Calculus of Variation: Extrema & Further Variations

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SUMMARY

The discussion focuses on the calculus of variations, specifically the conditions for determining extrema of functionals. It establishes that if the first variation, denoted as ##\delta I##, equals zero, further analysis is required to classify the extremum. The second variation, ##\delta^2 I##, indicates a minimum if positive and a maximum if negative. In cases where both the first and second variations are zero, the third variation, ##\delta^3 I##, is examined to ascertain the nature of the extremum, drawing parallels to single-variable functions and their derivatives.

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  • Understanding of functional analysis and variational principles
  • Familiarity with calculus, particularly derivatives and their significance
  • Knowledge of the concepts of maxima and minima in mathematical functions
  • Experience with the notation and symbols used in calculus of variations
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  • Study the implications of higher-order variations in calculus of variations
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LagrangeEuler
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If for some functional ##I##, ##\delta I=0## where ##\delta## is symbol for variation functional has extremum. For ##\delta^2 I>0## it is minimum, and for ##\delta^2 I<0## it is maximum. What if
##\delta I=\delta^2 I=0##. Then I must go with finding further variations. And if ##\delta^3I>0## is then that minimum? Or what?
 
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I suggest you ask the same question in a simpler situation: ordinary single-variable function.
 
Use the first non-vanishing derivative. If it is an odd derivative, than you have and inflection point, otherwise it will be either a maximum or a minimum depending on its sign
 

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