{ds},\qquad \frac{z-n}{r} = \frac{dz}{ds} ]

and, consequently,

[ \delta r = \frac{dx}{ds}\delta x + \frac{dy}{ds}\delta y + \frac{dz}{ds}\delta z. ]

If the resisting medium were in motion, one would have to compound this motion with that of the body to obtain the direction of the resistance force. Name (d\alpha, d\beta, d\gamma) the small spaces that the medium traverses parallel to the axes of the coordinates (x, y, z), while the body describes the space (ds); one need only subtract these quantities from (dx, dy, dz) to obtain the relative motions; and, as

[ ds = \sqrt{x^{2}+y^{2}+z^{2}}, ]

if one sets

[ d\sigma = \sqrt{(dx-d\alpha)^{2} + (dy-d\beta)^{2} + (dz-d\gamma)^{2}}, ]

one will have, in this case,

[ \delta r = \frac{dx-d\alpha}{d\sigma}\delta x + \frac{dy-d\beta}{d\sigma}\delta y + \frac{dz-d\gamma}{d\sigma}\delta z. ]

With regard to the resistance (\mathrm{R}), it is ordinarily a function of the velocity (\frac{ds}{dt}); but, in the case where the medium is in motion, it will be a function of the relative velocity (\frac{d\sigma}{dt}).

In this manner, one will be able to apply our general formulas to motions taking place in resisting media, without needing any particular consideration for these sorts of motions.

9. It is important to remark that the expression

[ d^{2}x,\delta x + d^{2}y,\delta y + d^{2}z,\delta z, ]

by which the general formula of Dynamics differs from that of Statics (art. 5), is independent of the position of the axes of the coordinates (x, y, z).

For, suppose that in place of these coordinates one substitutes other rectangular coordinates (x’, y’, z’) having the same origin, but referring to other axes. By the formulas for the transformation of coordinates, given in the first Part (sect. III, art. 10), one has

[ \begin{aligned} x &= \alpha x’ + \beta y’ + \gamma z’,\ y &= \alpha’ x’ + \beta’ y’ + \gamma’ z’,\ z &= \alpha’’ x’ + \beta’’ y’ + \gamma’’ z’. \end{aligned} ]

Differentiate these expressions for (x, y, z), considering all the coefficients (\alpha, \beta, \gamma, \alpha’, \ldots) as constants and the new coordinates (x’, y’, z’) as the only variables; one will obtain

[ \begin{aligned} d^{2}x &= \alpha d^{2}x’ + \beta d^{2}y’ + \gamma d^{2}z’,\ d^{2}y &= \alpha’ d^{2}x’ + \beta’ d^{2}y’ + \gamma’ d^{2}z’,\ d^{2}z &= \alpha’’ d^{2}x’ + \beta’’ d^{2}y’ + \gamma’’ d^{2}z’. \end{aligned} ]

One will similarly have

[ \begin{aligned} \delta x &= \alpha \delta x’ + \beta \delta y’ + \gamma \delta z’,\ \delta y &= \alpha’ \delta x’ + \beta’ \delta y’ + \gamma’ \delta z’,\ \delta z &= \alpha’’ \delta x’ + \beta’’ \delta y’ + \gamma’’ \delta z’. \end{aligned} ]

Substituting these values and having regard to the equations of condition given in the cited article, between the coefficients (\alpha, \beta, \gamma, \alpha’, \ldots), one will obtain

[ d^{2}x,\delta x + d^{2}y,\delta y + d^{2}z,\delta z = d^{2}x’,\delta x’ + d^{2}y’,\delta y’ + d^{2}z’,\delta z’. ]

If one makes the same substitutions in the expression of the rectilinear distances between the different bodies of the system, represented by (\mathrm{p, q, \ldots}), it is easy to see that the quantities (\alpha, \beta, \gamma, \alpha’, \ldots) will equally disappear and that the transformed expressions will retain the same form. Indeed, one has

