Calculus of Variations Intro

Key idea

Find functions that optimise integrals; Euler–Lagrange equation.

In depth

Instead of optimising a number, you optimise a functional, such as the length of a path. The Euler–Lagrange equation d/dx(∂L/∂y′) = ∂L/∂y determines the best function. Shortest path in a plane is a straight line.

Worked example

For L = √(1 + y′²), the equation forces y′ constant, so straight lines.

Real-world uses

• Finding the shortest path or the shape of a hanging chain in physics and optimal design

Common mistake

Treating y as a variable rather than a function.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

Expand all Collapse all

Related expert topics in Calculus

• Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
• Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
• Logarithmic Differentiation – Take logs before differentiating products and powers.
• Related Rates – Link rates of change of connected quantities.
• Linear Approximation – Approximate f near a using its tangent line.
• Differentials – Use dy = f′(x)dx for small changes.