Foundations — Kinetic energy in generalized coordinates

This is the ground floor. If the parent note ever wrote a symbol you couldn't say out loud, it lives here, defined and drawn.

1. A particle, its position, and the arrow $\mathbf{r}$

The picture: an arrow from a chosen corner of the room to the ball. Look at figure s01 — the arrow $\mathbf{r}$ and its three shadow-lengths $x,y,z$ on the walls.

Why the topic needs it: everything downstream is "where is the particle, and how fast is that arrow changing?" We cannot talk about energy until we can point at position.

The subscript in ${\mathbf{r}}_a$ just labels which particle: ${\mathbf{r}}_1$ is particle 1, ${\mathbf{r}}_2$ particle 2, and so on up to ${\mathbf{r}}_N$ for $N$ particles. The bold typeface always means "this is an arrow (a vector), not a single number".

2. The dot $\dot{}$ — velocity as a rate of change

Why a dot and not $d/dt$? Pure shorthand — in mechanics almost every derivative is with respect to time, so Newton's dot saves ink. $\ddot{\mathbf{r}}$ (two dots) would be acceleration, but the parent topic never needs it.

The picture: stand the particle at one instant, then a heartbeat later; the tiny arrow connecting the two positions, divided by that tiny time, is $\dot{\mathbf{r}}$. It points along the direction of travel and its length is the speed.

3. The dot product $\cdot$ — turning two arrows into one number

The special case we use most is an arrow dotted with itself: $\mathbf{v}\cdot \mathbf{v}=v_1^2+v_2^2+v_3^2=|\mathbf{v}{|}^2=v^2,$ the length-squared, i.e. speed-squared. That is why kinetic energy is written $T=\frac{1}{2}m\dot{\mathbf{r}}\cdot \dot{\mathbf{r}}$ — it is $\frac{1}{2}m{v}^2$ dressed in vector clothes.

The picture: figure s02 shows two arrows and the two things the dot product measures — how much one lies along the other, and (when equal) the plain squared length.

4. Kinetic energy $T$ and the $\frac{1}{2}m{v}^2$ we start from

The picture: every ball carries a little energy bag proportional to its mass times its speed-squared; $T$ is all the bags emptied into one pile. Nothing here is mysterious — it is Newtonian energy. The whole art of the parent note is rewriting the ${\dot{\mathbf{r}}}_a$ using better coordinates.

5. Generalized coordinates $q_i$ — the "natural" description

Why $n$ and not $3N$? A single free particle needs 3 numbers; $N$ of them need $3N$. But constraints (a bead trapped on a wire, a pendulum of fixed length) forbid most of those. The number $n$ that actually survives is often much smaller. See Generalized coordinates and constraints.

The picture: figure s03 — a bead on a bent wire. Its full room-position needs $x,y$, but along the wire one number $q$ (arc length) says everything. That one number is a generalized coordinate.

The index $i$ in $q_i$ just labels which generalized coordinate we mean; ${\dot{q}}_i$ (with the dot) is that coordinate's rate of change — the "speed along that natural direction".

6. The position map ${\mathbf{r}}_a(q,t)$ — the dictionary between worlds

The extra slot $t$ (time) appears only when a constraint is itself being moved on a schedule we impose from outside — a wire someone is spinning, a support being raised. Such a constraint is called rheonomic (moving). If nothing external moves the constraint, $t$ is absent and the constraint is scleronomic (rigid, time-frozen).

The picture: two dials ($q$ and, if present, a clock $t$) feeding into a machine that spits out the room-arrow ${\mathbf{r}}_a$.

7. Partial derivatives $\frac{\partial {\mathbf{r}}_a}{\partial q_i}$ and $\frac{\partial {\mathbf{r}}_a}{\partial t}$

The straight-$d$ derivative $d/dt$ changes everything at once; the curly-$\partial$ derivative changes one thing, holding the rest fixed. We need the curly one because our map has several inputs.

Two flavours matter:

• $\frac{\partial {\mathbf{r}}_a}{\partial q_i}$ — motion caused by you turning coordinate $i$.
• $\frac{\partial {\mathbf{r}}_a}{\partial t}$ — motion caused by the constraint moving on its own (only nonzero for rheonomic systems).

Why the topic needs both: the velocity of the particle is the sum of "motion I command through the $q_i$" plus "motion the moving constraint forces on me". That split is literally the parent note's Step 1.

8. The chain rule — how a velocity is assembled

Read it as a sentence: total room-velocity $=$ (for each dial: "which way it pushes me" $\times$ "how fast I'm turning it") summed up, $+$ ("which way the moving constraint drags me").

9. Double sums and the mass matrix $M_{ij}$

When we square the chain-rule velocity, products ${\dot{q}}_i{\dot{q}}_j$ appear for every pair $(i,j)$. Writing ${\sum }_{i,j}$ means "sum over all ordered pairs $i$ and $j$". Each such term carries a coefficient $M_{ij}={\sum }_a{m}_a\frac{\partial {\mathbf{r}}_a}{\partial q_i}\cdot \frac{\partial {\mathbf{r}}_a}{\partial q_j},$ a table of numbers (a matrix) indexed by two labels.

$M_{ij}$ is symmetric ($M_{ij}=M_{ji}$, because the dot product doesn't care about order) and positive-definite (real motion always costs positive energy).

Prerequisite map

Each arrow means "you must understand the tail before the head makes sense." The two streams — the energy stream (left) and the coordinate stream (right) — meet at the topic itself.

Equipment checklist

Test yourself: cover the right side, answer, reveal.