Averaged Lagrangian and the equations of motion

[nickthequick]

Hi,

Qualitatively: I am trying to decipher a method I've found in the literature, namely Whitham's method. It is a technique used to averaged out "fast variations" in the Lagrangian to then deduce governing equations for the system. I am trying to quantitatively deduce how accurate Whitham's method is, and whether or not it ignores relevant information at higher orders of the small parameter.

Quantitatively: Consider a Lagrangian $L=L(\phi,\eta;x,t)$ where $(\phi,\eta)$ represent the dependent variables of the system and (x,t) are the independent variables. We are going to take a WKB expansion of the dependent variables, so that

$\phi = \sum_n a_ne^{i\theta}$ and

$\eta = \sum_n b_n e^{i\theta}$.


We assume $a_n,b_n = f_n(a,a_t,a_x,...)$ so that our new dependent variables are $(a,\theta)$ and possible derivatives on these variables.

It is assumed that the coefficients are $O(\epsilon)$, for small parameter $\epsilon$ and vary slowly in space and time (for instance $a_n = a_n(\epsilon x, \epsilon t)$) while the phase is $\theta = kx-\omega t + \epsilon \sigma(\epsilon x, \epsilon t)$ for $(k,\omega) \in \mathbb{R}$, ie it has a 'fast scale' .

We now define

$\mathcal{L}=\frac{1}{2\pi} \int_0^{2\pi} L \ d\theta$.

It is conjectured that variations of this "averaged Lagrangian" $\mathcal{L}$ will then give us our governing equations.

My question is this: What, quantitatively, is the difference between the equations deduced via the condition $\delta L =0$ and $\delta \mathcal{L} = 0$ ?

Any suggestion are appreciated. Also, if this is vague or unclear, let me know and I will provide more information/examples.

Thanks,

Nick

 

[eljose79]

Hi Nick,

Whitham's method is a powerful technique for finding governing equations for systems with small parameters, such as in your case where the coefficients are O(\epsilon). It has been widely used in various fields of physics and has been shown to be accurate in many cases.

To quantitatively assess the accuracy of Whitham's method, you can compare the equations deduced from \delta L = 0 and \delta \mathcal{L} = 0. The difference between these equations would give you an idea of how much information is being ignored at higher orders of the small parameter. In general, the smaller the difference, the more accurate Whitham's method is.

However, it is important to note that Whitham's method is an approximation and may not capture all the dynamics of the system. In some cases, higher order terms may be necessary to fully describe the system. It is also possible that the averaged equations from \delta \mathcal{L} = 0 may not be valid for all values of the small parameter \epsilon. Therefore, it is always important to carefully consider the limitations and assumptions of Whitham's method in your specific system.

I hope this helps. Let me know if you have any further questions or need more clarification. Good luck with your research!