The free rigid body dynamics: generalized versus classic

TL;DR

This paper explores the normal forms of quadratic Hamiltonian systems on the dual of Lie algebra o(K), revealing that certain 3D systems with quadratic constants are equivalent to classical free rigid body dynamics with controls.

Contribution

It generalizes the classical free rigid body dynamics by analyzing systems on o(K) and establishing their affine equivalence under specific conditions.

Findings

Any 3D autonomous system with two quadratic constants and a definite linear combination is equivalent to controlled free rigid body dynamics.

The paper characterizes the normal forms of quadratic Hamiltonian systems on o(K).

It extends classical rigid body analysis to more general quadratic Hamiltonian systems.

Abstract

In this paper we analyze the normal forms of a general quadratic Hamiltonian system defined on the dual of the Lie algebra $\mathfrak{o}(K)$ of real $K$ - skew - symmetric matrices, where $K$ is an arbitrary $3\times 3$ real symmetric matrix. A consequence of the main results is that any first-order autonomous three-dimensional differential equation possessing two independent quadratic constants of motion which admits a positive/negative definite linear combination, is affinely equivalent to the classical "relaxed" free rigid body dynamics with linear controls.