Euler Lagrange and the Calculus of Variations

[Trying2Learn]

TL;DR

I cannot extract the Euler-Lagrange equation with different variation functions

Good Morning all

Yesterday, as I was trying to formulate my confusion properly, I had a series of posts as I circled around the issue.

I can now state it clearly: something is wrong :-) and I am so confused :-(

Here is the issue:

I formulate the Lagrangian for a simple mechanical system (let's go with 1D)

I formulate the variation of the position and the variation of the velocity, use the Gateaux derivative, work it though..

However, as I understand, I MUST vary the position and velocity INDEPENDENTLY!

And that is where the problem begins (for me)

Could someone read the attached, one page and tell me where I go wrong?

(And if someone can tell me how, I can post a Matlab code that actually tries DIFFERENT eta functions and shows that the correct solutoin produces a stationary Action -- and plots the area between KE and PE)

 

[Cleonis]

It is puzzling to me why you insist on an approach where position and velocity are varied independently

You appear to be referring to configuration space. For motion in the case of 1D spatial space: in terms of configuration space you have the position coordinate and the velocity coordinate.
Hamiltonian mechanics is formulated in terms of motion in configuration space.Lagrangian mechanics, the mechanics that uses the Euler-Lagrange equation, is formulated in terms of motion in spatial space.

To derive the Euler-Lagrange equation the applicable variation is variation of position.

 

[Trying2Learn]

It is puzzling to me why you insist on an approach where position and velocity are varied independently

You appear to be referring to configuration space. For motion in the case of 1D spatial space: in terms of configuration space you have the position coordinate and the velocity coordinate.
Hamiltonian mechanics is formulated in terms of motion in configuration space.Lagrangian mechanics, the mechanics that uses the Euler-Lagrange equation, is formulated in terms of motion in spatial space.

To derive the Euler-Lagrange equation the applicable variation is variation of position.

Ah ha! You are correct!

I also found this, just now

https://physics.stackexchange.com/q...it-make-sense-to-vary-the-position-and-the-ve

OK, so I am fine.

Sorry to have been a bother.

But thank you, everyone