Foundations — Applying E-L equations to various systems

Before you can use the parent recipe, every squiggle in it must mean something concrete. This page builds each one from nothing, in the order they depend on each other. If a term below already feels obvious, skim — but nothing here is assumed.

1. What is a coordinate?

The picture: a bead threaded on a straight wire. To say where the bead is, you need exactly one number — how far along the wire it sits. That number is a coordinate.

Why the topic needs it: the whole method starts with "pick coordinates". If you don't know what a coordinate is, step 1 of the recipe is a black box.

Notice two different bead-on-wire pictures give the same idea: one number = one position. The number can be a distance (metres along a rod) or an angle (degrees around a circle). Both are coordinates.

2. Degrees of freedom (DOF)

The picture:

• A bead on a straight wire → 1 DOF (one number: how far along).
• A free dot on a flat table → 2 DOF (two numbers: left-right and up-down).
• A free dot floating in a room → 3 DOF (three numbers).

Why the topic needs it: the parent's step 1 is "Count DOF → pick that many coordinates." Too few coordinates and you can't describe the motion; too many and they aren't independent. DOF tells you the exact right count.

3. Generalized coordinates $q_i$

Reading the notation: the letter $q$ is a stand-in for "some coordinate". The little subscript $i$ is a label — $q_1,q_2,q_3,\dots$ — one $q$ per degree of freedom. If there are 2 DOF you'll see $q_1$ and $q_2$.

The picture: the same pendulum bob, described two ways.

Why the topic needs it: the freedom to pick the cleverest coordinate is the entire payoff of the method. Choosing $\theta$ for the pendulum or $s$ along the incline is what makes the constraint force vanish. The symbol $q_i$ is deliberately generic so one recipe covers all these choices at once.

4. The dot: $\dot{q}$ means "rate of change"

Why a dot and not something else? It is just shorthand for $\frac{dq}{dt}$ — "change in $q$ divided by change in time". Physicists write the dot because time-derivatives appear on nearly every line and $\dot{q}$ is faster than $\frac{dq}{dt}$.

The picture: if $q=$ angle of the pendulum, then $\dot{q}$ is how fast the angle sweeps — the swing's angular speed. Two dots, $\ddot{q}$, means the rate of change of the rate of change — the acceleration.

Why the topic needs it: the Lagrangian depends on both position $q$ and its speed $\dot{q}$, and the final equation of motion is about the acceleration $\ddot{q}$. Without the dot you can't even read $L(q,\dot{q},t)$.

5. Kinetic energy $T$

Why the letter $T$? Historical convention (from older mechanics texts). It has nothing to do with time or temperature — it just means kinetic energy. Get used to it, because the parent uses $T$ everywhere.

The picture: a heavier ball or a faster ball is harder to stop — more $T$. Because $v$ is squared, doubling the speed quadruples the energy.

Why the topic needs it: $T$ is half of the energy score. Every example computes $T$ first.

6. Potential energy $V$

The picture: lift a book to a shelf — you've stored energy in it; let go and gravity converts that store back into motion (kinetic energy). Higher = more stored $V$.

Zero is your choice. Only differences in $V$ matter, so you pick where "$V=0$" is. In the pendulum we set the lowest point to zero, giving $V=mg\ell (1-\cos \theta )$ (zero at the bottom, maximum at the top).

Why the topic needs it: $V$ is the other half of the energy score. The full Lagrangian is $L=T-V$.

7. The Lagrangian $L=T-V$

Why minus, not plus? Total energy is $T+V$. The Lagrangian's minus sign is not an accident — it is exactly the combination whose "least action" reproduces Newton's laws. You'll see the deep reason when you meet 2.1.05-Deriving-the-Euler-Lagrange-equation; for now, accept $L=T-V$ as the definition.

8. Partial derivatives $\frac{\partial L}{\partial q}$ and $\frac{\partial L}{\partial \dot{q}}$

The picture: imagine $L$ as the height of a landscape, with two ground directions labelled $q$ (east) and $\dot{q}$ (north). Walking east measures the eastward slope $\partial L/\partial q$; walking north measures $\partial L/\partial \dot{q}$. Each partial is the steepness in one chosen direction.

Why the topic needs it: the E–L equation is built entirely from these two partials. If you can't compute them, you can't apply the recipe.

9. The full time-derivative $\frac{d}{dt}$

Difference from the partial: the partial freezes the other variables; the total $d/dt$ lets all of them ride along with time. In the E–L equation the term $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)$ first freezes-and-differentiates ($\partial$), then un-freezes and lets time flow ($d/dt$).

The picture: $\frac{\partial L}{\partial \dot{\theta }}=m{\ell }^2\dot{\theta }$ for the pendulum. Applying $\frac{d}{dt}$ to it gives $m{\ell }^2\ddot{\theta }$ — the $\dot{\theta }$ became $\ddot{\theta }$ because time kept flowing. This is where the acceleration finally appears.

10. Putting it together: the Euler–Lagrange equation

Each piece traces back to a section above: $L$ (§7), ${\dot{q}}_i$ and the dot (§4), the partials (§8), the total derivative (§9), and $q_i$ (§3). Nothing in it is now mysterious.

11. Cyclic coordinate & conserved momentum (preview)

The picture: in a central-force problem $L$ contains $\dot{\theta }$ but not $\theta$ itself — the physics doesn't care which way "north" is. That symmetry hands you a conserved quantity (angular momentum) for free. This deep link between symmetry and conservation is Noethers-theorem, expanded in 2.1.07-Cyclic-coordinates-and-conservation-laws.

Why the topic needs it: Example 4 in the parent uses exactly this to get $L_z=m{r}^2\dot{\theta }=\text{const}$ without any extra work.

Prerequisite map

Read it top to bottom: positions and rates feed the two energies, the energies form $L$, $L$ feeds the partials, and the partials plus the total time-derivative assemble the Euler–Lagrange equation — the engine of the whole parent topic.

Equipment checklist

Cover the right-hand side and test yourself. If any answer is fuzzy, re-read that section before tackling the parent's examples.

Ready? Head back to the parent topic and run the five-step recipe on the pendulum.