Proof of a conjecture by H. Dullin and R. Montgomery

TL;DR

This paper derives new, simpler formulas for periods in the planar Euler problem using complex analysis and Keplerian limits, and proves a conjecture about their monotonicity with respect to a first integral.

Contribution

It introduces alternative formulas for periods in the Euler problem and proves a conjecture on their monotonicity as functions of a first integral.

Findings

New formulas for periods are simpler on the opposite side of their singularity.

Proves the conjecture that periods and the rotation number are monotone functions of a first integral.

The formulas are derived using complex analysis and Keplerian limits.

Abstract

In the framework of the planar Euler problem in the quasi--periodic regime, the formulae of the periods available in the literature are simple only on one side of their singularity. In this paper, we complement such formulae with others, which result simpler on the other side. The derivation of such new formulae uses the Keplerian limit and complex analysis tools. As an application, we prove a conjecture by H. Dullin and R. Montgomery, which states that such periods, as well as their ratio, the {\it rotation number}, are monotone functions of their non--trivial first integral, at any fixed energy level.