Rigid body dynamics — Euler angles, Euler's equations of motion: Foundations

Before you can read a single line of the parent note, you need to own every symbol it throws at you. This page builds each one from nothing — a plain-words meaning, a picture, and the reason the topic can't live without it. Read top to bottom; every idea leans on the one above it.

1. Where things live: frames and axes

We use two frames throughout, and half the confusion in this topic is forgetting which one you're standing in.

Figure — Rigid body dynamics — Euler angles, Euler's equations of motion

In the figure: the black arrows on the left are the fixed ground frame $(X, Y, Z)$ ; the red arrows on the right are the body frame $(x, y, z)$ tilted and glued to a little cube. Notice the red set is rotated — that rotation is exactly what Euler angles will measure.

2. The angle symbols $ϕ, θ, ψ$

These are lowercase Greek letters. Say them: phi $ϕ$ , theta $θ$ , psi $ψ$ . Each is one "amount of turn," measured in radians or degrees.

Definition The three turns — the exact z–x'–z'' order (active rotations)

These are active rotations: we physically rotate the object (equivalently, its body frame) while the ground frame stays fixed. Starting with the body frame lying exactly on top of the space frame, apply in this order:

$ϕ$ about the space axis $Z$ (precession). This swings the old $x$ -axis to a new direction called the line of nodes.
$θ$ about that new line-of-nodes axis, the $x^{'}$ (nutation) — tips the spin axis over.
$ψ$ about the final body axis $z^{''} = z$ (spin) — twirls the object on its own axis.

Each turn is positive by the right-hand rule about its own axis: right thumb along the rotation axis, fingers curl the direction of increasing angle. This is the standard 3-1-3 (z–x'–z'') convention the parent note uses.

In the figure: the vertical black arrow is $Z$ ; the flat black arc is the $ϕ$ sweep; the black arc between $Z$ and the tilted red axis is $θ$ ; the small red arc at the tip is $ψ$ about the body axis. Follow them in that numbered order.

3. Rates of change: the dot $˙$

4. Vectors and the angular velocity $ω$

In the figure: the black disk spins the way the small rim arrows show; the red vertical arrow is $ω$ , sitting along the spin axis with length equal to the spin speed. The red arrow is what "spin" becomes once we pack it into a single vector.

Derivation WHY the turn-rates add:

$ω = \dot{ϕ} \hat{Z} + \dot{θ} \overset{n}{^} + \dot{ψ} \overset{z}{^}$ WHAT: the total spin-arrow is the sum of three separate spin-arrows, one per Euler-turn — each pointing along that turn's own axis with length equal to that turn's rate. WHY we may add them: for infinitesimal rotations (a tiny amount over a tiny time $d t$ ), rotations commute and behave like ordinary vectors, so their rates $\dot{ϕ}, \dot{θ}, \dot{ψ}$ add tip-to-tail as arrows. (Finite rotations do not commute — order matters — which is exactly why we needed the strict $z$ – $x^{'}$ – $z^{''}$ order in Section 2.) WHAT IT LOOKS LIKE: three arrows along $\hat{Z}, \overset{n}{^}, \overset{z}{^}$ (the tilted, non-perpendicular axes just defined) laid tip-to-tail; their single resultant is $ω$ . The sign convention: each rate is taken positive by the right-hand rule about its own axis, matching the positive sense of $ϕ, θ, ψ$ . Because $\hat{Z}, \overset{n}{^}, \overset{z}{^}$ are tilted relative to the body axes, projecting this sum onto the body frame is what produces the $sin θ, cos θ$ factors in the parent's $ω_{1}, ω_{2}, ω_{3}$ formulas — you cannot just read off $\dot{ϕ}, \dot{θ}, \dot{ψ}$ as the components.

