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Prove the Langrangian is not unique

  1. May 8, 2015 #1
    Question:
    If L is a Lagrangian for a system of n degrees of freedom satisfying Lagrange's equations show by direct substitution that

    http://qlx.is.quoracdn.net/main-74d090d14ee4fea0.png [Broken]

    also satisfies Lagrange's equations where F is any arbitrary but differentiable function of its arguments.



    Attempt at a solution:
    I'm not really sure how to solve this problem by direct substitution; I found a way to do it using the action integral but not direct substitution. Any clues or help on how to approach the problem?
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. May 8, 2015 #2

    robphy

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    Use L' as the new lagrangian in Lagrange's equation.
    Does L' satisfy that equation? Under what condition?
     
  4. May 8, 2015 #3
    I got that L' satisfies that equation when the derivative for Lagrange's equation of F is 0.
     
  5. May 8, 2015 #4

    robphy

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    You should probably show your work.
    You can use the LaTeX button below to help you typeset your equations.
     
  6. May 8, 2015 #5
    I don't know how to write LaTeX 11251701_10153310866401660_2140605752_n.jpg
     
  7. May 9, 2015 #6

    samalkhaiat

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    Your question is wrong. The function [itex]F[/itex] can not depend on the velocity [itex]\dot{q}(t)[/itex]. If it does, then [itex]d F /d t[/itex] will depend (at least linearly) on the acceleration [itex]\ddot{q}(t)[/itex]. This in turns mean that the two Lagrangians are not equivalent to each other. So, try to solve the exercise for the function [itex]F(t) = F( q(t) , t )[/itex].

    This is one part of well know theorem which states

    A function [itex]\Lambda[/itex] of [itex]q(t)[/itex], [itex]\dot{q}(t)[/itex] and [itex]t[/itex] satisfies Lagrange’s equations identically (i.e., independent of the [itex]q_{a}(t)[/itex]) if, and only if, it is the total time derivative [itex]d F / dt[/itex] of some function [itex]F ( q(t) , t )[/itex].
     
    Last edited by a moderator: May 7, 2017
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