The free rigid body dynamics: generalized versus classic

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.

💡 Research Summary

The paper investigates quadratic Hamiltonian systems defined on the dual of the Lie algebra 𝔬(K), where K is an arbitrary real symmetric 3×3 matrix. By exploiting the well‑known isomorphism between 𝔬(K) and ℝ³, the authors identify 𝔬(K)⁎ with ℝ³ equipped with the minus Lie‑Poisson bracket
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