Foundations — Torque-free rotation — Euler's equations, asymmetric top

Before you can read a single line of Euler's equations, you need to own a handful of pictures and symbols. This page builds each one from absolutely nothing. Read top to bottom — every idea leans on the one above it.

1. A rigid body — the thing that spins

The picture: imagine every atom of an object connected to every other atom by an unbreakable steel rod. The whole thing can move and turn, but it can never squish or bend.

Why the topic needs it: if the shape could change, its resistance to spinning would change moment to moment, and there would be no clean equations. Rigidity is what lets us describe the whole object's turning with just a few fixed numbers.

2. Two frames: the space frame and the body frame

A frame is just a set of three arrows (axes) you measure positions and directions against — think of the corner of a room where the floor and two walls meet.

The picture: look at the figure. On the left, the grey axes stay bolted to the wall while the object tumbles past them. On the right, the coloured axes are drawn on the object and flip over with it.

Why the topic needs both: in the space frame the object re-orients every instant, so its "resistance to spin" keeps changing — a nightmare to track. In the body frame that resistance is frozen constant, but the price is that the frame itself is now spinning. The whole topic is the trade between these two views. See Angular momentum in rotating frames for the deeper story.

3. The angular velocity vector $\vec{\omega }$

The right-hand rule picture: curl the fingers of your right hand the way the body spins; your thumb points along $\vec{\omega }$. A faster spin = a longer arrow.

What "radians per second" means: a radian is an angle measured by wrapping the radius around the circle — a full turn is $2\pi \approx 6.28$ radians. So $\vec{\omega }$ of length $6.28$ means one full turn every second.

Why we need it: the entire topic is the story of how this single arrow $\vec{\omega }$ moves around — sometimes steady, sometimes wobbling, sometimes flipping. The subscripts ${\omega }_1,{\omega }_2,{\omega }_3$ are just the shadows (components) of this arrow onto the three body axes.

4. Components and subscripts: ${\omega }_1,{\omega }_2,{\omega }_3$

The picture: shine a light straight down each axis; the shadow the arrow casts on that axis is that component. If $\vec{\omega }$ points purely along axis 3, then ${\omega }_1={\omega }_2=0$ and ${\omega }_3$ = the full length.

Why the topic needs it: Euler's equations are written component by component (${\omega }_1,{\omega }_2,{\omega }_3$) because in the body frame each principal axis behaves differently. You cannot mix them into one number.

5. Angular momentum $\vec{L}$ — the "keeper" arrow

The picture: think of $\vec{L}$ as a compass needle frozen pointing north in the room, no matter how the object tumbles under it.

Why the topic needs it: the single most important — and most surprising — fact of this whole subject is that $\vec{L}$ is conserved but $\vec{\omega }$ is not. They are two different arrows. Keep them mentally separate.

6. The inertia tensor $\mathbf{I}$ — the link between $\vec{L}$ and $\vec{\omega }$

Why not just a number? For a sphere, resistance is the same every direction — a single number would do. But a pencil is easy to spin about its long axis and hard to flip end-over-end: resistance depends on direction. A single number can't hold that. A tensor is the tool that stores "different resistance in different directions."

The picture (figure): feed in $\vec{\omega }$ (an arrow), the box $\mathbf{I}$ stretches and bends it, and out comes $\vec{L}$ — generally pointing a different way. That bending is the whole reason wobble exists.

Why "tensor" and not "matrix"? Written out in a chosen frame, $\mathbf{I}$ is a $3\times 3$ grid of numbers (a matrix). "Tensor" just signals that this grid transforms sensibly when you rotate your axes. For our purposes: read it as "the resistance-machine, written as a $3\times 3$ table." Full construction lives in Inertia tensor and principal axes.

7. Principal axes and $\mathrm{diag}(I_1,I_2,I_3)$

What $\mathrm{diag}(I_1,I_2,I_3)$ means: when we align our body axes with these three special directions, the $3\times 3$ resistance table collapses to just three numbers sitting on its diagonal — the rest are zero: $\mathbf{I}=\left(\begin{array}{ccc}{I}_1& 0& 0\\ 0& I_2& 0\\ 0& 0& I_3\end{array}\right)$

The picture: a table full of numbers becomes a table with entries only down the main diagonal (top-left to bottom-right). Off-diagonal zeros = "no cross-coupling in this frame."

Why the topic lives here: in this frame $\vec{L}=(I_1{\omega }_1,I_2{\omega }_2,I_3{\omega }_3)$ — clean, one number per axis. That simplicity is the entire reason we jump to the body frame.

8. The cross product $\times$

Why this tool and not the dot product? The dot product answers "how much do two arrows agree in direction?" (a number). We instead need "what new axis is created by these two arrows, and how strongly?" — a rotation-flavoured question whose answer is itself an axis-arrow. That is precisely the cross product. Its "zero when parallel" property is what makes a sphere never wobble.

The picture: point right-hand fingers along $\vec{a}$, curl toward $\vec{b}$; thumb gives $\vec{a}\times \vec{b}$. Component recipe used in the derivation: $(\vec{a}\times \vec{b}{)}_1=a_2{b}_3-a_3{b}_2$ and cyclically for the other two.

Why the topic needs it: the term $\vec{\omega }\times \vec{L}$ is the "my axes swept past the arrow" correction. It is the beating heart of both the transport theorem and Euler's equations.

9. Rate of change: the dot notation $\dot{\omega }$ and the derivative

Why a derivative here? A derivative answers exactly one question: "if I zoom in on this instant, how fast is the quantity changing?" Euler's equations are precisely statements about how fast each spin component changes — so a derivative is the only tool that fits.

The picture: on a graph of ${\omega }_1$ against time, ${\dot{\omega }}_1$ is the steepness (slope) of the curve at this instant. Flat curve → ${\dot{\omega }}_1=0$ (steady). Steep upward → large positive ${\dot{\omega }}_1$.

10. Two flavours of "the same equation": stable vs unstable

The stability question boils down to one tiny equation, ${\ddot{\omega }}_1=k{\omega }_1$ (a double-dot = rate-of-change of the rate-of-change). The sign of $k$ decides everything:

• $k<0$: the solution wiggles — a bounded oscillation (sine wave). A nudge stays small → stable.
• $k>0$: the solution is an exponential $e^{\sqrt{k}t}$ that blows up. A nudge explodes → unstable (the tennis-racket flip).
• $k=0$: no restoring, no growth — a marginal edge case (the sphere and symmetric cases).

Why we care about the sign, not the size: size sets how fast things happen; the sign sets whether a small wobble dies out or runs away. This single sign is the whole tennis-racket theorem. Deeper treatment: Stability analysis and linearization.

Prerequisite map

Read it top to bottom: the rigid body gives us frames, a spin arrow, and a resistance machine; those combine (through the cross product and the derivative) into Euler's equations, which then feed conservation and stability.

Equipment checklist

Cover the right side and answer aloud — if any stalls, re-read that section.