Equation of motion

“ It would have been scarcely possible to discover the law that every body in motion tends to continue moving in a straight line, by an induction from instances in which the motion is deflected into a curve, by being compounded with the effect of an accelerating force. ”

“ We might restate the first law of motion in the form "Every body tends to move in the track in which it actually does move, except in so far as it is compelled by material impacts to follow some other track than that in which it would otherwise move." ”

“ The question whether a method is conceivable by which absolute motion in space may be distinguished from relative motion must be answered definitely in the affirmative by one who accepts the above equations of non-Newtonian mechanics. A body is absolutely at rest when any motion imparted to it increases its mass, or when a certain force will give it the same acceleration in any direction. ”

“ Yet the law of the parabolic motion would be resolvable into the very same simple laws into which that of the elliptical motion is resolved, namely, the law of the permanence of rectilineal motion, and the law of gravitation. ”

“ To define the motion of a body which, in a similar medium, ascends or descends in a right line, and is resisted in the ratio of its velocity, and acted upon by an uniform force of gravity.
The body ascending, let the gravity be expounded by any given rectangle BACH; and the resistance of the medium, at the beginning of the ascent, by the rectangle BADE, taken on the contrary side of the right line AB. Through the point B, with the rectangular asymptotes AC, CH, describe an Hyperbola, cutting the perpendiculars DE, de, in G,g; and the body ascending will in the time DGgd describe the space EGge ”

“ Since the time enters to the second power, the product of mass and acceleration is smaller by the ratio than it would be if the system were at rest. And we conclude, therefore, that the unit of force, or the dyne, in a direction transverse to the line of motion is smaller in a moving system than in one at rest by this same ratio. ”

“ And therefore in clocks, if forces were impressed by some machine upon the pendulum which preserves the motion, and so compounded with the force of gravity, that the whole force tending downwards should be always as a line produced by applying the rectangle under the arc TR and the radius AR to the line TN, all the oscillations will become isochronous. ”

“ I understand that, besides the law of universal gravitation, there are two fundamental laws of motion. By one of these laws, if a body be set in motion and be acted on by no other than the projectile force, it will move forward in a straight line and with a uniform velocity forever. ”

“ Now we imagine the motion as altered by an external force , for instance by a homogeneous electric or magnetic field. Angular forces and rotation are excluded by us; we satisfy the equations of motion, when we accordingly alter the momentum vector of force , and let the motion be parallel to the respective direction of the momentum vector. ”

“ When a body has only a motion of translation all the particles of which it is composed move with equal swiftness, and also in parallel directions; and by the first law of motion, every particle put in such motion will constantly move with the same velocity in the same direction, unless it be prevented by some external cause.
Rotation.
By a motion of rotation, a body without changing its place, turns round on an axis passing through its centre of gravity. ”

“ In order that a body may fall, it is just as necessary that it be lifted up as that it should be heavy or possess gravity. The fall of bodies, therefore, ought not to be ascribed to their gravity alone. The problem of mechanics is to develop the equations which subsist between falling force and motion, motion and falling force, and between different motions. ”

“ The reason is that the absolute theory of space is not really abandoned, and the relative motion, which is all that can be observed, is treated as the differential effect of two absolute motions.
8.5 In the enunciation of Newton's Laws of Motion, the velocities and accelerations of particles must be supposed to refer to the space of some given consentient set. Evidently the acceleration of a particle is the same in all the spaces of a simple group of consentient sets—at least this has hitherto been the unquestioned assumption. ”

“ Now, from the law of conservation of momentum, A assumes that each ball experiences the same change in momentum, and therefore since he has already decided that the ball b has experienced a smaller change of velocity in the ratio , he must conclude that the mass of the ball in system b is greater than that of his own in the ratio .
In general, therefore, we must assume that the mass of a body increases with its velocity. We must bear in mind, however, as in all other cases, that the motion is determined with respect to some point arbitrarily chosen as a point of rest. ”

“ If is the velocity of the moving system, the speed of light, and if , then all dimensions parallel to the direction of motion shall be contracted in the ratio , the dimensions normal to the direction of motion shall remain unchanged. ”

“ Surely no more rigid, constant and unalterable cause could be conceived of than that of the normal motion of the particles of a medium among themselves, which, by an inevitable automatic adjustment arrange their motions so as to produce the effects of gravity. ”

“ Of course, if g were not projected horizontally, we should not have arrived at this result; all we assert is that whatever be the motion of a body, it will (when possible) be at the end of a second, sixteen feet nearer the earth than it would have been if gravity had not acted. If the body be projected horizontally, its descent is due to gravity alone, and is neither accelerated nor retarded by the horizontal velocity. What this experiment proves is, that the mere fact of a body having velocity does not affect the action of gravity thereon. ”

