Lord Kelvin’s gyrostat and its analogs in physics, including the Lorenz model

Abstract

A gyrostat is a system of bodies whose relative motion does not alter the intrinsic mass distribution of the system. A solid gyrostat is an arbitrary rigid body attached to one or more axisymmetric rotors with their axes fixed in the carrier. A liquid gyrostat is a cavity in a rotating rigid body filled with an inviscid, homogeneous fluid. For certain conditions, gyrostats obey the Volterra equations of motion, a generalization of the Euler equations for a rigid body rotating about a fixed point. The Lorenz model is equivalent to a special case of a gyrostat in a forced, dissipative regime, as is its laboratory analog, the Malkus water wheel. Over a century ago, Lord Kelvin, the gyrostat’s inventor, attempted to formulate mechanical models of the elasticity of matter and the ether, using lattices of linked gyrostats. Opportunities for using gyrostats in the teaching of undergraduate mechanics, nonlinear dynamics, and fluid dynamics are discussed.

Extension of a PID control theory to Lie groups applied to synchronising satellites and drones

“…A gyrostat is a system of bodies whose relative motion does not alter the intrinsic mass distribution of the system. A solid gyrostat is an arbitrary rigid body attached to one or more axisymmetric rotors with their axes fixed in the carrier [33].…”

Model Formulation Over Lie Groups and Numerical Methods to Simulate the Motion of Gyrostats and Quadrotors

“…The gyrostats are scientific models or instruments designed to illustrate experimentally the dynamics of a rotating body such as the spinning-top, the hoop and the bicycle, and also the precession of the equinox and the rotation of the earth. The original gyrostat is an instrument designed by Lord Kelvin to illustrate the complicated state of motion of a spinning body when free to wander about on a horizontal plane, like a top spun on the pavement, or a hoop or bicycle on the road [12]. It consists of a massive flywheel concealed in a metal casing.…”

“…In general, a gyrostat is a system of bodies whose relative motion does not alter the intrinsic mass distribution of the system. A solid gyrostat is an arbitrary rigid body attached to one or more axisymmetric rotors with their axes fixed in the carrier [12].…”

Saltzman’s Model. Part I: Complete Characterization of Solution Properties

“…There is a close connection between the class of energy conserving LOM (n) and the system of Volterra gyrostats and their generalization. Refer to Gluhovsky and Tong (1999), Wang (2008), andTong (2009) for details.…”