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Date: Mon, 8 Oct 2012 07:56:28 -0700 (PDT)
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Subject: Re: About the exchangeability of variational and differential.
From: Rupert <rupertmccal...@yahoo.com>
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On Oct 8, 3:52=A0pm, Hongyi Zhao <hongyi.z...@gmail.com> wrote:
> Hi all,
>
> I'm a physics teacher China. =A0Today, when I
> state the principle of least action to my students, I meet a basic
> problem which I cann't figure out:
>
> Under what condition can we exchange the sequence of variational and
> differential?
>
> Say for the following example:
>
> delta ( dq/dt ) =3D d ( delta q) / dt, =A0where, the q is a funtion of t.
>
> What's the condition for the above equality?
>
> I thinks it should have some conditions, but I cann't figure it out.
>
> Could you please give some hints? =A0Thanks a lot in advance for your
> time and patience :-)
>
> Regards,
> .: Hongyi Zhao [ hongyi.zhao AT gmail.com ] Free as in Freedom :.

We want (delta/delta h) ( d(q+hq')/dt ) =3D (d/dt) ( (delta/delta h) (q
+hq') ).

So the problem is about when you are allowed to swap two partial
derivative operators with respect to two different variables when
dealing with a function of two variables. Do you know a sufficient
condition for this?