Linearizing a differential equation

[eep]

This question popped up while trying to solve problem 7.41 from Taylor's Classical Mechanics book.

Basically we have the differential equation

$$(1+4k^2\rho^2)\ddot{\rho} = \rho \omega^2 - 4k^2\rho{\dot{\rho}}^2-2gk\rho$$

and we are looking if the equilibrium position $$\rho = 0$$ is stable.

After "linearizing" this you are supposed to end up with

$$\ddot{\rho} = (\omega^2 - 2gk)\rho$$

but I really don't understand the correct way to arrive at this. Particularly, how are you supposed to justify dropping the $$4k^2\rho{\dot{\rho}}^2$$ term?

 

[Mark44]

This question popped up while trying to solve problem 7.41 from Taylor's Classical Mechanics book.

Basically we have the differential equation

$$(1+4k^2\rho^2)\ddot{\rho} = \rho \omega^2 - 4k^2\rho{\dot{\rho}}^2-2gk\rho$$

and we are looking if the equilibrium position $$\rho = 0$$ is stable.

After "linearizing" this you are supposed to end up with

$$\ddot{\rho} = (\omega^2 - 2gk)\rho$$

but I really don't understand the correct way to arrive at this. Particularly, how are you supposed to justify dropping the $$4k^2\rho{\dot{\rho}}^2$$ term?

My guess is that at the equilibrium position, ρ ≈ 0, and $\dot{\rho}$ is either zero or close to it, which would make $\dot{\rho}^2$ negligible.