[ \mathrm{p} = \sqrt{(x-\mathrm{x})^{2} + (y-\mathrm{y})^{2} + (z-\mathrm{z})^{2}}, ]

(x, y, z) being the coordinates of one body (m) and (\mathrm{x, y, z}) those of another body (\mathrm{m}) referred to the same axes. By the change of axes, the first become (x’, y’, z’), and, if one designates by (\mathrm{x’, y’, z’}) what the latter become, one will also have

[ \begin{aligned} \mathrm{x} &= \alpha \mathrm{x}’ + \beta \mathrm{y}’ + \gamma \mathrm{z}’,\ \mathrm{y} &= \alpha’ \mathrm{x}’ + \beta’ \mathrm{y}’ + \gamma’ \mathrm{z}’,\ \mathrm{z} &= \alpha’’ \mathrm{x}’ + \beta’’ \mathrm{y}’ + \gamma’’ \mathrm{z}’. \end{aligned} ]

Substituting and having regard to the same equations of condition, one will obtain

[ \mathrm{p} = \sqrt{(x’-\mathrm{x}’)^{2} + (y’-\mathrm{y}’)^{2} + (z’-\mathrm{z}’)^{2}}, ]

and similarly for the analogous quantities (\mathrm{q, r, \ldots}).

10. It follows from this that, if the system is animated only by internal forces (\mathrm{P, Q, \ldots}) proportional to any functions of the distances (\mathrm{p, q, \ldots}) between the bodies, and if the conditions of the system depend only on the mutual arrangement of the bodies, so that the equations of condition are only between the different lines (\mathrm{p, q, \ldots}), the general formula of Dynamics (art. 5) will be the same for the transformed coordinates (x’, y’, z’) as for the primitive coordinates (x, y, z). Therefore, after having found, by the integration of the different equations deduced from this formula, the values of the coordinates (x, y, z) of each body (m), expressed in terms of time, if one takes these values for (x’, y’, z’), one will have, for the coordinates (x, y, z), these more general values

[ \begin{aligned} x &= \alpha x’ + \beta y’ + \gamma z’,\ y &= \alpha’ x’ + \beta’ y’ + \gamma’ z’,\ z &= \alpha’’ x’ + \beta’’ y’ + \gamma’’ z’, \end{aligned} ]

in which the nine coefficients (\alpha, \beta, \gamma, \ldots) contain three undetermined quantities, since there are only six equations of condition between them.

If the values of (x’, y’, z’) contain all the arbitrary constants necessary to complete the different integrals, the three indeterminates in question will merge into these same arbitrary constants; but they may supply those which might be lacking, and whose absence would render the solution incomplete. Thus, by means of these three new arbitrary constants which one can introduce at the end of the calculation, one will be free to suppose as many other arbitrary constants equal to zero or to determined quantities, which will often serve to facilitate and simplify the calculation.

11. Although one can always calculate the effects of impulse and percussion like those of accelerating forces, however, when one requires only the total velocity impressed, one can dispense with considering its successive increments; and one can, at once, regard the forces of impulse as equivalent to the impressed motions.

Let therefore (\mathrm{P, Q, R, \ldots}) be the forces of impulse applied to any body (m) of the system, along the lines (p, q, r, \ldots); suppose that the velocity impressed on this body is decomposed into three velocities represented by (\dot{x}, \dot{y}, \dot{z}), along the directions of the axes of the coordinates (x, y, z); one will have, as in article 5, changing the accelerating forces (\frac{d^{2}x}{dt^{2}}, \frac{d^{2}y}{dt^{2}}, \frac{d^{2}z}{dt^{2}}) into the velocities (\dot{x}, \dot{y}, \dot{z}), the general equation

[ \sum m(\dot{x},\delta x + \dot{y},\delta y + \dot{z},\delta z) + \sum (\mathrm{P},\delta p + \mathrm{Q},\delta q + \mathrm{R},\delta r + \ldots) = 0. ]

This equation will give as many particular equations as there remain independent variations after having reduced all the variations marked by (\delta) to the smallest possible number, according to the conditions of the system.