# Aliasing question for a spectrally derived Lagrangian

Consider two functions

$$X=\sum_{n=-N}^N i\ sign(n) y_n e^{-in\xi}; \quad \quad Y = \sum_{n=-N}^N y_n e^{-in\xi}.$$

where $y_n$ are the dependent variables of the system, and are functions of time. sign(n) is $1,0,-1$ for n positive, 0 and negative respectively.

Based on the physics of my system, I find that the Lagrangian contains terms like

$$\frac{1}{2\pi}\int_0^{2\pi} \frac{\partial X}{\partial \xi} Y^2 \ d\xi = \sum_{j+k+l=0} |j|y_jy_ky_l$$

with $(j,k,l)\in (-N,N)$.

I then apply the Euler-Lagrange equations, which take the form

$$\frac{d}{dt} \frac{\partial L}{\partial \dot{y_k}} - \frac{\partial L}{\partial y_k}=0 \quad \quad k=\pm 1,\pm 2,... ,\pm N,$$

for $L$ my Larangian to get my governing system of couples ODEs in the dependent variables $y_k$.

These are then solved numerically.

My question is this - does one need to worry about aliasing errors in the sample terms shown above as one normally does for pseudospectral models (even though here my adventure occurs entirely in spectral space)? e.g. should I be cleaving away the highest 1/3 of my modes?

• I'm unclear what are the givens here and what are the quantities you are seeking? Commented Apr 28, 2015 at 4:01
• @WolfgangBangerth I've edited the question a bit. Hopefully this helps clarify the point. Commented Apr 28, 2015 at 5:03