A rigid body has finitely many degrees of freedom: three translational and three rotational in the generic case. Rotational kinetic energy is $T_{\mathrm{rot}}=\tfrac12 \boldsymbol{\omega}^\mathsf{T} I \boldsymbol{\omega}$, where $I$ is the inertia tensor. Diagonalizing $I$ reveals principal axes and simplifies the dynamics dramatically.

In a body-fixed principal-axis frame, the torque-free rigid body satisfies Euler’s equations: $I_1\dot{\omega}_1+(I_3-I_2)\omega_2\omega_3=0$, and cyclic permutations. These equations are nonlinear and integrable, producing rich phenomena such as free precession and the intermediate-axis instability.

In accelerating or rotating frames, fictitious forces appear: centrifugal, Coriolis, and Euler forces. The Coriolis force $-2m\boldsymbol{\Omega}\times\mathbf{v}'$ is especially important in geophysical flows, planetary atmospheres, and rotating experimental platforms.