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Euler derived equations for rigid body rotations in the body reference frame and the stationary reference frame by considering an infinitesimal part of the rigid body. Another derivation is possible, and it is widely used: transforming torque-angular momentum relation to the body reference frame. However, their equivalence is not shown explicitly. In this work, for a rigid body with different moments of inertia, we calculated Euler equations explicitly in the body reference frame and the stationary reference frame and torque-angular momentum relation. We also calculated equations of motion from Lagrangian. These calculations show that all four of them are equivalent.

Equivalence of Euler equations and torque-angular momentum relation