For a double pendulum we can consider 2 generalised coordinates $\theta_1$ (angle between first mass and vertical axis) and $\theta_2$ (angle between second mass and vertical axis).
The Lagrangian to this system is:
I found here , that for small oscillations we can assume the following approximations:
For $T$: $\cos(\theta_1+\theta_2)\approx 1 $ (working in zeroth order)
For $V$: $\cos(\theta_1)\approx 1-\theta_1^2/2$, as for $\cos(\theta_2)$ (working in second order)
Why can we work with different orders on the same system for small oscillations?
If we assume an $n$ order, shouldn't we maintain that order independently if it's $T$ or $V$?