The Exact Solution of Lagrange's Gyroscope

TL;DR

This paper derives an exact, closed-form solution for the nonlinear dynamics of a fixed-point gyroscope using advanced mathematical functions, enabling precise analysis of its nutation behavior.

Contribution

It presents the first exact solution for the fully nonlinear gyroscope equations, utilizing Cardano's formula and elliptic integrals to analyze nutation.

Findings

Exact solutions expressed in Jacobi's elliptic integrals.

Identification of thresholds for different nutation behaviors.

Analytical expressions for energy, momentum, and inertia effects.

Abstract

The closed form solution is found for the fully nonlinear dynamics of the gyroscope with a fixed point at the tip. The solution is found by using Cardano's formulae to factor a cubic, in the case where all roots are known to be real. From this, the nutation angle is solved first in terms of Jacobi's elliptic integral of the first kind. A simple change of variables then transforms the dependent variable of the remaining equations from time to the projection of the nutation angle onto the vertical axis. After this transformation, the remaining equations can be integrated exactly, giving solutions expressed in Jacobi's elliptic integrals of the third kind. Reduced energy, angular momentum, moment of inertia and Cardano's discriminant are defined. The thresholds are found, separating looping, cuspidial, and unidirectional domains of nutation.