The astronom Johannes Kepler (1571 - 1630) was the first to discover that the planet's motions is described by ellipses. Later, he proposed the famous three Kepler's laws, which describe (in good approximation) the basic constants of motions of our solar system. These laws are:
Some mathematics:The following provides a (very rough) introduction into the physics and mathematics behind the Kepler problem and how it was applied to the planetsun sample applications:
The Kepler problem means to solve the equation of motion for two masses with
mass The Lagrange function of the problem is given by
with
Inserting the Lagrange function in the Lagrange equation, we get a coupled system of six second order differential equations. By using the symmetries of the problem (1. translation symmetry 2. rotation symmetry 3. time symmetry) this problem can be reduced to two first order differential equation:
where
After some computation the solution is given by
where the two constants
Unfortunately, the equation cannot be solved in a closed form like
We can write the solution for the radius
The position of the planet and the sun at any time is now calculated by first finding
the corresponding parameter That's the basic mathematics of the planetsun program! |