### arXiv:1909.04652 [math.NA]AbstractReferencesReviewsResources

#### Relativistic changes to particle trajectories are difficult to detect

Jean-Pierre Eckmann, Farbod Hassani

Published 2019-09-10Version 1

We study the sensitivity of the computed orbits for the Kepler problem, both for continuous space, and discretizations of space. While it is known that energy can be very well preserved with symplectic methods, the semi-major-axis is in general not preserved. We study this spurious shift, as a function of the integration method used, and also as a function of an additional interpolation of forces on a 2-dimensional lattice. This is done for several choices of eccentricities, and semi-major axes. Using these results, we can predict which precisions and lattice constants allow for a detection of the relativistic perihelion advance. Such bounds are important for calculations in N-body simulations, if one wants to meaningfully add these relativistic effects.

**Comments:**14 pages, 8 figures