Foundations — Rigid body — definition, degrees of freedom

Before we can count anything, we have to agree on what each symbol on the parent page actually means — as a picture, not as a squiggle. We build them in order, so nothing is used before it is earned.

1. A point in space — and the numbers $(x,y,z)$

Pick a corner of your room as "zero". Point one arrow right, one forward, one up — three arrows all at right angles. To reach any speck of dust in the room you walk some amount right, some amount forward, some amount up. Those three amounts are $x$, $y$, $z$.

Why the topic needs this: the parent's "3 DOF per particle" is literally these three numbers. Everything else is copies of this idea with rules subtracted.

2. The position vector $\vec{r}$ — one arrow instead of three numbers

Instead of saying "3 right, 2 forward, 5 up", we draw one arrow from the corner straight to the dust speck. That arrow is $\vec{r}$. It secretly still carries the three numbers $(x,y,z)$ — they are its shadow-lengths along the three arrows of Section 1 — but bundling them into one symbol keeps equations short.

The subscript $i$ in ${\vec{r}}_i$ just means "the arrow for particle number $i$" — particle 1 has ${\vec{r}}_1$, particle 2 has ${\vec{r}}_2$, and so on. It is a name tag, nothing more.

Why the topic needs this: the rigidity rule is written with ${\vec{r}}_i$ and ${\vec{r}}_j$. We must know these are just labelled arrows to particles before that rule can mean anything.

3. The difference of two arrows ${\vec{r}}_i-{\vec{r}}_j$ and its length $|\cdot |$

Here is the heart of the parent's definition, built one piece at a time.

Look at the figure: two arrows come out of the origin, one to $i$, one to $j$. The green arrow joining their tips is ${\vec{r}}_i-{\vec{r}}_j$. Subtraction of arrows means "tip minus tip," and geometrically it is the bridge from one particle to the other.

So $|{\vec{r}}_i-{\vec{r}}_j|$ is the straight-line distance between particle $i$ and particle $j$. That is all the scary symbol says.

4. Counting notation: $N$, $3N$, and "constraints"

5. The choosing symbol $\left(\genfrac{}{}{0ex}{}{N}{2}\right)$

For $N=4$: $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=\frac{4\cdot 3}{2}=6$ pairs. Each pair has a fixed-distance rule — but the parent warns these rules are not all independent once $N$ grows. That subtlety is why we fix 3 points instead of blindly counting pairs.

The figure shows the $3+2+1=6$ construction: particle 1 free (3), particle 2 on a sphere around it (2 left), particle 3 on a circle (1 left). Freezing three points freezes the whole body.

6. The angle symbol $\theta$ and "rotation axis"

For a door on a hinge, $\theta$ is how far open it is. It is the one leftover number when everything else is pinned — the "1 DOF" of a wheel on an axle.

Why the topic needs this: the split $6=3_{\text{trans}}+3_{\text{rot}}$ uses $(x,y,z)$ for the three slides and three axis-angles for the three turns. Both halves are now fully defined.

How these foundations feed the topic

Where these tools go next

Each symbol you just built reappears downstream. A few threads worth keeping in view:

• The reference point whose $(x,y,z)$ we track is usually the Centre of Mass.
• Once the body spins, how hard it is to spin is the Moment of Inertia, and how the angle $\theta$ changes in time is Rotational Kinematics.
• Counting DOF by subtracting constraints is exactly the language of Constraints and Lagrangian Mechanics.
• The "5 DOF for a diatomic" fact powers the Equipartition Theorem for gases.
• A rigid body that both slides and spins with a matching condition is doing Rolling Motion.

All of these live under the parent Rigid body — definition, degrees of freedom.

Equipment checklist

Cover the right side and answer each before moving on.