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I have been reading a little about calculus of variations. I understand the basic method and it's proof. I also understand Lagrange multipliers with regular functions, ie since you are moving orthogonal to one gradient due to the constraint, unless you are also moving orthogonal to the other gradient you have some component of movement along the gradient and it will change the value of the function, so the gradients have to be multiples of each other.

But when reading about the use of Lagrange multipliers in calculus of variations with an integral constraint, I haven't found an explanation which I could understand. If anyone has a good link or an explanation I would be glad.

For example, when I read this one:

https://netfiles.uiuc.edu/liberzon/www/teaching/cvoc/node38.html [Broken]

Why are the Lagrange multipliers introduced? Both of the integrals have to be zero, when n is such a function that preserves the constraint, but why does the Euler-Lagrange equation of one have to be a multiple of the other? Wouldn't they have to be both zero, or what is going on?

But when reading about the use of Lagrange multipliers in calculus of variations with an integral constraint, I haven't found an explanation which I could understand. If anyone has a good link or an explanation I would be glad.

For example, when I read this one:

https://netfiles.uiuc.edu/liberzon/www/teaching/cvoc/node38.html [Broken]

Why are the Lagrange multipliers introduced? Both of the integrals have to be zero, when n is such a function that preserves the constraint, but why does the Euler-Lagrange equation of one have to be a multiple of the other? Wouldn't they have to be both zero, or what is going on?

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