Dictionary
The law of universal attraction


Newton's law: the law of universal attraction
Isaac Newton is the inventor of modern dynamics, having formulated the three laws of dynamics between '700 and '800. He also was the first to formulate the law of universal attraction.
In fact, observing the motion of the Moon around the Earth, he deduced that a force acting between the two bodies was needed to keep the Moon in its orbit, making it continuously deflect from the straight line predicted by the principle of Inertia (the first law of dynamics). This force was identified by Newton as a force acting upon any two bodies in the Universe, called the universal attraction or gravity, proportional to the two masses and inversely proportional to the square value of the distance, in other words:

This principle, applied to the solar system, marked the end of the tolemaic system. In fact, being the force exerted by a body proportional to its mass, it seemed impossible that the Earth could make the Sun orbit around it.


From the force to the concept of field
When formulating his principle of universal attraction, Newton was aware of some conceptual problems that affected his theory: what is a force, how can a force act at a distance, what is the nature of this force? These questions remained without answer until the theory of general relativity was formulated by Einstein in 1916. The main idea this theory is based upon, is the fact that the force can be substituted by the concept of a field of forces.
For the general relativity, an empty space can be represented as a geometrical two dimensional grid where an euclidean geometry can be applied. In this empty space, a body initially moving will keep its state of motion, respecting the principle of inertia (Newton's first law of motion).
The presence of a mass in this space modifies the geometrical structure, creating a sort of deformation of the grid or in other words a gravitational field where the geometry is no longer euclidean. This gravitational field replaces the concept of force of gravitational attraction. In fact, any body that comes in proximity of this deformation will tend to fall inside it, modifying its initial orbit into a new one. With this intuitive explanation of a field, it is easy to have a new vision of the conics, the orbits solution of the 2-body problem.

An empty space can be represented as an euclidean 2 dimensional grid (that is easily extended to a 3 dimensional space). The presence of a massive body (the red ball) produces a deformation of this grid, giving birth to a sort of hole in the field, also called "potential well".
When other bodies move in the proximity of this mass (in this case the green smaller ball) their trajectories will be modified from the initial straight ones.