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Hope this is not a stupid "what is X" question.

I read a book from applied mathematics and I failed to find any reference for the concept "conserved". I am not sure if this is a mathematical jargon or if there is a definition for it.

What does it mean when one says that "the $L^2$-norm of $u$ is conserved"?

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  • $\begingroup$ possible duplicate of the minimization of a functional from stochastic differential equations $\endgroup$ – PrimeNumber Apr 12 '11 at 22:35
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    $\begingroup$ It means that the value is unchanged. To quote Wikipedia, "In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves." $\endgroup$ – Dave Radcliffe Apr 12 '11 at 23:17
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    $\begingroup$ It would be useful if you'd provide more context, but I'd guess that the is some transformation $t$ acting on $u$, in which case this means that the $L^2$-norm of $t(u)$ is the same as the $L^2$ norm of $u$. $\endgroup$ – Alex Becker Apr 12 '11 at 23:17
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    $\begingroup$ @Alex @Dave: I think both of those comments are reasonable answers to the question. May I convince you to post them as such? $\endgroup$ – Willie Wong Apr 12 '11 at 23:42
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It would be useful if you'd provide more context, as the precise meaning varies from case to case. In general, the term "conserved" occurs when you are studying something which is being changed, yet some property of the thing remains the same. We then say that this property is "conserved".

In your case, I'd guess that you have some transformation $t$ acting on $u$, in which case "the $L^2$-norm of $u$ is conserved" means that the $L^2$-norm of $t(u)$ is the same as the $L^2$-norm of $u$.

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    $\begingroup$ A beautiful demonstration of how a non-question is well answered by a non-answer. As Lenny Susskind would say : information is conserved! $\endgroup$ – uniquesolution Dec 18 '17 at 10:37

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