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Time-Dependent Classical Lagrangian with variation of time

  1. Jun 12, 2014 #1

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    Hello everyone!

    I was reading the following review:

    http://relativity.livingreviews.org/open?pubNo=lrr-2009-4&page=articlesu23.html [Broken]

    And I got stuck at the first equation; (10.1)

    So how I understand this is that there are two variations,

    [itex]\tilde{q}(t)=q(t)+\delta q(t) \hspace{1cm} \text{and} \hspace{1cm} \tilde{t}=t+\delta t [/itex]

    Further we also have a `total variaton' for q at first order:
    [itex]\tilde{q}(\tilde{t})=q(t)+\delta q(t)+\dot{q}(t)\delta t [/itex]

    and its derivative,
    [itex]\dot{\tilde{q}}(\tilde{t})=\dot{q}(t)+\delta\dot{q}(t)+\ddot{q}(t) \delta t [/itex]

    So now how is [itex]\delta L(q,\dot{q},t)[/itex] defined?
     
    Last edited by a moderator: May 6, 2017
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  3. Jun 13, 2014 #2

    UltrafastPED

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    Is your question about the paper, or about the fundamentals of the calculus of variations?

    If the latter, see the attachment at https://www.physicsforums.com/showthread.php?t=752726#2

    It contains a leisurely introduction to the calculus of variations, followed by derivations of Lagrangians & Hamiltonians.
     
  4. Jun 13, 2014 #3

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    The question is about the fundamentals of calculus of variations. I know how to derive the usual Euler-Lagrange equations without the extra variation in t [itex] \tilde{t}=t+\delta t [/itex]. But I am having trouble incorporating this extra variation.

    if i define:
    [itex] \delta L = \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial q} \frac{\partial q}{\partial t}\delta t + \frac{\partial L }{\partial \dot{q} } \delta \dot{q} +\frac{\partial L}{\partial \dot{q}} \frac{\partial \dot{q}}{\partial t}\delta t + \frac{\partial L}{\partial t} \delta t [/itex]

    then it gets a similar result as (10.1) but everywhere there is [itex] \dot{q} [/itex] they have [itex] -\dot{q} [/itex].

    Its driving me pretty crazy. Any help would be greatly appreciated.
     
  5. Jun 13, 2014 #4

    UltrafastPED

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    Then download the attachment ...
     
  6. Jun 13, 2014 #5

    586

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    Unfortunately the attachment does not treat coordinate variations.
     
  7. Jun 13, 2014 #6

    UltrafastPED

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    Then you will need to find a specialized text book.
     
  8. Jun 15, 2014 #7

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