Cubic equations with 2 Roots in the interval $[-1, 1]$

TL;DR

This paper establishes and visualizes the conditions under which cubic equations have exactly two roots within the interval [-1, 1], relevant to physics applications like the theory of tops.

Contribution

It provides explicit conditions and visualizations for cubic equations with two roots in [-1, 1], addressing a specific mathematical problem with physical relevance.

Findings

Conditions for two roots in [-1, 1] are derived and visualized.

The results are applicable in physics, such as in the theory of tops.

Abstract

The conditions for cubic equations, to have 3 real roots and 2 of the roots lie in the closed interval $[-1, 1]$ are given. These conditions are visualized. This question arises in physics in e.g. the theory of tops.