= Quadratic rotational Lie-Poisson dynamics
{title2=$\{x_i,x_j\}=-\epsilon_{ijk}x_k,\quad H=Ax_1^2+Bx_2^2+Cx_3^2$}
With the displayed signed <Lie-Poisson bracket> and evolution $\dot F=\{F,H\}$, the equations are $\dot x_1=2(C-B)x_2x_3$, $\dot x_2=2(A-C)x_3x_1$, and $\dot x_3=2(B-A)x_1x_2$. The energy and the <Casimir function of a Poisson manifold> $x_1^2+x_2^2+x_3^2$ are conserved. For $A=1/(2I_1)$, $B=1/(2I_2)$, $C=1/(2I_3)$ with positive inertias, these are the <Euler equations for a torque-free rigid body>. Reversing the bracket while fixing $H$ reverses time orientation.
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