Perihelion Precession in a Schwarzschild Effective Potential

What you are seeing: a planet on an elliptic orbit around a massive central body. In pure Newtonian gravity the ellipse is closed: the point of closest approach (perihelion) stays exactly in place. General relativity adds a small correction that breaks this closure. Each orbit rotates the entire ellipse by a tiny angle. For Mercury the real value is about 43 arcseconds per century, way too slow to render; here you can dial the GR strength up so it is visible within a handful of orbits.

The acceleration is $\vec a = -\dfrac{GM}{r^3}\,\vec r\left(1 + \dfrac{\alpha}{r^2}\right)$ , with $GM = 1$ and semi-major axis $a = 1$ in code units. The first term is Newton; the second is the orbit-averaged 1PN correction from the Schwarzschild metric. The orange ellipse is the current orbit; faint blue traces are the four most recent orbits, showing the precession. Red dots are perihelion passages.

Figure 1. Precessing Keplerian orbit in the 1PN-corrected effective potential of the Schwarzschild metric. Method: velocity-Verlet from shared/js/engine/symplectic.js, fixed dt = 0.005.

alpha0.020

e0.40

speed0.5

WHAT TO TRY

Watch the red perihelion dots: each marks the closest approach, and they march steadily around the Sun. That rotation of the orbit's long axis is the relativistic perihelion precession, the dashed apsidal line tracks its current direction.
Turn alpha up: the general-relativistic correction strengthens and the orbit precesses faster, sweeping the perihelion dots around in fewer revolutions. At alpha = 0 the ellipse closes and there is no precession.
Raise the eccentricity: a more elongated orbit makes each precession step easier to see, the same effect that made Mercury (the Solar System's most eccentric inner planet) the classic test of GR.