- #1
nickthequick
- 39
- 0
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
\mathcal{L}=\frac{1}{2\pi}\int_0^{2\pi}Ld\theta
where \theta is defined as \theta_x=k and \theta_t=-\omega 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 L=L(\theta;x,t) and the Euler-Lagrange equation then becomes
\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
If I integrate the above equation over \theta from 0 to 2\pi 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, \mathcal{L}. So my question is this, how does the term
\frac{1}{2\pi}\int_0^{2\pi}\frac{\partial }{\partial x }\frac{\partial L}{\partial \theta_x} \ d\theta
relate to
\frac{\partial }{\partial x }\frac{\partial }{\partial \theta_x} \mathcal{L}
Thanks!
Nick
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
\mathcal{L}=\frac{1}{2\pi}\int_0^{2\pi}Ld\theta
where \theta is defined as \theta_x=k and \theta_t=-\omega 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 L=L(\theta;x,t) and the Euler-Lagrange equation then becomes
\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
If I integrate the above equation over \theta from 0 to 2\pi 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, \mathcal{L}. So my question is this, how does the term
\frac{1}{2\pi}\int_0^{2\pi}\frac{\partial }{\partial x }\frac{\partial L}{\partial \theta_x} \ d\theta
relate to
\frac{\partial }{\partial x }\frac{\partial }{\partial \theta_x} \mathcal{L}
Thanks!
Nick