# Newtonian Gravitation

Newton's laws of mechanics state:

1. A body always preserves a state of rest, or of uniform movement in a fixed direction, unless some force acts on it;
2. The rate at which a body's state of movement changes is proportional to the force acting on it, and in the same direction; and
3. All forces occur in pairs; and these two forces are equal in magnitude and opposite in direction.

The first (a.k.a. Galileo's Principle of Inertia) tends to be re-stated, by modern physicists, as saying that there are frames of reference with respect to which it holds true; these are known as inertial frames of reference. The interesting property is then that, if the points which a frame of reference deems to be at rest are deemed to be moving at constant velocity (i.e. in a straight line at constant speed) by some inertial frame, then the original frame is also inertial. The second and third laws implicitly take an inertial frame of reference for granted.

The second law is commonly restated to equate force with rate of change of momentum (which is mass times velocity). Either way, the second is essentially the quantitative definition of force, ready for use by the third. For angular momentum to also be conserved, one must strengthen the third law to assert that the two forces in a pair act along a common line. Interestingly, electromagnetism can appear to violate this principle; but the principle is preserved by the electromagnetic field itself carrying the stray angular momentum.

Newton also developed the mathematical infrastructure, called the differential calculus, for actually solving equations of motion of bodies governed by the above laws and acted on by forces such as gravitation, whose effect Newton characterized by the law:

• Between any two massive bodies, there acts an attractive force on each towards the other, which is proportional to the product of their masses divided by the square of the distance between them.

Newton demonstrated that this leads to orbits obeying Kepler's three laws.

Indeed, from Newton's three laws and Kepler's three laws, it is fairly straightforward to infer the inverse-square law, at least if you have Newton's differential calculus at your disposal.

From Kepler's first law, infer that a planet could follow an actually circular orbit, since a circle is simply an ellipse with zero eccentricity (one may do the following with a general elliptical orbit, but it is messier). The two foci of a circle coincide at its centre, so Kepler's first law puts The Sun at the centre; thus the line from such a planet to The Sun would have constant radius. Kepler's second law now tells us that the angular velocity (and, indeed, the speed) of our hypothetical planet is constant. A fairly straightforward application of the differential calculus yields the acceleration of a body in circular orbit at constant angular velocity; it's proportional to the radius divided by the square of the period (actually, one can probably infer that without overtly using differential calculus). Kepler's third law can be restated to say that the square of the period is proportional to the cube of the semi-major axis (in our circular case, this is just the radius), with all planets sharing the same constant of proportionality; our planet's acceleration is proportional to radius divided by square of period, hence to radius divided by cube of radius, i.e. to the inverse of the square of the radius.

Now, Kepler's laws don't involve the mass of the planet at all, so our planet's acceleration doesn't depend on its mass; thus, applying Newton's second law, the gravitational force acting on each planet is simply proportional to its mass. By Newton's third law the same force is likewise acting, in the opposite direction, on the Sun as the gravitational influence of the planet. By an implicit symmetry argument, the force acting on The Sun must be proportional to its mass. We thus infer Newton's law of gravitation from Newton's laws of mechanics and Kepler's laws of planetary motion.

Thus, while Robert Hooke may actually have conjectured every fragment of Newton's laws – he almost certainly knew of the first, from Galileo, and there are reasonable grounds to suppose he came up with the inverse square law (he and Newton fell out over a fierce priority dispute on this exact point), given which he quite likely did suppose something resembling Newton's second and third laws – the supreme achievement of Newton was really the development of a mathematical too, the differential calculus, without which such laws could serve no practical purpose. Hooke could at best conjecture and speculate; he lacked the mathematical skill to turn his intuitions into a usable physical theory. Given Galileo's work on mechanics, and Hooke's extensive (and highly valuable) experimental work, Newton's laws are reasonable intuitions and conjectures that we may fairly suppose several investigators may have independently come to. Given that the natural philosophers of the day (including both the gregarious Hooke and even the reclusive Newton) participated in a lively discourse, it is likely that these ideas emerged collectively from discussions among many – the exact details of who first articulated each fragment are relatively unimportant. Furthermore, the study of infinitesimals had been pursued vigorously (if not particularly rigorously) for much of the preceding century, so that it is not especially surprising that Newton and Leibniz came up with the infinitesimal calculus (of which the differential calculus is one half) at roughly the same time. Thus both halves of what Newton achieved were ideas whose time had come; yet it is a tribute to his intelligence that he did indeed bring them forth, credibly as independently as anyone ever brings forth an idea; and the progress of natural philosophy (which we now call science) was doubtless brought forward significantly by his combination of mathematical talent and physical insight. Leibniz was doubtless his peer in the former, and Hook likely his peer in the latter, but we owe much to the combination of the two in one mind.

It is worth noting that Newton understood gravitation's action at a distance as not being mediated by any intervening thing. Others subsequently described it in terms of a gravitational field which propagated according to a field equation and acted locally on each body. None the less, until Maxwell's triumph with electrodynamics gave greater legitimacy to field theories, many scientists regarded the field theory of gravity purely as a computational technique to simplify the complexities of computing the dynamics of systems: the vast respect in which Newton was held long after his death ensured that his belief in action at a distance (as likewise the corpuscular nature of light) was an orthodoxy that contrary descriptions had to overcome before gaining acceptance.

Written by Eddy.