- published: 03 Nov 2014
- views: 601160
In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other (a binary star), and a classical electron orbiting an atomic nucleus (although to solve this system correctly a quantum mechanical approach must be used).
The two-body problem can be re-formulated as two independent one-body problems, a trivial one and one that involves solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved. By contrast, the three-body problem (and, more generally, the n-body problem for n ≥ 3) cannot be solved,[citation needed] except in special cases.
Let x1 and x2 be the positions of the two bodies, and m1 and m2 be their masses. The goal is to determine the trajectories x1(t) and x2(t) for all times t, given the initial positions x1(t = 0) and x2(t = 0) and the initial velocities v1(t = 0) and v2(t = 0).
Verse 1
Too many days go by
When we don't see eye to eye
We hold the key that will open all the doors
I just wish that we could give just a little more
Chorus
This time we've gotta stick together
Can't let our hearts turn into stone
We'll win this fight if we never say never
As two bodies, one mind one soul
Verse 2
I swore that I'd never give up on you
And you said that you would see this through
So let's light the fire that will burn the wall to the ground
Let's find what we thought could never be found