Time-Dependent Classical Lagrangian with variation of time

[586]

Hello everyone!

I was reading the following review:

http://relativity.livingreviews.org/open?pubNo=lrr-2009-4&page=articlesu23.html

And I got stuck at the first equation; (10.1)

So how I understand this is that there are two variations,

$\tilde{q}(t)=q(t)+\delta q(t) \hspace{1cm} \text{and} \hspace{1cm} \tilde{t}=t+\delta t$

Further we also have a `total variaton' for q at first order:
$\tilde{q}(\tilde{t})=q(t)+\delta q(t)+\dot{q}(t)\delta t$

and its derivative,
$\dot{\tilde{q}}(\tilde{t})=\dot{q}(t)+\delta\dot{q}(t)+\ddot{q}(t) \delta t$

So now how is $\delta L(q,\dot{q},t)$ defined?

 

[UltrafastPED]

Is your question about the paper, or about the fundamentals of the calculus of variations?

If the latter, see the attachment at https://www.physicsforums.com/showthread.php?t=752726#2

It contains a leisurely introduction to the calculus of variations, followed by derivations of Lagrangians & Hamiltonians.

 

[586]

The question is about the fundamentals of calculus of variations. I know how to derive the usual Euler-Lagrange equations without the extra variation in t $\tilde{t}=t+\delta t$. But I am having trouble incorporating this extra variation.

if i define:
$\delta L = \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial q} \frac{\partial q}{\partial t}\delta t + \frac{\partial L }{\partial \dot{q} } \delta \dot{q} +\frac{\partial L}{\partial \dot{q}} \frac{\partial \dot{q}}{\partial t}\delta t + \frac{\partial L}{\partial t} \delta t$

then it gets a similar result as (10.1) but everywhere there is $\dot{q}$ they have $-\dot{q}$.

Its driving me pretty crazy. Any help would be greatly appreciated.

 

[UltrafastPED]

Then download the attachment ...

 

[586]

Unfortunately the attachment does not treat coordinate variations.

 

[UltrafastPED]

Then you will need to find a specialized textbook.

 

[586]

solved:

http://physics.stackexchange.com/qu...mechanics/119339?iemail=1&noredirect=1#119339

thanks!