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Calculus of variation

  1. Feb 26, 2014 #1
    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?
     
  2. jcsd
  3. Feb 26, 2014 #2
    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.
     
  4. Feb 28, 2014 #3
    So then how I could know? Is it minimum or maximum?
     
  5. Feb 28, 2014 #4
    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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