Foundations — Generalized coordinates — choosing them, degrees of freedom

Before you can count degrees of freedom, you must be fluent in every piece of notation the parent note fires at you. This page assumes you know nothing and builds each symbol from a picture. Read top to bottom — each block only uses symbols defined above it.

0 — What a "system" and a "configuration" even are

Look at the figure. On the left, one bead sits somewhere in a room. To say where it is, you point along three directions. On the right, the exact same idea but the bead is threaded on a wire — most of the room is now forbidden, and only the positions on the wire are real configurations. The whole topic is about throwing away the forbidden greyed-out region.

1 — A point, and the number line behind each direction

Why numbers at all? Because "somewhere over there" cannot be calculated with. A number can be added, subtracted, and plotted. The letter we use for such a number is usually $x$, $y$, or $z$.

2 — The position vector $\mathbf{r}$

Why bundle them? Because writing $x$, $y$, $z$ separately every time is noisy. When the parent note writes ${\mathbf{r}}_i$, the bold letter says "all three numbers of one particle, packaged." The subscript $i$ is next.

Picture: $N$ separate arrows, one per particle, each with its own little tag ${\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_N$.

3 — Counting the raw numbers: $3N$

Why does the parent note sometimes use $2N$? Because if the whole problem lives in a flat plane (a pendulum swinging in one vertical sheet), the "up out of the page" direction never changes — it contributes no number. Dropping a frozen direction is your first taste of the whole game.

4 — Functions: the notation ${\mathbf{r}}_i(q_1,\dots ,q_n,t)$

Picture: a box with dials on the front. Turn the dials $(q_1,q_2,\dots )$ and a marble drops out of the chute — that marble is the position ${\mathbf{r}}_i$.

So when the parent writes ${\mathbf{r}}_i={\mathbf{r}}_i(q_1,q_2,\dots ,q_n,t)$ it is saying: "give me the values of the generalized coordinates $q_1\dots q_n$ (and the clock reading $t$), and I will hand you back where particle $i$ actually is." The knobs $q$ are the honest coordinates; the box hides the geometry of the wire or rod.

5 — Angles as coordinates: $\theta$, $\varphi$, sin, cos

Why does the parent note love angles? Because a pendulum bob on a rigid rod cannot change its distance from the pivot — only its direction. The one number that is genuinely free is the tilt angle. Using $\theta$ as the coordinate builds the "fixed length" rule into the coordinate itself.

Look at the figure: for every angle $\theta$ you can dial, the tip stays exactly a distance $\ell$ from the pivot — Pythagoras gives $x^2+y^2={\ell }^2{\sin }^2\theta +{\ell }^2{\cos }^2\theta ={\ell }^2$. That is why the length constraint disappears when you switch to $\theta$: it is satisfied automatically by the shape of sine and cosine.

6 — The equation $f(\dots )=0$ and the word "constraint"

Why "$=0$"? Any rule "left side $=$ right side" can be shoved into the form "(left minus right) $=0$." So $x^2+y^2={\ell }^2$ becomes $f=x^2+y^2-{\ell }^2=0$. Writing it as "something $=0$" gives every constraint the same shape, which is what lets us count them uniformly.

7 — Putting the symbols together: $n=3N-k$

Now every symbol in the headline formula is defined. Read it as an English sentence:

Picture: start with $3N$ empty boxes, cross out one box for every independent constraint, count what survives. That survivor count is $n$.

8 — The prerequisite map

The solid arrows are the build order of this page. The dotted arrows point to where you go next: Constraints — holonomic vs non-holonomic deepens the "$f=0$" idea, and Configuration space and phase space gives a home to the surviving $n$ numbers. From there the machinery of Lagrangian mechanics — the Lagrangian L = T - V and the Euler–Lagrange equations takes over, with orientation angles handled by Rigid body kinematics — Euler angles.

Equipment checklist

Test yourself — cover the right side and answer aloud.