Commuting derivative/Integral (not FTC or Leibniz)

  • #1

Main Question or Discussion Point

Hi,

I'm concerned with finding the Euler-Lagrange equations for slowly modulated surface gravity waves and, as is custom in this type of physical problem, I would like to consider the averaged Lagrangian defined as

[tex]\mathcal{L}=\frac{1}{2\pi}\int_0^{2\pi}Ld\theta [/tex]

where [tex]\theta[/tex] is defined as [tex]\theta_x=k [/tex] and [tex]\theta_t=-\omega [/tex] where k and $\omega$ represent wave number and frequency respectively, which can also be functions of space and time. My Lagrangian, which comes from the physics of the problem, is [tex]L=L(\theta;x,t)[/tex] and the Euler-Lagrange equation then becomes

[tex]\frac{\partial L}{\partial \theta}-\frac{\partial }{\partial x }\frac{\partial L}{\partial \theta_x}-\frac{\partial }{\partial t}\frac{\partial L}{\partial \theta_t}=0 [/tex]

If I integrate the above equation over [tex]\theta[/tex] from 0 to [tex]2\pi[/tex] and normalize, I know the first term goes to 0 because L is periodic (which again comes from the physics). So I'm trying to see if I can write the rest of equation in terms of the averaged Lagrangian, [tex]\mathcal{L}[/tex]. So my question is this, how does the term

[tex]\frac{1}{2\pi}\int_0^{2\pi}\frac{\partial }{\partial x }\frac{\partial L}{\partial \theta_x} \ d\theta[/tex]

relate to

[tex]\frac{\partial }{\partial x }\frac{\partial }{\partial \theta_x} \mathcal{L}[/tex]

Thanks!
Nick
 

Answers and Replies

  • #2
336
0
I don't see any simple relation between the two.Although L is periodic, its second order derivatives may not be so.
 
  • #3
What if we make the additional assumption that L and all of its derivatives are periodic?
 

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