5. The cross product $\times$

In the figure: the vertical black arrow is $ω$ , the slanted black arrow is $A$ frozen in the body, the dashed ellipse is the circle its tip traces, and the red arrow is $ω \times A$ — pointing sideways along that circle, perpendicular to both. That red sideways drag is the whole mechanism.

Recall Why

$\times$ and not ordinary multiplication? Because spinning doesn't stretch $A$ , it rotates it. Rotation moves the tip sideways (perpendicular), which is precisely what $\times$ produces. Ordinary multiplication points along $A$ — the wrong direction entirely. ::: The cross product is the tool for "perpendicular sideways drag," exactly what rotation does.

6. Two meanings of $d / d t$ — which frame is watching?

7. Mass distribution: the inertia tensor $I$

8. Angular momentum $L$ and torque $τ$

9. How it all feeds the topic

Each foundation on the left builds the box to its right; everything funnels into Euler's equations. If any left-hand box is fuzzy, that fuzziness will surface as confusion later — patch it now.

Equipment checklist

Test yourself. Cover the right side; you should be able to answer each in one breath.

What does a frame physically mean?

Three right-angle direction-arrows glued to something, used as rulers for measuring directions.

Difference between space frame

(X, Y, Z)

and body frame

(x, y, z)

Space = nailed to the ground, never moves; body = glued inside the object, tumbles with it.

Are these frames left- or right-handed?

Right-handed: fingers

X \to Y

give thumb

Z

(same for the body frame).

State the exact Euler rotation sequence and whether it is active.

Active:

ϕ

about space

Z

, then

θ

about the new line of nodes

x^{'}

, then

ψ

about the final body axis

z

(3-1-3, z–x'–z'').

What are the domains of

ϕ, θ, ψ

0 \leq ϕ < 2 π

0 \leq θ \leq π

0 \leq ψ < 2 π

What is gimbal lock and when does it strike?

θ = 0

π

, axes

Z

and

z

coincide, so

ϕ

and

ψ

merge — one angle is lost and

1/ sin θ

terms blow up.

What does a dot mean, as in

\dot{ϕ}

; and

\ddot{θ}

Rate of change per second; two dots = rate of the rate (angular acceleration).

What is the angular velocity

ω

and its sign rule?

The spin-arrow: direction = spin axis by right-hand rule, length = spin speed.

Why do the Euler rates add:

ω = \dot{ϕ} \hat{Z} + \dot{θ} \overset{n}{^} + \dot{ψ} \overset{z}{^}

Infinitesimal rotations commute and act as vectors, so their rates add tip-to-tail along the (non-perpendicular) axes

\hat{Z}, \overset{n}{^}, \overset{z}{^}

What are

\hat{Z}, \overset{n}{^}, \overset{z}{^}

and are they perpendicular?

Unit axes of

ϕ, θ, ψ

; NOT mutually perpendicular — that tilt causes the

sin θ, cos θ

factors.

What does

ω \times A

give you and how is its direction fixed?

An arrow perpendicular to both — the tip's sideways velocity; direction by right-hand rule, order flips the sign.

What are the two meanings of

d / d t

and how are they linked?

Space (ground) vs body (bug) rate; linked by

(d A / d t)_{space} = (d A / d t)_{body} + ω \times A

Which frame does Newton's

τ = d L / d t

require?

The inertial (space) frame.

Is the inertia tensor

I

symmetric, and how many independent numbers?

Yes, symmetric; 6 independent (3 moments + 3 products of inertia).

How does

I

act on

ω

Matrix multiply:

L_{i} = \sum_{j} I_{ij} ω_{j}

;

L

need not be parallel to

ω

Why choose principal axes?

There

I

becomes diagonal,

L_{i} = I_{i} ω_{i}

, killing all products of inertia.

State the master rotation law.

τ = (d L / d t)_{space}

— torque equals the inertial-frame rate of change of angular momentum.

What does "torque-free" mean?

τ = 0

; nothing is twisting the body.

Once every line is instant, go back to the parent topic and the notation will read like plain sentences.