Newton as
a remedy

Michele Diego

This is the power and beauty of physics itself: the ability to abstract from the specific case and arrive at a simple, analytical and universal formulation.

Among the many unwritten rules of physics, that I have heard from the professors who have marked my journey in science so far, is one everyone seems to agree with: in physics, beauty resides in simplicity. The greatness of a discovery, an equation or a theory lies in being able to encapsulate a wide variety of phenomena in a few simple concepts. Very complicated phenomena, described by unwieldy mathematical laws full of parameters, are unattractive to physicists. In every physicist’s heart there is hope to combine seemingly disconnected phenomena into a single formula of staggering simplicity.
In 1923, during his Nobel Prize lecture, Albert Einstein said: “The mind striving after unification of the theory cannot be satisfied that two fields should exist which, by their nature, are quite independent. A mathematically unified field theory is sought in which the gravitational field and the electromagnetic field are interpreted only as different components or manifestations of the same uniform field”1. Einstein was not the only one. Many distinguished physicists applied themselves to the unification of previously disconnected phenomena: Michael Faraday and André-Marie Ampère unified electricity and magnetism into electromagnetism, to which James Clerk Maxwell incorporated light by interpreting it as an electromagnetic wave; Abdus Salam unified the electromagnetic force with the weak force (responsible for the decay of certain atoms) into the electro-weak force, etc.
The highest task of physics, in short, is to discover the first principles of reality, the fundamental mechanisms with which Nature constructs more complex and sometimes apparently dissociated phenomena, the few simple notes with which the universe plays its cosmic symphony.

Yet today, this belief seems to be slowly fading away in the name of a new paradigm. With the dawn of increasingly powerful computers, a new frontier of physics is emerging day by day. While until now the tools that man had for understanding Nature automatically led him to look for the basic and simple components of reality, today a supercomputer can consider an infinity of different parameters and variables.
Let’s take a practical example, like the collision of two galaxies. It is clear that, because of its complexity and the number of parameters involved, such a phenomenon is beyond the reach of a description formulated through a simple set of analytical equations. It is inconceivable to consider the billions of interactions and collisions between stars and celestial bodies that make up galaxies in a single formula. A traditional physicist would be inclined to describe the fundamental mechanisms underlying the collision of celestial bodies (gravitational attraction, dynamics of forces, conservation of total energy, etc.), but would certainly not be able to describe the chaotic nature of the collision of billions of stars. On the other hand, by taking into account the enormous number of parameters and events involved a supercomputer can do so.
This new trend in physics, from simplicity to complexity, from ideal cases to real ones, from a mathematically analytical treatment to a computational one, brings with it – at least in the eyes of the author of this article – the risk of forgetting the golden rule of physics: simplicity is beauty. In a world where any phenomenon can be described through hundreds of equations, each of which has hundreds of variables, there may be a lack of interest in the search for simple, elegant equations that can untangle the complexities of the world and reveal its purest components.

A similar scenario, taken to its extreme consequences, was already predicted by Isaac Asimov. In the short novel “The Feeling of Power”, Asimov imagines a world in which nobody knows the rules of any mathematical calculation anymore. Calculators do all the calculations, which in the eyes of men are devoid of any logical reason; the calculator has become a black box into which numbers are inserted, obtaining an indisputable result, devoid of any rational basis. One man, however, seems to have rediscovered the mechanism behind calculators from scratch and is therefore able to carry out simple mathematical operations independently. Having overcome the initial scepticism of those who consider him to be a sort of illusionist, his ability appears to be revolutionary. The ability to do mathematical calculations is the first step towards liberation from the machines on which humans are now intrinsically dependent.
The story ends on a bitter note. The liberation from machines is not intended to elevate humanity and put it at the centre of the intellectual process once more, but for utilitarian reasons: if it is true that mankind can learn to do mathematical arithmetic, they can then be used instead of a calculating instrument; for example, inside an intelligent missile, thus making the weapon less expensive.

