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Lagrangian symmetry problem

  1. Sep 28, 2015 #1
    1. The problem statement, all variables and given/known data
    Given L (q, dq/dt, t).
    translation: q ---> q + e (e is infinitesimal constant)
    show that if ∂L/∂q = 0, then L is symmetry under the above translation.
    then find conserved quantity.

    2. Relevant equations
    S = ∫ L dt

    3. The attempt at a solution
    My attempt is nothing... because I don't know the proper procedure to prove a symmetry. Do I simply prove L' - L = 0? But then the next part askes me to find conserved quantity. I have no idea how to find it...
    So, can anyone spare me some hints...?
     
    Last edited: Sep 28, 2015
  2. jcsd
  3. Sep 28, 2015 #2

    CAF123

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    Gold Member

    Let ##L_1=L_1(q, \dot{q}, t)## and ##L_2## that with the replacement ##q\rightarrow q+ \epsilon##. Now consider the expansion of ##L_2## in the infinitesimal parameter ##\epsilon##.
     
  4. Sep 28, 2015 #3
    do you mean taylor expansion??
     
  5. Sep 28, 2015 #4
    A conserved quantity means null time derivative of that quantity. Look at the Euler-Lagrange equation.
     
  6. Sep 28, 2015 #5

    CAF123

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    Yup :)
     
  7. Sep 28, 2015 #6
    cool, I will try it first!
     
  8. Sep 28, 2015 #7
    the taylor series has a term (x-a), what should be the a?
     
  9. Sep 28, 2015 #8
    What should be the x?
     
  10. Sep 28, 2015 #9
    x is q?
     
  11. Sep 28, 2015 #10
    That's not what CAF123 suggested you to do. L is a function of phase space: you are interested in a small perturbation of the position variable, which changes the whole function L. So this means taking a McLaurin expansion in the infinitesimal translation parameter.
     
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