“ Any kind of centripetal force being supposed, and the centre of force, and any plane whatsoever in which the body revolves, being given, and the quadratures of curvilinear figures being allowed; it is required to determine the motion of a body going of from a given place, with a given velocity, in the direction of a given right line in that plane. ”

“ Therefore, a homogeneous and perfect globe will not retain several distinct motions, but will unite all those that are impressed on it, and reduce them into one; revolving, as far as in it lies, always with a simple and uniform motion about one single given axis, with an inclination perpetually invariable. And the inclination of the axis, or the velocity of the rotation, will not be changed by centripetal force. ”

“ And therefore the parts equally press each other by their weights, that is, equally attract each other, according to the third Law of Motion; and, if separated and let go, would fall towards each other with velocities reciprocally as the bodies. All which we may try and see in the load-stone, whose attracted part does not propel the part attracting, but is only stopped and sustained thereby. ”

“ And when a body is thrown upwards, its uniform gravity impresses forces and takes off velocities proportional to the times; and the times of ascending to the greatest heights are as the velocities to be taken off, and those heights are as the velocities and the times conjunctly, or in the duplicate ratio of the velocities. And if a body be projected in any direction, the motion arising from its projection as compounded with the motion arising from its gravity. ”

“ Wherefore the particles of the times being taken in that ratio, the bodies will always lose parts of their motions proportional to the wholes, and therefore will retain velocities always proportional to their first velocities. ”

“ The mention of the laws of motion raises another point where the prevalent theory has nothing to say and the new theory gives a complete explanation. It is well known that the laws of motion are not valid for any axes of reference which you may choose to take fixed in any rigid body. You must choose a body which is not rotating and has no acceleration. ”

“ The manner in which you compound these different momenta to obtain their resultant strikes me as so novel that my mind is left in no small confusion. I do not refer to the composition of two uniform motions, even when unequal, and when one takes place along a horizontal, the other along a vertical direction; because in this case I am thoroughly convinced that the resultant is a motion whose square is equal to the sum of the squares of the two components. The confusion arises when one undertakes to compound a uniform horizontal motion with a vertical one which is naturally accelerated. ”

“ Every point in the body describes a circle about the axis, and returns to its original position after each complete revolution. But if the axle itself be in motion, the paths of the different points of the body will no longer be circular or re-entrant. Even the velocity of rotation about the axis requires a careful definition, and the proposition that, in all motion about a fixed point, there is always one line of particles forming an instantaneous axis, is usually given in the form of a very repulsive mass of calculation. ”

“ The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. ”

“ More important astronomically—and probably intrinsically also—is what he calls the science of local motion,80 which deals with the motion of bodies. He builds up on the basis of his early experiments (§ 116) a theory of falling bodies, in which occurs for the first time the important idea of uniformly accelerated motion, or uniform acceleration, i.e. motion in which the moving body receives in every equal interval of time an equal increase of velocity. ”

“ The axis of angular momentum would have an invariable position in space, and would travel with respect to the earth round the axis of figure with a velocity where is the sidereal angular velocity of the earth. ”

“ In the ninth section Newton dealt with the motion of particles in orbits which are revolving about the centre of force, and on the motion of the apses of such orbits: this introduced the theory of disturbing forces. The tenth section is concerned with constrained motion, and particularly with the motion of pendulums. The eleventh section deals with the motion of particles under their mutual attractions and incidentally with the problem of three bodies. These three sections afford a notable illustration of Newton’s analytical powers. ”

“ The effects which distinguish absolute from relative motion are, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion, they are greater or less, according to the quantity of the motion. If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water ”

“ In my view of equations (C') , the quantities E, M mean the field strengths, true coordinates and true times. They are identical with the measured coordinates and times. The moving system is not deformed, doesn't possess a specific measure of time, and hasn't become isotropic by motion. ”

“ The relative motion of the different parts of each body is not to occur. But each mass is seen to rotate by an observer at rest relative to the other mass round the connecting line of the masses with a constant angular velocity (definite relative motion for both the masses) . ”

“ In its second half, however, every theorem includes a statement about mechanics; the elastic forces that determine the shape of the body, must be influenced by motion so that they exactly cause this contraction.
As the last kinematic consequence, the transformation formula for the velocity of a material point shall be derived. If its coordinates in system are functions of time , then the velocity components are:
The corresponding definition applies to its velocity with respect to . ”

“ The third law of motion being once established, the equality of action and reaction, in the sense of momentum gained and lost, necessarily follows. Thus, if a weight hanging by a string over the edge of a smooth level table draw another weight along the table, the hanging weight moves more slowly than it would do if not so connected, and thus loses velocity by the connexion; while the other weight gains by the connexion all the velocity which it has, for if left to itself it would rest. ”