It seems like X. He found another conserved integral for Euler (the “alignment”):

http://www.sciencedirect.com/science/article/pii/S0893965999000348

It appears the same author later found a sufficient condition for blowup, using an unrelated technique.

]]>When I was a freshman, we set the derivative to zero.

And we drew elaborate charts keeping track of the sign of the second derivative.

All discussions I seen using calculus of variations fall into extremes. Either they justifying the critical points or omitting those details entirely.

Also, I highly recommend using Einstein summation convention — if you wish.

]]>Yes, in the infinite-dimensional Lie group of diffeomorphisms on . For instance, if is the identity map (identifying both and with ), and is a fixed vector field, then the integrated flow map is often denoted , which is consistent with the interpretation of as a logarithmic derivative of .

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