

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 2body 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. 