1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What is a Lagrangian

  1. Jul 24, 2014 #1
    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:
    [itex]I = \int L(q(t), \dot q(t), t) dt[/itex]

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

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

    It is easily generalized to multiple independent variables [itex]x_i[/itex] and multiple dependent variables [itex]q_a(x)[/itex]:
    [itex]\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[/itex]

    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):
    [itex]m \frac{d^2 q}{dt^2} = - \frac{\partial V}{\partial q}[/itex]

    It can easily be derived from this Lagrangian with the Euler-Lagrange equations:
    [itex]L = T - V[/itex]

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

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: What is a Lagrangian
  1. What is Lagrangian ? (Replies: 2)

  2. Lagrangian for fluids (Replies: 1)

  3. Lagrangian Mechanics (Replies: 4)

Loading...