Lagrangian

Imagine you have a big land piece and you want to build a beautiful mansion there. For this task you have to hire designers, masons, painter and all sort of workers. getting this work done from them is a pretty difficult task. And you suddenly you came across a machine that can do all sort of work alone. This is more or less same in the case of a lagrange

The word Lagrangian can refer to a mathematical quantity or a description of a way/method to look at physical events. The Lagrangian method is normally contrasted with the Eulerian method. If you follow the motion of few moving things and devise equations to describe the motion and how the other attributes change along the path, then you are using the Lagrangian method. If you stick to ground and observe these moving things and again devise equations to describe the motion and the changes in the attributes of the moving objects, then you are using the Eulerian method. Like being in a plane and putting rules for its flight, or sitting in the control room and observe the same plane.

In the simple case, L is a function of velocity and position. Assume there's only one particle (or object, or field). There could be multiple objects, each with their velocities and positions; and more variables, like time, and second derivative. Ignore those.

The function tells how much "Action" there is at the point X, at velocity V. The integral gives the total Action over the path. Instead of going into the physics further, let's get straight to the layman's explanation.

Action can be considered analogous to the gasoline in a car. If a car is going at speed V at a point X, it requires a certain amount of gas. Skipping details, it depends on the square of V and the grade (inclination) of the road at X. So if you want to go fast uphill, you must press the accelerator to the floor, using a lot of gas per second. (Gas per second is analogous to Energy.) If you go downhill slow, you might need no gas at all. In fact you might build up speed (sort of like potential energy) which will take you part way up the next hill for free. Basically, the faster the V, and the steeper the road at point X, the more gas you need. The L-function tells how much.

If you drive from point A to point B, you'll use a certain amount of gas. That's the L-integral: the L-function (gas at each instant) integrated over the trip route.

Suppose there are a lot of different ways to get from point A to point B. Usually we want to get there as fast as possible. Sometimes we want to minimize the distance traveled. But pretty often, if we're low on funds, we want to minimize gas consumption. In that case, how do you find the best route? Well, you could drive them all, noting how much gas each route takes, but that's extremely wasteful! Instead - assuming we know the L-function, and know how to do the math - we can integrate the L-function to find the L-integral (gas consumed) for each path. Then select the one which minimizes that integral.

It turns out that's what nature does. It uses the minimum amount of gas