# What is a Lagrangian

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

The Lagrangian is a function that summarizes equations of motion. It appears in the action, a quantity whose extremum (minimum or maximum) yields the classical equations of motion by use of the Euler-Lagrange equation. In quantum mechanics, the action, and thus the Lagrangian, appears in the path integral, and the eikonal or geometric-optics limit of the path integral is finding the extremum of the action.

Though originally developed for Newtonian-mechanics particle systems, Lagrangians have been used for relativistic particle systems, classical fields, and quantum-mechanical systems. Lagrangians are often more convenient to work with than the original equations of motion, because one can easily change variables in them to make a problem more easily tractable.

Lagrangians are also easier for testing for symmetries, since unlike equations of motion, they are unchanged by symmetries.

Equations

Action principle: the action I for time t and system coordinate q(t) is the integral of the Lagrangian L:
$I = \int L(q(t), \dot q(t), t) dt$

The extremum of the action yields the Euler-Lagrange equation, which gives:
$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q(t)}\right) - \frac{\partial L}{\partial q(t)} = 0$

with appropriate terms for any higher derivatives which may be present.

It is easily generalized to multiple independent variables $x_i$ and multiple dependent variables $q_a(x)$:
$\sum_i \frac{\partial}{\partial x_i}\left(\frac{\partial L}{\partial (\partial q_a(x) / \partial x_i)}\right) - \frac{\partial L}{\partial q_a(x)} = 0$

Extended explanation

Let us start with a Newtonian equation of motion for a particle with position q and mass m moving in a potential V(q,t):
$m \frac{d^2 q}{dt^2} = - \frac{\partial V}{\partial q}$

It can easily be derived from this Lagrangian with the Euler-Lagrange equations:
$L = T - V$

where the kinetic energy has its familiar Newtonian value:
$T = \frac12 m \left( \frac{dq}{dt} \right)^2$

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