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Mathematica in Celestial Mechanics: A Sampling

The advanced symbolic and other capabilities of Mathematica make it the ideal tool for solving problems in the field for which calculus and much of modern mathematics was originally invented.

Zero Velocity Curves in the Restricted Three Body Problem

Let  and  be bipolar coordinates and Mu the smaller mass.

Series Solution of Newton's Equation

Let R[t] be the radius vector, r[t] the length. Then we can write  . f[t] and g[t] can be calculated iteratively by series expansion using the equation of motion.

Here is the actual computation.

Poisson Series Expansion for the Center in Kepler Motion

The motion of the so-called center in Kepler motion cannot be expressed in closed form, but a series approximation can be calculated in the following way.

Here is the 2,2 approximation.

Here is the more accurate 6,6 approximation.

Riemann Surface for the Kepler Equation

One of the most important equations in celestial mechanics is the Kepler equation:

Here is a picture of a part the Riemann surface of the function  over the complex m-plane. The branch points at  are clearly visible .