Classical Mechanics

Why Lagrange’s Equations?

In classical mechanics, Newton’s second law describes all the dynamics of a particle / system of particles, and it seems that we just need to turn the mathematical crank (i.e. F = ma) to grind out the motions of everything.

However in many real world problems, Newton’s second law is way too simple, making it not applicable. This is especially true when there are restriction conditions, say, a rigid circle rolling on a surface. The molecules on the circle will exert electro-magnetic force against each other (which makes the circle ‘rigid’). It’s nearly impossible to use Newton’s second law to solve the motion of this circle.

To solve these problems, we need a more powerful mathematical device: Lagrange’s equations. It’s a generalized version of Newton’s second law, in that it uses generalized coordinates instead of the traditional Cartesian coordinates.