A study of the apsidal angle and a proof of monotonicity in the logarithmic potential case

TL;DR

This paper investigates the apsidal angle in central force systems with various potentials, deriving formulas and proving the monotonicity of the angle with respect to angular momentum specifically for the logarithmic potential case.

Contribution

It provides a new integral formula for the apsidal angle and proves its monotonicity in the logarithmic potential case, extending understanding of orbital dynamics.

Findings

Derived a fixed-end points integral formula for the apsidal angle.

Proved the monotonicity of the apsidal angle in the logarithmic potential case.

Analyzed the derivative of the apsidal angle with respect to angular momentum.

Abstract

This paper concerns the behaviour of the apsidal angle for orbits of central force system with homogenous potential of degree $-2\leq \alpha\leq 1$ and logarithmic potential. We derive a formula for the apsidal angle as a fixed-end points integral and we study the derivative of the apsidal angle with respect to the angular momentum $\ell$. The monotonicity of the apsidal angle as function of $\ell$ is discussed and it is proved in the logarithmic potential case.