Lagrangian Mechanics

Generalized coordinates and the Euler-Lagrange equations

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The Brachistochrone Challenge

A ball rolls down from A to B under gravity. What path gets it there fastest?

This 300-year-old puzzle stumped mathematicians until they discovered a revolutionary new way to think about physics - Lagrangian mechanics.

The Challenge: Find the Fastest Path

A frictionless bead slides down a wire from the start point to the finish point. Which wire shape gets the bead to the bottom fastest? (Hint: the shortest path isn't necessarily the quickest!)

Which path is the fastest?

Take a guess! This problem challenged Newton, Leibniz, and the Bernoulli brothers in 1696.

The Lab

The Lab

The Lab

The Lab

Lagrangian Mechanics

Generalized coordinates and the Euler-Lagrange equations

The Brachistochrone Challenge

A ball rolls down from A to B under gravity. What path gets it there fastest?

The Challenge: Find the Fastest Path

Which path is the fastest?

Lagrangian Mechanics

Generalized coordinates and the Euler-Lagrange equations

The Brachistochrone Challenge

A ball rolls down from A to B under gravity. What path gets it there fastest?

The Challenge: Find the Fastest Path

Which path is the fastest?