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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.

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# Taylor expansion

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