Calculus of variation

  • #1
660
16
If for some functional ##I##, ##δI=0## where ##δ## is symbol for variation functional has extremum. For ##δ^2I>0## it is minimum, and for ##\delta^2I>0## it is maximum. What if
##δI=δ^2I=0##. Then I must go with finding further variations. And if ##δ^3I>0## is then that minimum? Or what?
 

Answers and Replies

  • #2
166
1
Finding further variations is useless from this point of view.
The stationarity of the functional, i.e. δI=0 , occurs for maxima, minima and saddles.
 
  • #3
660
16
So then how I could know? Is it minimum or maximum?
 
  • #4
166
1
Here you will find a better explanation than I could give on sufficient and necessary conditions for minima http://www.math.utah.edu/~cherk/teach/12calcvar/sec-var.pdf
If you have the book "introduction to Calculus of Variations" by Fox you will find there a thorough discussion of the second variation: yes further variations are to be computed.
I really do apologise for my previous reply which was wildly inaccurate due to a misunderstanding of mine.
 

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