Foundations — Euler-Lagrange equation — derivation

This page assumes you have seen nothing. We build every letter, slash, and squiggle the parent Euler–Lagrange derivation uses, in an order where each piece rests on the one before it. Skip nothing — the parent moves fast because it trusts that you have all of this in your hands.

0. The picture that ties everything together

Look at the figure. Two fixed dots — a start point and an end point. Between them you could draw infinitely many curves. Each curve gets a score (a single number). The calculus of variations is the machine that finds the one curve with the best score. Keep this picture in mind: every symbol that follows is a label on some part of it.

1. A function $y(x)$ — a curve, not a number

• Plain words: "tell me where you are left-to-right ($x$), I'll tell you how high you are ($y$)."
• Picture: any single wiggly line in the figure above.
• Why the topic needs it: the unknown we are solving for is a whole curve $y(x)$, not a single number. This is the big leap from school calculus.

The letters $a$ and $b$ mark the fixed endpoints on the $x$-axis. The two heights we insist on are written $y(a)=y_a$ and $y(b)=y_b$ — "the curve must start here and finish there".

2. The slope $y^{\prime }$ — how steep the curve is

• Plain words: stand on the curve, take an infinitesimally small step to the right of size $dx$; you rise by $dy$. The ratio $dy/dx$ is the steepness.
• Picture: the little right triangle hugging the curve in the figure — horizontal leg $dx$, vertical leg $dy$. Its slant is the tangent line.
• Why the topic needs it: arc length, travel time, and energy all depend on how steep the curve is, so the score of a curve depends on $y^{\prime }$ as well as $y$. That is why the Lagrangian carries a $y^{\prime }$ slot.

3. The integral ${\int }_a^b(\cdots )dx$ — adding up a running total

• Plain words: "sum of height × width over every sliver from $a$ to $b$."
• Picture: the sum of the thin blue rectangles under a curve in the figure; as they get thinner the staircase becomes the smooth shaded region.
• Why the topic needs it: the score of a curve is a running total accumulated along it — total length, total time, total energy. Accumulation is exactly what an integral does. The symbol $\int$ is a stretched "S" for "Sum".

The pieces:

• $\int$ — the summation sign.
• $a,b$ — where the sum starts and stops (the limits).
• $dx$ — the width of each infinitesimal strip.
• The thing between $\int$ and $dx$ — the integrand, the height being summed.

4. The Lagrangian $L(x,y,y^{\prime })$ — the score-per-step rule

• Plain words: at any point on the curve, $L$ says "the local price of being here, this high, tilted this much".
• Picture: imagine a colour painted on each tiny strip of the curve — dark = expensive, light = cheap. $L$ is that colour rule; the integral sums the colours up.
• Why the topic needs it: it is the single object you choose to encode your problem. Shortest path? $L=\sqrt{1+y^{\prime 2}}$. Fastest slide? $L=\sqrt{(1+y^{\prime 2})/y}$. Change $L$, change the question.

5. Two kinds of derivative: $\frac{\partial }{\partial y}$ vs $\frac{d}{dx}$

• Picture: partial = turning one dial on a mixing desk with the others taped down. Total = pushing the master fader; every channel that is wired to it moves too.
• Why the topic needs it: the Euler–Lagrange equation literally contains BOTH — $\frac{\partial L}{\partial y}$ (freeze) minus $\frac{d}{dx}\frac{\partial L}{\partial y^{\prime }}$ (chain-react). Confusing them is the #1 error, so we separate them here.

The curly $\partial$ ("partial d") signals "one slot only". The straight $d$ signals "let the chain rule run".

6. The functional $J[y]$ — a machine that eats a curve, spits a number

• Plain words: "feed me a curve, I'll integrate its score and hand you the total."
• Picture: the whole figure at the top — each candidate curve drops into the box $J[\cdot ]$ and out comes a scoreboard number.
• Why the topic needs it: $J[y]$ is the quantity we minimize. Ordinary calculus minimizes $f(x)$ over numbers; variations minimize $J[y]$ over curves. The square brackets are the visual reminder of that jump.

7. The variation: $\eta (x)$, $\epsilon$, and the perturbed curve $Y$

• Plain words: take the true curve $y$, add $\epsilon$ times some bump $\eta$. Turn $\epsilon \to 0$ and the bump vanishes — you are back on the true curve.
• Picture: the amber true curve plus dashed nearby curves in the figure, each one $y+\epsilon \eta$ for a different $\epsilon$.
• Why the topic needs it: to test "is this curve optimal?", you must be able to nudge the whole curve and check the score doesn't drop. $\eta$ and $\epsilon$ are the nudge.

Because every competitor must still hit the fixed endpoints, the wiggle must die at the ends: $\eta (a)=\eta (b)=0.$ In the figure the dashed curves all pinch back to the two dots — that pinching is this condition. It is exactly what kills the boundary term later in the derivation.

8. Stationary — "flat to first order"

• Plain words: you're standing where the ground is momentarily level; step a hair in any direction and, to first approximation, you neither rise nor fall.
• Picture: the flat bottom of a valley, or the flat top of a hill.
• Why the topic needs it: we do NOT demand a true minimum — only that $J$ is stationary. That is the "$=0$" in ${\frac{d\Phi }{d\epsilon }|}_{\epsilon =0}=0$, where $\Phi (\epsilon )=J[y+\epsilon \eta ]$ is the ordinary function that the score becomes once we fix a wiggle.

9. Integration by parts — trading a derivative between factors

• Plain words: if one factor carries the derivative, you can move the derivative onto the other factor, at the cost of a boundary term $[uv{]}_a^b$.
• Why the topic needs it: in the derivation the wiggle appears as ${\eta }^{\prime }$ (differentiated). We need a bare $\eta$ so we can factor it out. Integration by parts shifts the derivative off $\eta$ and onto $\frac{\partial L}{\partial y^{\prime }}$, and the leftover boundary term dies because $\eta (a)=\eta (b)=0$. See Integration by Parts for the full drill.

Prerequisite map

Read it top-down: raw ideas (function, slope, integral) feed the Lagrangian and the functional; the wiggle plus stationarity plus integration by parts assemble into the Euler–Lagrange equation. Everything routes into the parent derivation, which then powers Lagrangian Mechanics, the Brachistochrone Problem, Geodesics, the Beltrami Identity, and Noether's Theorem — all built on the Principle of Least Action and the Calculus of Variations.

Equipment checklist

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