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 [itex]L=L(\phi,\eta;x,t)[/itex] where [itex](\phi,\eta)[/itex] 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

[itex]\phi = \sum_n a_ne^{i\theta}[/itex] and

[itex]\eta = \sum_n b_n e^{i\theta}[/itex].


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

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

We now define

[itex]\mathcal{L}=\frac{1}{2\pi} \int_0^{2\pi} L \ d\theta [/itex].

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

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

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

Thanks,

Nick