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!

Space-time symmetry (Langrangian Mechanics)

  1. Dec 22, 2009 #1
    When deriving the conserved quantity in the case of space-time symmetry, a line in my notes goes from:

    [tex]\int{dt.(1+\epsilon\dot{\xi}).L[q(t+\epsilon\xi)+{\delta}q(t+\epsilon\xi)]} - \int{dt.L[q(t)+{\delta}q(t)]}[/tex]

    where L is the Lagrangian and [tex]\xi[/tex] is a function of time and both integrals are over the same time interval, to:


    I can't see how these two lines equal one another.

    How does the [tex]O(\epsilon^{2})[/tex] come about?

  2. jcsd
  3. Dec 22, 2009 #2


    User Avatar
    Homework Helper

    That would be the result of a Taylor expansion in [itex]\epsilon[/itex]. Also I think you're missing a factor in the last expression, and that it should be
    [tex]\int \mathrm{d}t\,\epsilon\biggl(\dot{\xi}L + \xi\frac{\mathrm{d}L}{\mathrm{d}t}\biggr) + O(\epsilon^2)[/tex]
  4. Dec 22, 2009 #3
    thanks diazona, much appreciated:

    I tried Taylor expansion as you suggested (and I did get the right answer) but I am not sure whether I have done so mathematically correctly or not. Can you check whether or not I am on the right lines...

    We have [tex]\int{dt.(1+\epsilon\dot{\xi(t)}).L[q(t+\epsilon\xi(t))+{\delta}q(t+\epsilon\xi(t))]} - \int{dt.L[q(t)+{\delta}q(t)]}[/tex]

    start with [tex]L[q(t+\epsilon\xi(t))][/tex]

    if we first Taylor expand the bit in parenthesis (ie. [tex]q(t+\epsilon\xi(t))[/tex]) we get



    This is equal to:



    [tex]\frac{{\partial}L}{{\partial}q}\frac{{\partial}q(t)}{{\partial}t} = \frac{{\partial}L}{{\partial}t}[/tex] (CANCELLING the 'dq's)


    [tex]L[q(t+\epsilon\xi(t))]=L(q)+\frac{{\partial}L}{{\partial}t}\epsilon\xi(t)+smaller terms][/tex]

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook