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!

Homework Help: Simplifying Lagrangian's Equations (Classial Dynamics)

  1. Aug 14, 2010 #1
    1. The problem statement, all variables and given/known data
    [PLAIN]http://img96.imageshack.us/img96/9288/lagrangiansimplifcation.jpg [Broken]
    "[URL [Broken]

    If you require more info in the derivation, it's on page 1:

    2. Relevant equations
    L = T - V

    3. The attempt at a solution

    I understand exactly the process of obtaining the Lagrangian... But I do not understand his simplifying process at all.

    Could somebody please help/forward me into the right direction?

    Thanks in advance!
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 14, 2010 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    If you think in terms of the action, you can always add a constant such that, when varied, the action will give the same equations of motion. This translates into meaning that you can always add to the Lagrangian (a) a constant, (b) a function that depends on time but not on the coordinates in the Lagrangian or (c) a total time derivative of a function.

    The first two in the list fall into category (b). If we change the Lagrangian so that


    then the action is

    [tex]\tilde{S}=\int_{t_1}^{t_2}{\tilde{L}}dt=\int_{t_1}^{t_2}{L}dt+\int_{t_1}^{t_2}F(t)dt=S+\int_{t_1}^{t_2}F(t)dt=S+{\rm const.}[/tex]

    In the third point on that list, a term is rewritten in terms of a total time derivative plus some other term; the total time derivative then being discarded. To see why, let's look at the change of the action due to the Lagrangian changing to


    which gives

    =S+G(\varphi(t_1),t_1)-G(\varphi(t_2),t_2)=S+{\rm const.}[/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook