Three-Body Inertia Tensor

TL;DR

This paper derives a comprehensive formula for the inertia tensor of a three-body system, linking it to triangle geometry and mass distribution, with specific results for symmetric mass cases.

Contribution

It introduces a covariant expression for the inertia tensor of three particles with different masses using Lagrange multipliers, connecting inertia with triangle geometry.

Findings

Derived a general inertia tensor formula for three-body systems.

Established a relation between inertia and Heron's formula.

Provided specific inertia expressions for symmetric mass distributions.

Abstract

We derive a general formula for the inertia tensor of a three-body system. By employing three independent Lagrange undetermined multipliers to express the vectors corresponding to the sides in terms of the position vectors of the vertices, we present the general covariant expression for the inertia tensor of the three particles of different masses. If $m_a/a=m_b/b=m_c/c=\rho$, then the center of mass coincides with the incenter of the triangle and the moment of inertia about the normal axis passing the center of mass is $I=\rho abc$, where $m_a$, $m_b$, and $m_c$ are the masses of the particles at $A$, $B$, and $C$, respectively, and $a$, $b$, and $c$ are the lengths of the line segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively. The derivation and the corresponding results are closely related to the famous Heron's formula for the area of a triangle.