Newton–Euler equations and free rigid bodies

Learning outcomes

Write translational and rotational momentum balances.
Explain body-frame gyroscopic terms.
Check torque-free energy and angular momentum.

Prerequisites: Linear algebra, ODEs, and earlier lectures in this sequence.

Translation and rotation

For a body of mass m, center of mass position p, and world force F, $m\ddot p=F+mg^W$ . About the center of mass, use a body-frame inertia tensor $I_B$ :

I_B\dot\omega^B+\omega^B\times(I_B\omega^B)=\tau^B.

The cross product appears because the body frame itself rotates. It is present even when

I_B

is constant in that frame.

In world coordinates, angular momentum is $L^W=R_{WB}I_B\omega^B$ , and $\dot L^W=\tau^W$ . Mixing body inertia with world angular velocity without rotating one of them gives an inconsistent equation.

Inertia and energy

A physical inertia tensor is symmetric and positive definite for a nondegenerate three-dimensional body. Principal axes diagonalize it. Rotational kinetic energy is $T=\frac12(\omega^B)^\top I_B\omega^B$ . Moving the reference point uses the parallel-axis theorem, $I_O=I_C+m(\|r\|^2I-rr^\top)$ .

The center of mass, inertia frame, and visual geometry origin need not coincide. A model can look correct while its inertial properties are wrong.

Worked Euler equations

With principal inertia $(I_1,I_2,I_3)=(1,2,3)$ and no torque,

\dot\omega_1=-\omega_2\omega_3,\quad \dot\omega_2=\omega_3\omega_1,\quad \dot\omega_3=-\tfrac13\omega_1\omega_2.

\omega=(1,1,1)

, the acceleration is

(-1,1,-1/3)

. Its nonzero value does not violate momentum conservation: body-frame components change while world momentum remains fixed.

Steady rotation about the intermediate principal axis is unstable to small perturbations. The other principal-axis rotations are stable in the torque-free idealization. This gives a qualitative check alongside quantitative invariants.

Numerical experiment

Integrate angular velocity and a scalar-first quaternion together. Monitor energy, quaternion norm, and world angular momentum. Preserving quaternion norm alone cannot establish correct rigid-body dynamics.

Practice 3 uses RK4 on the coupled state with quaternion normalization after each completed step. This is a teaching method with approximate conservation, not a variational integrator. Refine the timestep and compare drift over the same final time.

Exercise

A student removes the gyroscopic cross product because no external torque acts. What happens? Answer: body angular velocity becomes artificially constant. Unless rotation is along a principal axis, the predicted world angular momentum will change. The missing term is necessary to express conservation in rotating coordinates.

The multibody notes connect this single-body calculation to generalized coordinates MIT multibody dynamics. Next we derive the same style of mass and bias terms from energy.