Abstract
The Kepler problem, which is a basic two-body problem of mechanics, is used to introduce simple integrators. This problem can be solved analytically and is, thus, a very interesting candidate to benchmark numerical methods of integration. First of all, the analytic solution of the problem is discussed in considerable detail. As possible numerical solutions the Euler methods (explicit, implicit, and implicit mid-point) are introduced. They are all based on the finite difference approximation. All these methods violate due to their methodological error energy conservation, an important requirement of physics. This, finally, guides us to the so-called symplectic integrators.
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Notes
- 1.
In particular, we substitute
$$\displaystyle{ w = \left (u -\frac{m\alpha } {\ell^{2}} \right )\left (\frac{2mE} {\ell^{2}} + \frac{m^{2}\alpha ^{2}} {\ell^{4}} \right )^{-\frac{1} {2} }. }$$
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Stickler, B.A., Schachinger, E. (2016). The Kepler Problem. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-27265-8_4
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DOI: https://doi.org/10.1007/978-3-319-27265-8_4
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