A Place of Ideas

Lagrangian Mechanics

Lagrangian Mechanics is a formalism for understanding the behavior of physical systems. It leads to exactly the same results as the more familiar Newtonian mechanics, but gets there in a completely different way.

Newtonian mechanics is about forces. To understand how an object moves, we analyze the forces that act on it. Through Newton's laws, this gives us the acceleration, which tells us how the velocity is changing over time.

Lagrangian mechanics, on the other hand, is about a quantity called action.

Say you know where an object started and where it is now, but you want to know how it got there. You could imagine all sorts of possible trajectories. Each of these trajectories has a corresponding action; the correct one is the one whose action is smallest.1

The frameworks sound completely different. But try the two on any problem, and the results are always the same. Lagrangian mechanics is just as correct as Newtonian mechanics is; it's just a different route to the same answers.

Demo

Choose a potential field.


On the left you have a possible particle trajectory, drawn on a background that describes a potential field. On the right you have plots of the kinetic and potential energy. The distance between these plots is the Lagrangian, defined as L = T - V. Integrating that over time gives you the action, which is the area of the (highlighted) region between the plots.

Lagrangian mechanics says that if you find a trajectory that makes the action as small as it can get without changing the endpoints, then that trajectory you've found is a physically realistic one.

If your browser has Javascript enabled (the default), you can try this out for yourself! Drag the control points to try out various trajectories. Look for the ones with small action, and check that they make sense.