Foundations — Constraints — holonomic vs non-holonomic, rheonomic vs scleronomic

This is the D1 Foundations page for the parent topic. We assume you have seen nothing. We list every symbol the parent note leans on, give each a plain-words meaning, a picture, and the reason the topic needs it — built in an order where each rung stands on the one below it.

0 — A particle and where it lives

To say "where" a dot is, we need an address. In flat space that address is three numbers.

Why the topic needs this: every constraint equation is a sentence about these numbers. Before we can restrict a dot's position we must be able to name it.

1 — Bundling three numbers: the position vector $\mathbf{r}$

Writing $(x,y,z)$ every time is clumsy. We give the whole address one bold name.

Why the topic needs it: the parent writes constraints like $f(\mathbf{r})=0$. One symbol $\mathbf{r}$ hides the trio $(x,y,z)$ so formulas stay short.

2 — Many particles: the subscript $i$ and the count $N$

Real systems have more than one dot: two masses on a rod, a gas of molecules.

Why the topic needs it: this count $3N$ is the starting stockpile of numbers. In §6 we will subtract the rules from it to find how much freedom is actually left.

3 — Change over time: the dot on top, $\dot{\mathbf{r}}$

A constraint can restrict not just where you are but how fast and in which direction you move. So we need a symbol for velocity.

Why the topic needs it: the distinction holonomic vs non-holonomic is exactly "is the rule about $\mathbf{r}$ alone, or does it truly involve $\dot{\mathbf{r}}$?"

4 — The rule itself: the function $f$ and "$=0$"

Why the topic needs it: holonomic (defined formally in §9) means exactly "the rule can be written as $f(\mathbf{r},t)=0$." Everything hinges on recognizing this shape of statement.

5 — Distance and length: $|\mathbf{r}|$ and $|{\mathbf{r}}_1-{\mathbf{r}}_2|$

The most common constraint fixes a distance. We need the symbol for it.

Why the topic needs it: the dumbbell constraint $|{\mathbf{r}}_1-{\mathbf{r}}_2{|}^2-{\ell }^2=0$ is this idea. Here ==$\ell$== is the fixed rod length (a single number); squaring the distance avoids the square root and keeps the algebra clean. Notice we could only write this constraint now — after ${\mathbf{r}}_1$, ${\mathbf{r}}_2$, the magnitude bars, and $\ell$ each had a meaning.

6 — Counting freedom: degrees of freedom $n$, constraint count $k$

Why the topic needs it: shrinking $3N$ down to $n$ is the entire payoff of studying constraints; it feeds straight into Generalized Coordinates.

7 — Tiny changes: the differential $d$ and step-rules

Velocity rules are often written with $d$ instead of a dot. Same idea, different dress.

Before we write the general step-rule, we need names for its pieces.

Why the topic needs it: the rolling-disk rule (built in §8) is written in exactly this step form precisely because it cannot be collapsed into an $f(\mathbf{r})=0$.

8 — Tools that appear inside the rules

The parent uses a few named math tools. Each answers a specific question. We meet the angle-and-heading symbols here, the first time they are needed, and we set up the two example systems they belong to.

Why the topic needs them: these are the only extra math tools the parent invokes, and each is chosen because it answers one precise question — slope-with-time ($\tan$), splitting a heading into components ($\cos ,\sin$), or testing whether a step-rule hides a position rule ($\partial$).

9 — The two central terms, defined at last

Now every symbol they use is in place, so we can state the topic's headline definitions cleanly.

The prerequisite map

Each foundation feeds the topic: coordinates and vectors let us name positions, the function $=0$ shape lets us write the rule, velocity + differentials let us tell holonomic from non-holonomic, and the count $3N-k$ lets us cash in the rules as fewer variables. From here you are ready for Generalized Coordinates and Lagrangian Mechanics; the integrability machinery leads to Lagrange Multipliers, and the "does the surface move" question leads to Kinetic Energy and the Hamiltonian. The stepwise reasoning about tiny allowed moves connects to Virtual Displacements and D'Alembert's Principle.

Equipment checklist

Give the plain meaning of each before you move on — cover the right side and test yourself.