Blog on Applied Mathematics

# 1. Introduction.

Mathematics has always had a tremendous impact on reality, bigger than you might think, lowering itself perfectly in the present but also anticipating and outlining the future. Even with less advanced than today's technological means, past discoveries changed our perception of the world, simply using theories and formulations.

In this post we won't see complicated formulas (or at least nothing that we haven't already seen at high school) or theorems on the limit of human understanding, I would rather tell a story that struck me deeply: how humans discovered Neptune. It's a story explaining what scientific activity is: a constant challenge to extend the boundaries of our understanding and our knowledge of reality.

# 2. Newton, Laplace and Lagrange.

Without starting from Adam and Eve (although even there at the beginning there was an apple), I would say that our story can start in 1687 with the publication of the formula of universal gravitation by Isaac Newton: in the universe every material point attracts every other material point with a force that is directly proportional to the product of their masses and inversely proportional to the square of their distance, or, in other words,

### $F = G \frac{m_1 m_2}{r^2}$

where:

$F$ is the Intensity of the Force between two masses;

$G$ is the the Gravitational Constant;

$m_1$ is the first mass;

$m_2$ is the second mass;

$r$ is the distance between the centers of masses.

###### Isaac Newton (1642 - 1727)

From this result, it was immediately clear that the motion of the planets of the Solar System is largely determined by the attraction of the sun (just think that the mass of the Sun is 1,000 times the mass of Jupiter, the largest planet of the Solar System).

No time to celebrate this historic achievement, which criticism of Newton's law immediately set off; substantially, this formula gave a good approximation for the motion of the planets, but to describe correctly the dynamics of the Solar System it's needed to take into account the Forces of mutual attraction between all the planets, thus extending the gravitational problem to the case of n-bodies. Newton himself could not resist the obvious objections, feeling so necessary that God put sometimes planetary orbits into place.

Eighteenth century Physicists and mathematicians dedicated themselves to the study of "perturbations" exerted by the planets in the n-body problem:

• Laplace introduced mathematical methods able to treat the problem of planetary perturbations and to show that the observed motion of Jupiter and Saturn could be explained by the mutual attraction of the two planets.

• Although he did considerable simplifications to the overall problem, Joseph Louis Lagrange in 1772 demonstrated the existence of a stable system for the case of three bodies.

These and other achievements in the late 700 and early 800 have laid the foundation for the discovery of the planet Neptune.

# 3. The discovery of Neptune.

In the late eighteenth century Uranus was discovered (from a musician!) and what better time to apply the results obtained by Laplace & Co.? From the beginning it was realized that the actual trajectory of Uranus in the sky was different from the one calculated mathematically; indeed, it was obvious that these diverged.

With the newest scientific knowledge, astronomers concluded that the orbit of Uranus was disturbed by the presence of another body, perhaps a planet, in a farer zone of the Solar System.

It was 1846. The astronomer Galle looked through the telescope in the position suggested by Le Verrier's mathematical formulas and with only 3 nights of observations (a record at the time) was identified Neptune ... eureka! A planet discovered on the table with paper and pen!

Now this story ends here as we arrived to the discovery of Neptune, but it should be stressed how this event, the result of two centuries of research (from Newton Le Verrier, via Laplace and Lagrange), has been another starting point for many of the important scientific discoveries that followed in the coming centuries: chaos theory, relativity, quantum mechanics, and so on; for each of these there is much to tell, but the beauty of every single result is its preceding story and its ability to open a new horizon of discoveries.

Making a final overview, you realize centuries later how all progresses are closely related to each other; as for athletics, scientific research (in which mathematics plays a key role) is like a relay race where the baton passes endless more or less quickly between coats, not falling down ever.

And who knows if someone will get to explain how and why it all began. But...it's another story...