Wikipedia:Reference desk/Archives/Mathematics/2019 December 16
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December 16
editProving the isoperimetric inequality with Lagrange multipliers applied to calculus of variations
editI'm having a little bit of a conundrum here.
I am formulating the problem as maximizing subject to . If I take the functional derivative of the inside of each integral, multiply the right hand side's functional derivative by a Lagrange multiplier, equate components, and divide the equations, I end up with a tautology. More explicitly, I have which gets simplified to , which is satisfied for all x and y such that the numerator and denominator aren't zero. This leads me to think that one must instead look at where they are zero, which occurs when . This is satisfied only in the case which, however, does not exclude an ellipse.--Jasper Deng (talk) 03:49, 16 December 2019 (UTC)
- I think the problem is you eliminated lambda. You get two equations but they are redundant, so just look at the first equation
- Cancel y' and divide by -λ to get
- in other words curvature is constant, i.e. the curve is a circle. Btw, for those following along at home, the functional derivative in question is given in Euler–Lagrange equation#Several functions of single variable with single derivative but without the "=0". --RDBury (talk) 07:40, 16 December 2019 (UTC)
For this,
why its reverse direction,
is not included? --Ans (talk) 14:02, 16 December 2019 (UTC)
- Possibly it's because that is not a separate law, but rather a result obtained from the second law:
- And by the law of negation of disjunction applied to the part in outermost brackets:
- CiaPan (talk) 14:13, 16 December 2019 (UTC)
- In your third step, Are you are using to conclude ? RoxAsb (talk) 15:31, 16 December 2019 (UTC)
- @CiaPan:, your last step can imply from if , but the law of negation of disjunction in sequent notation (in the article) is in the form, , not the form, . is not sufficient to imply from --Ans (talk) 05:25, 18 December 2019 (UTC)
If no any other opposed comments, I will add the reverse rules in the article, then. --Ans (talk) 05:49, 18 December 2019 (UTC)