Introduction to Lagrangian Mechanics

Photo: Rhett Allain

I’ll be honest. There is a lot of stuff to talk about when we are talking about the Lagrangian. It’s not super simple — but it is super powerful.

Newtonian Mechanics

Before jumping into Lagrangian mechanics, let’s think about Newtonian mechanics. At the most basic level, this uses the momentum principle and the work-energy principle:

Newtonian mechanics works best if you can calculate the forces directly. Some examples of calculated forces would be:

• The gravitational force
• Air resistance
• The force from a spring
• Electromagnetic forces

For example, consider a mass hanging vertically on a spring.

Image: Rhett Allain

Since you can get an expression for both the spring force and the gravitational force, you can use the momentum principle to find the acceleration and find an equation of motion (yes, you need to solve a differential equation).