2.1 Euler’s Rotational Equations
Euler’s rotational equations describe how a rigid body spins when forces and torques act on it. When expressed in the body frame, the equations become particularly simple because the inertia matrix remains constant. This makes them a foundation for spacecraft attitude dynamics, gyro motion, aircraft rotational motion, and planetary rotation modeling.
We begin by developing angular momentum in the body frame, then derive Euler’s equations, and finally examine useful integrals and small-angle approximations.
2.1.1 Angular Momentum & Inertia
Angular momentum about the center of mass
For a rigid body, the angular momentum about the center of mass $C$ is
\[ \mathbf{H}_c = \int_B \mathbf{r} \times (\boldsymbol{\omega} \times \mathbf{r})\, dm . \]
Because the body is rigid, each particle moves according to
\[ \dot{\mathbf{r}} = \boldsymbol{\omega} \times \mathbf{r}. \]
After expanding the triple product and collecting terms, the angular momentum becomes linear in $\boldsymbol{\omega}$:
\[ \mathbf{H}_c = [I_c]\,\boldsymbol{\omega}, \]
where $[I_c]$ is the inertia tensor about the center of mass.
Inertia tensor
The inertia tensor is defined as
\[ [I_c] = \int_B \left(\lvert\mathbf{r}\rvert^2 I - \mathbf{r}\mathbf{r}^T \right)\, dm . \]
It summarizes how mass is distributed relative to the center of mass. In a general body-fixed frame:
\[ [I_c] = \begin{bmatrix} I_{11} & -I_{12} & -I_{13} \\ -I_{12} & I_{22} & -I_{23} \\ -I_{13} & -I_{23} & I_{33} \end{bmatrix}. \]
Principal axes
If we choose the body frame such that the axes align with the eigenvectors of the inertia tensor, the matrix becomes diagonal:
\[ [I_c] = \mathrm{diag}(I_1, I_2, I_3), \]
where $I_1$, $I_2$, and $I_3$ are the principal inertias of the body.
A physical body must satisfy triangle-inequality-like constraints:
\[ I_1 + I_2 \ge I_3,\quad I_1 + I_3 \ge I_2,\quad I_2 + I_3 \ge I_1, \]
ensuring the inertia values are physically realizable. In practice, spacecraft coordinate axes are often chosen approximately along principal axes, giving a nearly diagonal inertia tensor.
2.1.2 Euler’s Equations in the Body Frame
We now derive Euler’s rotational equations. The starting point is the vector torque equation:
\[ \mathbf{L}_c = \frac{d\mathbf{H}_c}{dt}\Big|_N , \]
where the derivative is taken in an inertial frame $N$.
Using the transport theorem,
\[ \frac{d\mathbf{H}_c}{dt}\Big|_N = \frac{d\mathbf{H}_c}{dt}\Big|_B + \boldsymbol{\omega} \times \mathbf{H}_c . \]
Since $\mathbf{H}_c = [I_c]\boldsymbol{\omega}$ and $[I_c]$ is constant in the body frame,
\[ \frac{d\mathbf{H}_c}{dt}\Big|_B = [I_c]\,\dot{\boldsymbol{\omega}}. \]
Substituting these expressions yields
\[ \mathbf{L}_c = [I_c]\dot{\boldsymbol{\omega}} + \boldsymbol{\omega} \times ([I_c]\boldsymbol{\omega}), \]
which is Euler’s vector equation of rotational motion.
Component form for diagonal inertia matrix
When the body axes align with principal axes,
\[ [I_c] = \mathrm{diag}(I_1, I_2, I_3). \]
Then Euler’s equations reduce to the familiar scalar form:
\[ \begin{aligned} I_1 \dot{\omega}_1 &= (I_2 - I_3)\,\omega_2 \omega_3 + L_1, \\ I_2 \dot{\omega}_2 &= (I_3 - I_1)\,\omega_3 \omega_1 + L_2, \\ I_3 \dot{\omega}_3 &= (I_1 - I_2)\,\omega_1 \omega_2 + L_3. \end{aligned} \]
These nonlinear, coupled equations describe how body angular velocity evolves in time.
