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Taylor expansion

  1. Dec 26, 2009 #1
    [I posted this in the Classical Mechanics sub forum - but I have not received any responses as yet -I think it is better placed here!]

    I have an expression that I need to approximate. I Taylor expanded a bracket twice over and got the result I wanted to get, however I am not sure if what I have done is mathematically correct or not.

    This is the expression:

    [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]

    I 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:

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

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

    This is equal to:

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

    but

    [tex]\frac{{\partial}L}{{\partial}q}\frac{{\partial}q(t)}{{\partial}t} = \frac{{\partial}L}{{\partial}t}[/tex] ('CANCELLING' the 'dq's - I know this will have the mathematicians up in arms)

    So:

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

    Is it okay to Taylor expand in this way?

    Thanks.
     
  2. jcsd
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