The 2-body problem

Let's consider the gravitational interaction between two isolated objects which don't interact with anything else. This is a simple dynamics problem (called the 2-body problem) that can be resolved easily. 

  The simplest formulation of the 2-body problem: Newton's solutions

A first approximated example of the 2-body problem is the case when the smallest of the two bodies has a negligible mass. In this case, the heaviest body can be considered as having an infinite mass, and so being still, while the lighter body evolves around it.
Newton showed that in the hypothesis formulated above, using the second principle of dynamics, and knowing that the two bodies are attracted by each other by the force of universal attraction , it is possible to determine the orbit of the lighter object which comes near the Sun from an infinite distance. The orbits resulting from this calculations are four curves called conics, represented on the drawing on the right.  


Ellipse


Parabola

The orbits found by Newton as a solution to the 2-body problem are very different from each other: some of them are closed and periodic, meaning that the body will periodically continue following that trajectory (circle and ellipse) while the others are opened orbits (parabola and hyperbola).
But, in which of the possible orbits determined by Newton the body will lie? This depends on the initial binding energy of the two bodies, or in other words the energy required to bring the two bodies at an infinite distance.
Even if the two assumptions made in this case (the fact that there are only 2 bodies, isolated from the universe, and the assumption that the heaviest of the two bodies is still) seem fairly restrictive, they allow us to give a first explanation of  why Kepler's three laws describe, in a first approximation, the dynamics of the solar system.


A more complicated 2-body problem

In the reality, it is difficult to imagine a body having an infinite mass. If the mass of the lighter body is not negligible (or the two bodies have comparable masses) the problem is a little more complicated. In fact, in this case, it is necessary to define the  center of mass of the system that will be the point around which the two bodies will evolve.
 

In the animations, for simplicity the orbits have been chosen as circles. On the right, we see the real motion of the system around its center of mass.