The Elegance of Variational Calculus

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Prerequisites

If you'd like to read this text, you should be fluent in single variable calculus. Without knowing what a derivative or integral is, you probably won't get far here. The main results are in form of differential equations.

The Brachistochrone Problem

Function Spaces & Functionals

Deriving the Euler-Lagrange Equation

The Fundamental Lemma of Variational Calculus

Before being able to rigorously finish our journey to the main result of this piece of mathematics, we need to take a small detour.
To derive the actual useful formula, there is one more Lemma we need to prove. But stay with me, it's fairly doable.
The Fundamental Lemma

Proof

Suppose ..

The Euler-Lagrange Equation

After applying the Fundamental Lemma to Equation X, we receive the following: $$ $$