Foundations — Calculus of variations — functionals, functional derivative

This page assumes nothing. If a symbol appears on the parent note and you are not 100% sure what it means, it is defined here, in order, each anchored to a picture.

0. What a function even is (our starting stone)

The one fact we will reuse endlessly: at the lowest point of the curve the tangent line is flat — its slope is zero. That "slope $=0$" is the seed of the whole subject.

• WHY the topic needs it: the entire calculus of variations is one giant analogy — "flat slope at the bottom," but for curves-of-curves instead of points. If the picture of slope zero at a minimum is solid, everything else is translation.

1. The derivative $f^{\prime }(x)$ and the slope

• Picture: in the figure above, the flat orange line at the valley floor has slope $0$; a tilted line anywhere else has non-zero slope.
• WHY the topic needs it: "$f^{\prime }(x)=0$ at a minimum" is the sentence the whole subject upgrades. We will meet the symbols $\partial L/\partial y$ and $d/dx$ — both are kinds of derivative, so this is the ancestor of every symbol on the parent page.

2. Two flavours of derivative: partial $\partial$ vs total $d$

The parent page writes both $\frac{\partial L}{\partial y}$ and $\frac{d}{dx}$. They are different tools. You must not mix them.

• WHY the topic needs it: the Euler–Lagrange equation is literally partial minus total: $\frac{\partial L}{\partial y}-\frac{d}{dx}\frac{\partial L}{\partial y^{\prime }}$. If you cannot tell the two apart, the equation is gibberish.

3. The integral ${\int }_a^b$ — adding up a whole curve

• Picture: the shaded band in the figure below is the running total the integral collects.
• WHY the topic needs it: a functional adds up a cost along the entire curve — length, time, energy. That "add up along the curve" is exactly an integral. No integral, no functional.

4. The unknown is now a curve: $y(x)$, $y^{\prime }(x)$, and endpoints

• WHY the topic needs it: we are choosing among all wires that start and end at the pins. Fixing the ends is what later makes a pesky boundary term vanish (Section 6).

5. The Lagrangian $L$ and the functional $J[y]$

• WHY the topic needs it: this is the object we minimise. Everything else on the parent page exists to answer "which curve $y$ makes $J[y]$ smallest?"

6. The wiggle: $\epsilon$, the test function $\eta (x)$, and $\Phi (\epsilon )$

This is the single genuinely new idea. Master this picture and the parent's derivation is inevitable.

• WHY the endpoints matter now: integration by parts (parent, Section 2) spits out a boundary term $[L_{y^{\prime }}\eta {]}_a^b$. Because $\eta (a)=\eta (b)=0$, this term is $0$ and drops out — that is the whole reason we demanded fixed endpoints.

7. Two named results the parent leans on

• WHY the topic needs it: this lemma is what lets the parent go from "$\int (\dots )\eta dx=0$ for all $\eta$" to the actual Euler–Lagrange equation "$(\dots )=0$." Without it we would be stuck with an integral, not an equation.

Prerequisite map

Where these feed onward: the machinery here powers Lagrangian mechanics, Geodesics and differential geometry, the Brachistochrone problem, and connects to Constrained optimization & Lagrange multipliers, Ordinary differential equations (the Euler–Lagrange equation is an ODE), and the broader setting of Functional analysis.

Equipment checklist

Cover the right side; you are ready for the parent note only if each reveal matches your own words.