With this article, unlike Asimov’s story, I hope to leave the readers with a note of beauty. I have purposely not yet mentioned the man who, perhaps more than any other in the history of physics, was able to unify apparently disconnected phenomena in a simple, elegant and very general law. Isaac Newton, through his law of universal gravitation, was able to condense years of hard work of astronomical observation by great scientists. From Kepler’s empirical laws he explicitly demonstrated why they had been formulated in this way. He then united terrestrial mechanics to celestial mechanics, making humanity understand how the force that makes an apple fall to the ground is the same that makes the Moon orbit the Earth.
In the remainder of the article, therefore, I will try to show the thinking process of a physicist in the attempt to extract a very elegant mathematical formula such as that of universal gravitation. This will not be a historical derivative, following Newton’s process in full, but rather a simple and comprehensible process for all to follow. The aim is to give a taste of what has been considered beauty within the world of physics to date.

Let’s start with the second principle of dynamics, formulated by Newton himself:


that is, given an object of mass m, in order to move it with acceleration a, a force f must be applied to it. Therefore, where there is an acceleration of an object, there must also be a force responsible for that acceleration. Since all objects fall to the ground with a (gravitational) acceleration that we will call g, there must be a (gravitational) force F responsible for their fall. So, taking an object of mass m in free fall, we can write:


Since it had already been shown during Newton’s time that all objects fall to the ground with the same acceleration g, we have:

F∝ m

(force is directly proportional to mass)

This is because if the force were not directly proportional to the mass, bodies with greater mass would fall more slowly than lighter bodies (it is difficultly convincing, but it is true). For the same force, a larger object is accelerated less than a lighter one. On the other hand, since the acceleration of free-falling bodies is always the same for all bodies, this implies that the force responsible for their fall must be proportional to their mass.
At this point, the third principle of dynamics (again formulated by Newton), also known as the principle of action-reaction, states that for every action there is an equal and opposite reaction. This means that if an object is attracted by the Earth with a force F, the Earth is also attracted by the object with the same force. Then, by following the same reasoning made a few lines above, this implies that the force is also proportional to the Earth’s mass M:


Proceeding further, one last fundamental ingredient is missing. In order to understand it, we need to rely on Newton’s genius, he realised that the force that makes an object fall to the ground is the same force that makes the Moon revolve around the Earth.
So, let us take the Moon in orbit around the Earth. In order to keep it spinning, a force is needed to provide it with a centripetal force. This mean, that the same force F that we have talked about so far must be equal to the centripetal force of the Moon, i.e.:


where m in this case is the mass of the Moon, r is the Earth-Moon distance and T is the period it takes the Moon to complete an orbit around the Earth.
The equality can be rewritten as:


The term on the right (4π²·m) is a constant, while on the left we have a term that does not seem to be constant and depends on the Earth-Moon distance and the force the Earth exerts on the Moon. However, Newton knew from Kepler’s third law that the term T²/r³ is a constant. Thus, for the term on the left to also be constant, we must have F such that the expression on the left is proportional to T²/r³, and so:


Summarising what has been demonstrated so far, we can conclude that:


which combined all together gives:


At this point we call G the constant of proportionality between the force and the terms on the right, obtaining:


which is the law of universal gravitation: the force between two objects of masses m and M is given by the ratio of their masses (m-M), divided by the squared distance between the two objects () and multiplied by the universal gravitation constant G.
It encapsulates the work of great scientists who dedicated their lives to looking at the night sky and transcribing the position of stars, attempting to find their place in a universe that did not seem to correspond to the Aristotelian-Ptolemaic dictates of the past. With his law, Newton summarised all these observations in a single formula, which also included the Earth’s gravity. From this simple equation, one can calculate the gravitational force between any pair of objects in the universe, whether it the Moon and the Earth, or an apple and the Earth, or even between us and a distant star in the universe. This is the power and beauty of physics itself: the ability to abstract from the specific case and arrive at a simple, analytical and universal formulation.

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