Lagrangian for small oscillations

Question

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:

$$L=T-V.$$

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 $V$?

Answer 1

The small oscillation approximation considers terms in the Lagrangian to quadratic order in $\theta_i$ and $\dot{\theta}_i$. The reason to only work to zeroth order in $\theta_i$ in one term is because the pertinent term in the Lagrangian is already quadratic in $\dot{\theta}_i$.