Inertia tensor — principal axes, principal moments: Foundations

This is a prerequisites page. Before you can read the parent note, you need to own every symbol it throws at you. We go one at a time, each anchored to a picture, each building on the last.

0. The absolute starting point: a rigid body

Picture a brick glued together from millions of tiny mass-dots. If the brick turns $10°$ , every dot turns $10°$ . That single fact is what makes the whole theory possible — see Rigid body rotation.

The figure below draws this: faint dots are the "before" brick, bright dots the "after". Look at the yellow and green arrows — a corner dot and an inner dot swing through the same angle around the fixed red origin. Nothing stretches; only orientation changes.

Figure — Inertia tensor — principal axes, principal moments

1. Position vector $r$ , basis vectors, and length $r$

The little arrow on top always means "this is a vector — it has a direction, not just a size."

Why the topic needs this: the inertia tensor is built from where each particle sits, i.e. from its components $x, y, z$ . No positions, no tensor.

2. Mass: $m_{a}$ , the sum $\sum$ , the density $ρ$ , and the integral $\int d m$

$\sum_{a} m_{a} = m_{1} + m_{2} + m_{3} + \dots = total mass.$

For a smooth solid body there are infinitely many infinitely-small chunks. Adding infinitely many tiny pieces is exactly what an integral does. To make sense of the mass of an "infinitely small chunk," we need one more idea: density.

Why the topic needs this: the entries of $I$ are mass-weighted sums of coordinate products. For a solid body you compute them as volume integrals over $ρ d V$ — exactly how the parent's cube example gets $\frac{1}{6} M L^{2}$ .

3. Angular velocity $ω$ — the spin arrow

The figure shows the disc spinning (green circulation arrows) while the single yellow $ω$ arrow stands straight up the axis — its direction is the axis, its length is the rate.

Why the topic needs this: $ω$ is the input to the inertia tensor. $L = I ω$ reads "feed the spin arrow in, get the momentum arrow out."

4. The cross product $A \times B$ — geometry and components

Two arrows can be multiplied in a way that produces a third arrow. We need it twice: for velocity ( $v = ω \times r$ ) and for angular momentum ( $L = r \times m v$ ).

In the figure, blue $A$ and green $B$ lie in the page; the yellow shaded parallelogram is their $∣ A ∣∣ B ∣ sin θ$ area; the red dot-in-a-circle is $A \times B$ pointing straight out of the page — perpendicular to both.

Recall Quick check: cross product of parallel arrows

If $A ∥ B$ then $θ = 0$ , so $sin θ = 0$ , so $A \times B = 0$ . A particle sitting on the spin axis ( $r ∥ ω$ ) therefore has $v = 0$ — it doesn't move. Correct: the axis is stationary.

5. The dot product $A \cdot B$

Why the topic needs it: the derivation of $I$ produces the term $r (r \cdot ω)$ . The dot $r \cdot ω$ collapses to one number that then re-scales $r$ .

6. Angular momentum $L$ — the spin's "amount of turning"

Why the topic needs it: $L$ is the output of the inertia tensor. The entire topic exists to answer: given the spin $ω$ , what is $L$ ?

7. The tensor components $I_{ij}$ — the general formula, up front

The parent note fills a $3 \times 3$ grid with entries labelled $I_{xx}, I_{x y}, I_{y z}$ , and so on. Here is exactly what each label means, before you ever meet it in the derivation. The two subscripts $i$ and $j$ each stand for one of the axes $x, y, z$ ; $I_{ij}$ is the entry in row $i$ , column $j$ .

Unpacking that one formula gives every entry the parent uses:

r

mean and what is

r

without an arrow?

r

is a direction+length (a vector, position); plain

r = x^{2} + y^{2} + z^{2}

is just the distance, a single number.

What do the basis vectors

\hat{i}, \hat{j}, \hat{k}

mean and what does a hat signal?

They are unit (length-1) arrows along the

x, y, z

axes; a hat marks "this vector has length 1, direction only." Any vector is

x \hat{i} + y \hat{j} + z \hat{k}

When do you use

\sum_{a} m_{a}

versus

\int ρ d V

\sum

for discrete point masses;

\int ρ d V

(with

d m = ρ d V

) for a continuous solid body — same "add it all up" idea.

What two facts does the single arrow

ω

encode?

Its direction = the rotation axis (right-hand rule); its length = the spin rate.

What kind of object does the cross product output, and the dot product?

Cross → a new vector perpendicular to both (components

(A_{y} B_{z} - A_{z} B_{y}, \dots)

); dot → a single number measuring alignment.

Why is

v = ω \times r

perpendicular to the axis?

The cross product is always perpendicular to both inputs, and a spinning particle moves in a circle around the axis.

Write the general inertia-tensor component

I_{ij}

I_{ij} = \sum m (δ_{ij} r^{2} - x_{i} x_{j})

; diagonal → moments

\sum m (y^{2} + z^{2})

etc., off-diagonal → products

- \sum m x_{i} x_{j}

What is special about a diagonal matrix acting on a vector?

It only stretches each component — it never tilts the arrow.

Why is the inertia tensor symmetric?

The off-diagonal

- \sum m x_{i} x_{j}

is unchanged if you swap

i

and

j

, so

I_{ij} = I_{j i}

State the eigenvector equation, including what the hat and

λ

mean.

I \overset{e}{^} = λ \overset{e}{^}

;

\overset{e}{^}

is a unit-length principal-axis direction,

λ

the principal moment — the spin direction where

L ∥ ω

Once every line is easy → go to the parent note.

Foundations — Inertia tensor — principal axes, principal moments

0. The absolute starting point: a rigid body

1. Position vector $r$ , basis vectors, and length $r$

2. Mass: $m_{a}$ , the sum $\sum$ , the density $ρ$ , and the integral $\int d m$

3. Angular velocity $ω$ — the spin arrow

4. The cross product $A \times B$ — geometry and components

5. The dot product $A \cdot B$

6. Angular momentum $L$ — the spin's "amount of turning"

7. The tensor components $I_{ij}$ — the general formula, up front

8. A matrix and matrix–vector multiply

9. Diagonal, symmetric, identity — three matrix shapes

10. Eigenvalue, eigenvector — the special arrows

11. The prerequisite map

Equipment checklist