Special case: torque-free motion
For torque-free motion, $\mathbf{L}_c = \mathbf{0}$, so
\[ \begin{aligned} I_1 \dot{\omega}_1 &= (I_2 - I_3)\,\omega_2 \omega_3, \\ I_2 \dot{\omega}_2 &= (I_3 - I_1)\,\omega_3 \omega_1, \\ I_3 \dot{\omega}_3 &= (I_1 - I_2)\,\omega_1 \omega_2. \end{aligned} \]
This case reveals important geometric behavior such as precession, nutation, and stability of rotation about maximum and minimum inertia axes.
Axisymmetric bodies
If $I_1 = I_2 = I_T$ and $I_3$ is distinct (a common spacecraft model), Euler’s equations become
\[ \begin{aligned} I_T \dot{\omega}_1 &= -(I_3 - I_T)\,\omega_2 \omega_3, \\ I_T \dot{\omega}_2 &= (I_3 - I_T)\,\omega_3 \omega_1, \\ I_3 \dot{\omega}_3 &= 0. \end{aligned} \]
Thus $\omega_3$ (spin about the symmetry axis) stays constant, while $\omega_1$ and $\omega_2$ follow simple harmonic motion with frequency
\[ \omega_p = \left(\frac{I_3}{I_T} - 1 \right)\omega_3 . \]
A convenient closed-form solution is
\[ \begin{aligned} \omega_1(t) &= \omega_{10}\cos(\omega_p t) - \omega_{20}\sin(\omega_p t), \\ \omega_2(t) &= \omega_{20}\cos(\omega_p t) + \omega_{10}\sin(\omega_p t), \\ \omega_3(t) &= \omega_{30}, \end{aligned} \]
so the transverse components trace a circle of constant radius,
\[ \omega_{12}^2 = \omega_1^2 + \omega_2^2 = \text{constant}. \]
Geometrically, the angular velocity vector precesses around the symmetry axis, as in the standard axisymmetric rigid-body picture.
2.1.3 Energy & Momentum Integrals
Kinetic energy
For a rigid spacecraft, the total kinetic energy is
\[ T = \frac{1}{2} M \dot{\mathbf{R}}_c \cdot \dot{\mathbf{R}}_c + \frac{1}{2} \boldsymbol{\omega}^T [I_c]\boldsymbol{\omega}. \]
In pure rotational motion (center of mass fixed),
\[ T_{\text{rot}} = \frac{1}{2}\boldsymbol{\omega}^T [I_c]\boldsymbol{\omega}. \]
This scalar energy is conserved in torque-free motion.
Angular momentum conservation
If $\mathbf{L}_c = \mathbf{0}$,
\[ \dot{\mathbf{H}}_c = \mathbf{0} \quad \Rightarrow \quad \mathbf{H}_c = \text{constant}. \]
Combined with energy conservation, the dynamics are constrained to the well-known Poinsot construction: rolling without slipping of the inertia ellipsoid on a fixed angular-momentum plane.
Work–energy relation
The power delivered by torques is
\[ \dot{T}_{\text{rot}} = \boldsymbol{\omega} \cdot \mathbf{L}_c. \]
For torque-free bodies, rotational energy remains constant.
2.1.4 Linearization & Small-Angle Models
Small-angle and small-disturbance approximations are fundamental for control design and stability analysis.
Small disturbance about a nominal spin
Assume the spacecraft spins mainly about principal axis 3:
\[ \boldsymbol{\omega} \approx (\omega_1,\, \omega_2,\, \omega_3), \qquad \omega_3 \gg \omega_1, \omega_2. \]
Linearizing Euler’s equations about this motion gives
\[ \dot{\omega}_1 \approx \frac{I_2 - I_3}{I_1}\,\omega_2 \omega_3, \qquad \dot{\omega}_2 \approx \frac{I_3 - I_1}{I_2}\,\omega_1 \omega_3. \]
These equations have the same harmonic form as the axisymmetric case and directly predict the nutation frequency used in spacecraft attitude stability analysis.
In summary:
- Spin about the maximum and minimum inertia axes is stable.
- Spin about the intermediate inertia axis is unstable (the classic “tennis-racket theorem”).
These linearized models form the basis for reaction wheel sizing, momentum-bias stability, and spin-stabilized spacecraft dynamics (e.g., early spin-stabilized satellites).