Motion of a planet around a sun

Euler

modularize

Improved Euler

Runge-Kutta 4th order

animation

Newton's cannon

adaptive step control

Notes

1. Please do implement and use RK4 because Improved Euler just won't be accurate enough.

(2) Here is my adaptive step-size selection idea. We want to avoid the velocity changing a lot from one step to the next. This is particularly important when the planet has a close encounter with a sun - the direction of the force of gravity changes very rapidly with position, hence doubly so with time since the planet moves fast when it's close to a sun. The acceleration, which is f(X,t)[2:] happens to hand us exactly how fast the velocity vector is changing. So we can put a cap on the magnitude of the change in velocity over a single step by setting h = c/|| a || for every step ( i.e. c/numpy.linalg.norm( f(X,t)[2:] ) ) where c is some suitably chosen constant. I have found that this works really well, taking lots of steps when passing close to a star and striding out when far away.