A Derivation of Euler Lagrange, variations

[cosmic onion]

What is wrong with the simple localised geometric derivation of the Euler Lagrange equation. As opposed to the standard derivation that Lagrange provided.

Sorry I haven't mastered writing mathematically using latex. I will have a look at this over the next few days.

More clarification. I seen a simple derivation that looked at the change in position of a length of rope at a single point and the increase in the gradient to the left and decrease of the gradient to the right at the same point, adding up these variations gave a neat and easy derivation of euler lagrange and made the terms make sense

What's wrong get with the simple euler derivation.

Sorry may have to rewrite this once I have practiced latex.

 

[blue_leaf77]

What is wrong with the simple localised geometric derivation of the Euler Lagrange equation. As opposed to the standard derivation that Lagrange provided.

By 'localized geometric derivation', do you mean something like in this site?
The derivation given there is not wrong, but it's also not general. Euler-Lagrange equation gives you the differential equation for solving the function which makes certain functional stationary, it does not pertain only to the shape of a hanging rope under gravity or to only physical problems, instead it's one of the disciplines in math just like differential and integral calculus, linear algebra etc.

 

[cosmic onion]

Thanks for the reply .
Yes that's exactly the derivation I was thinking about.

So your saying this derivation is not general as it pertains to this particular problem as opposed to the accepted lag range derivation that pertains to all variational problems of this type. ? And that's the answer.

Hope I got this right.

 

[blue_leaf77]

So your saying this derivation is not general as it pertains to this particular problem as opposed to the accepted lag range derivation that pertains to all variational problems of this type. ?

I don't think "all variational problems" is the right phrase here. I believe the variational problem should satisfy a set of conditions, one of which is the differentiability behavior, that must be satisfied by the functions before it can be treated using Euler-Lagrange equation. You should be able to find these conditions in calculus of variation literature.

 

[cosmic onion]

Thank you for this insight. Only started to learn the subject. Find it very interesting and it also seems to have an interesting past.