I always like to do the variational calculations in rigor way for example like this(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

0 = D_{\alpha} \Big(\int dt\; L(x(t)+\alpha y(t))\Big)\Big|_{\alpha=0} = \cdots

[/tex]

because this way I understand what is happening. However in literature I keep seeing the quantity

[tex]

\delta x(t)

[/tex]

being used most of the time. What does this delta mean really? Does it have a rigor meaning? It seems to be same kind of mystical* quantity as the [itex]df[/itex], but this time an... infinite dimensional differential?

*: mystical in the way, that even if the rigor meaning exists, it is not easily available, and the concept is usually used in non-rigor way.

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# What is the variation

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