# Closed-form equation for orientation and angular velocity over time

If a rigid body, rotating freely in 3d, experiences no friction or other external forces and has an initially diagonal inertia matrix $\mathbf{I}_0$ (with $I_{11}>I_{22}>I_{33}>0$) and initial angular velocity $\omega_0$ relative to an external inertial reference frame, it seems that the orientation and angular velocity as a function of time must be deterministic from the initial conditions.

I believe that conservation of angular momentum tells us:

$\tau = \frac{dL}{dt} = \mathbf{R}_t\mathbf{I}_0\mathbf{R}^\text{T}_t\dot{\omega}_t+\omega_t \times \mathbf{R}_t\mathbf{I}_0\mathbf{R}^\text{T}_t\omega_t$

Computing this iteratively is possible, but not very accurate. Explicit integration tends to gain energy like crazy. Is there a known, closed-form solution for $\mathbf{R}_t$ and $\omega_t$ as a function of time, assuming that $\mathbf{R}_0$ is the identity matrix?

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