In this question we will consider a damped mass-on-a-spring system whose spring exhibits cubic deviations from Hooke’s law. We will consider a damped spring with a restoring force F such that
where β is the “Hookian” part and α is a new nonlinear term. Unlike the usual spring constant k, β can have either sign. Use γ (not 2γ) as the damping parameter that multiplies x'.
1.) What does the potential energy function look like for this system? Consider cases where α and β are both positive and negative.
2.)Solve for the fixed points of this system and determine how their stability depends on α and β.
The Attempt at a Solution
So the general equation of motion with damping and restoring force is: x'' + γx' + βx + αx^3 = 0
So I know the net force is F = -γx' - mβx - mαx^3. And in general F = - dU/dx (however this is true for conservative forces -hence I don't think it would apply here). So can I just integrate the for the Force? If not, I unsure how to approach this problem to get the potential energy function.