# Daily Archives: December 25, 2007

## Kepler’s Symmetries and Newton’s DE

This is the first of a series of posts about gravity, and the java applet I wrote that simulates Newtonian and relativistic orbits around a (non rotating) black hole. We begin by discussing the relationship between Kepler’s observations and Newton’s universal law of gravitation.

Usually when you write a scientific applet, most of the effort is in the java coding rather than in finding the equations. That is very much true about Newton’s equations of motion around a black hole, which are very easy. His Law of universal gravitation is: where F is the force, G is the universal constant, M is the mass of the black hole (or other spherically symmetric gravitating body), and m is the mass attracted. We will simplify this. We will use units with GM = 1, let m << M, rewrite Force as mass x acceleration, choose Cartesian coordinates (x,y,z) so that $r^2 = x^2 + y^2 + z^2$, only consider motion in 2 dimensions so z=0. The resulting simplified differential equation (DE) is: 