Exact calculation of the number of degrees of freedom of a rigid body constituted by n particles

TL;DR

This paper derives formulas to accurately calculate the degrees of freedom of a system of n particles with fixed distances, applicable in any number of spatial dimensions.

Contribution

It provides a generalized method for computing degrees of freedom of constrained particle systems in arbitrary dimensions.

Findings

Derived explicit formulas for degrees of freedom in D dimensions.

Validated formulas for systems with fixed inter-particle distances.

Applicable to theoretical models in physics and engineering.

Abstract

In this work we discuss how to correctly obtain the number of degrees of freedom of a system constituted by n particles with fixed relative distances and which are immerse in a three-dimensional space. As a result of our analysis, we develop expressions which yield the number of degrees of freedom of an analogous system, not only in three, but in any arbitrary number D of dimensions.