Derivation of Euler Lagrange, variations

In summary, the conversation discusses the differences between a simple localized geometric derivation and the standard derivation of the Euler-Lagrange equation. The localized geometric derivation is specific to a certain problem, while the standard derivation is applicable to all variational problems that meet certain conditions. The conversation also mentions the importance and interesting past of the subject of calculus of variation.
  • #1
cosmic onion
27
2
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.
 
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  • #2
cosmic onion said:
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.
 
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  • #3
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.
 
  • #4
cosmic onion said:
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.
 
  • #5
Thank you for this insight. Only started to learn the subject. Find it very interesting and it also seems to have an interesting past.
 

Related to Derivation of Euler Lagrange, variations

1. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a fundamental equation in classical mechanics that describes the motion of a system based on its energy and constraints. It is used to derive the equations of motion for a system by considering the variations of a functional.

2. How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived using the calculus of variations, which involves finding the functional derivative of a given functional. This derivative is then set to zero, resulting in the Euler-Lagrange equation.

3. What are the applications of the Euler-Lagrange equation?

The Euler-Lagrange equation has many applications in physics, engineering, and mathematics. It is commonly used in classical mechanics, optimal control theory, and in the study of partial differential equations.

4. Can the Euler-Lagrange equation be extended to higher dimensions?

Yes, the Euler-Lagrange equation can be extended to higher dimensions, such as in the case of a functional with multiple independent variables. This is known as the multi-dimensional Euler-Lagrange equation.

5. What is the relationship between the Euler-Lagrange equation and the principle of least action?

The Euler-Lagrange equation is directly related to the principle of least action, which states that the actual path of a system between two points is the one that minimizes the action integral. The Euler-Lagrange equation is used to derive the equations of motion that satisfy this principle.

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