The Lagrangian

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How a special function, called the "Lagrangian", can be used to package together all the steps needed to solve a constrained optimization problem.

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This man is bringing me single handedly through my economics undergraduate degree!!! Keep it up and thank you!

flomoose

Perhaps the best explanation on the Internet. Thanks 3B1B !

Vaibhav

Arguably the best explanation of Lagrangian on the internet! Thanks @Khan Academy!

RaviShankar-jmqw

I just...can't believe that I took a whole course in Optimization and nobody ever told me this is where the Lagrangian comes from. It's so clear.

NoahTopper

The usefulness of the Lagrangian equation is not only because it's easier to program into a computer. The lambda can also be interpreted as the amount that R(x, y) would increase if the constant b in the constraint function could be relaxed by one unit. That can also be explained given only the geometric setup, but it's a little easier to calculate it given the Lagrangian.

WigglyHedgehog

Awesome explanations! I've been struggling to figure this out for 3 years in my Ph.D. and this is the best one!

joezhang

The art of creatively adding zero: B(x, y)=4 means B(x, y)-4 = 0 means lambda(B(x, y)-4) = 0 means R(x, y) - lambda(B(x, y)-4) = R(x, y) provided the constraint holds. So lambda only has an effect when the constraint does not hold. Minimising the magnitude of influence of lambda (i.e. how much effect changing lambda has) is equivalent to making that influence zero (since influence=0 can be attained when the constraint is), and making that influence zero for nonzero lambda means obeying the constraint. I always enjoy cleverness like this in mathematics. Zero is a creature with many disguises.

Chalisque

Khan academy is the GOAT of mathematical/physics topics!

zachmiles

Very clear and concise explanation. Thanks a lot. I had the lagrange function in many courses at college but this is the first time I fully understand the concept behind it.

vambire

thanks. I am studying the mechanical aspect of the lagrangian, applied to a mechanical system that is conservative. This approach is useful to widen my view of the probelam!

cesaaargm

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moussatouhami

Turning some calculus into geometry is helpful to those who visualize. Thank you!

silverlining

I just clicked on this video, because I trust Khan Academy as a knowledge source. But Grant was a welcome surprise.

peterfisher

Simplest way of explaining the most complicated looking concept. Hats off to you Sir 👌👌👌

principaljpnce

Thank you! I have this issue in lectures where I frequently zone-out/forget to turn up so I appreciate your thorough clear explanation which leaves no knowledge gaps :)

eughyuck