On the monotonicity criteria of the period function of potential systems

TL;DR

This paper investigates the conditions under which the period function of potential systems is monotonic, introduces new criteria that can be more precise than existing ones, and illustrates these with examples.

Contribution

It introduces new monotonicity criteria for the period function of potential systems, expanding the understanding beyond previously known conditions.

Findings

New criteria (C_n) can be more precise than older conditions

Examples demonstrate the effectiveness of the new criteria

Enhanced understanding of period function behavior in potential systems

Abstract

The purpose of this paper is to study various monotonicity conditions of the period function $T(c)$ (energy-dependent) for potential systems $\ddot x + g(x)=0$ with a center at the origin 0. We had before identified a family of new criteria noted by $(C_n)$ which are sometimes thinner than those previously known ({\it Period function and characterizations of Isochronous potentials}\quad arXiv:1109.4611). This fact will be illustrated by examples.