Foundations — 6DOF equations — translational (Newton), rotational (Euler's equations)

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

0. What is a "rigid body"?

Why the topic needs it: the clean split into "point moves" + "body twists" is a theorem that only holds for rigid bodies. A noodle would need infinitely many numbers.

1. Vectors and the arrow picture

A vector in 3D is really three numbers stacked — one for each axis. Look at the figure: the arrow $\vec{A}$ casts a shadow onto each axis, and those three shadow-lengths $A_x,A_y,A_z$ are its components. Writing $\vec{A}=A_x{\hat{e}}_x+A_y{\hat{e}}_y+A_z{\hat{e}}_z$ just means "walk $A_x$ along $x$, then $A_y$ along $y$, then $A_z$ along $z$, and you land at the arrow tip."

Why the topic needs it: every quantity in 6DOF (force, velocity, spin) is a vector, and the transport theorem is derived by asking how the basis vectors ${\hat{e}}_i$ themselves move when the rocket turns.

2. Rate of change: the dot and $\frac{d}{dt}$

Why the topic needs it: Newton's law and Euler's law are both statements about a rate of change — momentum's rate equals the push. Without the dot there is no law.

3. Velocity $\vec{v}=(u,v,w)$ and angular velocity $\vec{\omega }=(p,q,r)$

Why the topic needs it: $\vec{v}$ is the star of Newton's equation, $\vec{\omega }$ the star of Euler's. Both equations carry a "$\vec{\omega }\times$" term, so you must first know what $\vec{\omega }$ is.

4. The cross product $\vec{\omega }\times \vec{A}$

Look at the figure: a point at position $\vec{A}$ on a body spinning at $\vec{\omega }$ swings along the green arrow $\vec{\omega }\times \vec{A}$ — tangent to its circle, at right angles to both the axis and the radius.

Why the topic needs it: the whole reason the body-frame equations differ from the ground-frame ones is this $\vec{\omega }\times$ term. It is the transport theorem's punchline.

5. Two frames: inertial $I$ and body $B$

Why the topic needs it: this is the central tension the parent note resolves. Measure in $B$, but the law lives in $I$ — the transport theorem is the bridge. See Reference frames and rotation matrices and Transport theorem (rotating frames) to go deeper.

6. Mass $m$, momentum $\vec{p}$, and force $\vec{F}$

Why the topic needs it: these three are the entire cast of the translational equation.

7. Inertia tensor $\mathbf{I}$ and angular momentum $\vec{H}=\mathbf{I}\vec{\omega }$

Why the topic needs it: $\vec{M}=\dot{\vec{H}}{|}_I=\mathbf{I}\dot{\vec{\omega }}{|}_B+\vec{\omega }\times (\mathbf{I}\vec{\omega })$ is Euler's equation. Every symbol in it — $\mathbf{I}$, $\vec{\omega }$, $\vec{H}$, $\vec{M}$, the cross product, the two frames — you now own.

8. How it all feeds the topic

Everything funnels into the transport theorem, which then produces both the Newton and Euler equations of the parent note the 6DOF topic.

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