From Isaac Newton and Centripetal Forces to Universal GravityJune 27, 2021
In order to grasp the novelty of the Principles of Mathematics (Principia) in all its dimensions and what they reveal, it is necessary to look at the attitude of this study on proving.
Introducing the theory of centripetal forces and the assumption of universal gravitation is a good example in this respect.
One of Kepler’s three laws, the law of fields, played a very important role in the development of celestial mechanics (the first law states that planets draw an ellipse with the Sun at one of their foci; the second is the famous law of fields, and the line segment joining a planet to the Sun takes equal lengths of time. claims that it sweeps equal areas; the third law bitches the sizes of ellipses with their rotation times, that is, connects planets with the time they take to complete their orbits (the square of the time a planet takes to complete its orbit is proportional to the cube of its mean distance from the Sun).
Newton recognized the importance of the law of fields in the study of the motion of bodies subject to central forces. For this reason, II of the 1st book of the Principia. section opens with two propositions; The aim of the second proposition, which is the counterpart of the first, is to reveal that the characteristic feature of an accelerated or centrally strong motion is that the area scanning speed in a plane is invariant or that the area scanned by such a motion is proportional to time. Knowing such a property makes a very important contribution to establishing the theory of central forces, as this field can be used to determine time.
Accordingly, the 1st proposition of the 1st book of the Principia becomes: «The areas drawn by curvilinear bodies with lines directed towards the stationary center of forces are inside the stationary planes and are directly proportional to time».
This first result led Newton to put forward proposition 4; The proposition gave a general expression of the intensity of central forces at a point. This proposition allowed Newton to derive the expression of force determined as a function of the distance between moving bodies and the center of the given force. Thus, to take an example, proposition 11 (“The centripetal force is activated when the object revolving on an ellipse approaches one of the foci of the ellipse”) would lead to the well-known conclusion that the central force is “inversely proportional to the square of the distance from the center of force”.
This proposition foreshadows the studies of celestial mechanics that the center of force is no longer a mathematical point but a body with mass and can interact (in proposition 75 of chapter XII, Newton showed that the masses of attracting bodies can be thought of as concentrated at their centers).
This new situation is the XI of book 1. has been delicately emphasized by Newton in the chapter; «On the motion of bodies mutually attracted by central forces».
Thus, Newton first deals with the ‘two-body problem (propositions 57 to 65), then the ‘three-body problem (66 to 68), on which work has been done with the utmost care. In fact, it deals with the three-body problem (from 66 to 68), while the two-body problem has been competently solved today, on which work has been done with extreme care. In fact, while the two-body problem is competently solved today, there is no general solution for three-body and above, although there are highly accepted approximate solutions; The calculations made by astronomers during the movements of the planets or the ejection of spacecraft are proof of this. Newton, on his own account, came up with a very clever solution to the two-body problem by artificially taking one of the bodies (the one with the greater mass) fixed. Xl of the 1st book. This departure, which was developed in the third section, has all its meaning and content. will find in the book; that is, the motion of celestial bodies has been shown to be governed by the law of universal gravitation, based on astronomical observations. This means that all bodies attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
After understanding the results obtained from the study of the motions of the planets and their moons and, in particular, the motion of the Moon – these results gave the possibility of establishing the identity between the central force and gravity (in this sense, it can be said that the Moon is at every moment, just like a thrown stone or a dropped apple which makes it fall). it falls towards the center of the Earth for the same reason; but the Moon is also actuated by a tangential motion, and the combination of the two motions causes curvilinear motion, as in the oblique shot problem) – Newton describes the motions of comets (propositions 42), the tides (propositions 42). proposition 46) and the oblateness of the Earth at the poles (proposition 19). XVIII. In the 19th century, the arcs of the meridian, respectively, Maupertius, were used to predict with great precision the return of the comet observed by Haley in 1681-1682 and to determine the Earth’s oblateness and equatorial position.