Non-linear Oscillator: Understanding Bound Motion through Conservation of Energy

[Adorno]

Homework Statement

A non-linear oscillator consisting of a mass on a spring has a potential energy of the form $\frac{1}{2}kx^2 - \frac{1}{3}\alpha x^3$, where $k$ and $\alpha$ are positive constants, and $x$ is displacement. Using conservation of energy, show that the motion is oscillatory if the initial position $x_0$ satisfies $0 < x_0 < \frac{k}{\alpha}$ and the initial velocity satisfies $v_0 < \frac{k}{\alpha}\sqrt{\frac{k}{m}}$.

Homework Equations

$E = T + U = \mathrm{constant}$

$F = -\frac{dU}{dx}$

The Attempt at a Solution

By conservation of energy, the quantity $E = \frac{1}{2}mv^2 + U(x)$ must be constant. So if the motion is oscillatory then the velocity will be zero at two (and only two) different positions, i.e. we have $E = U(x_1) = U(x_2)$. Since $E$ is the maximum value of the potential energy, this is equivalent to saying that the potential energy must reach the value of its local max/min at two (and only two) positions. In other words, the $x$ values must lie between the two critical points of $U$. By setting $dU/dx = 0$, we get $x = 0$ and $x = \frac{k}{\alpha}$ as the two critical points, and so we must have $0 < x < \frac{k}{\alpha}$, as required.

I think the above is OK, but feel free to correct me if you see a problem. What I don't understand is how the initial velocity comes into it. The question is saying that if the initial velocity is greater than $\frac{k}{\alpha}\sqrt{\frac{k}{m}}$, then the motion won't be oscillatory, but I don't know how to derive this. Presumably this has to do with the conservation of energy as well -- I guess I have to use the kinetic energy term $\frac{1}{2}mv^2$ somehow. Could anyone help with this part?

 

[Spinnor]

Draw the potential, and with that draw the force. For small initial velocity the particle can be trapped by the potential.

 

[dynamicsolo]

What I don't understand is how the initial velocity comes into it. The question is saying that if the initial velocity is greater than $\frac{k}{\alpha}\sqrt{\frac{k}{m}}$, then the motion won't be oscillatory, but I don't know how to derive this. Presumably this has to do with the conservation of energy as well -- I guess I have to use the kinetic energy term $\frac{1}{2}mv^2$ somehow. Could anyone help with this part?

Consider also that a system is only bound for E < 0 . When the mass is at the extreme of displacement $x = \frac{k}{\alpha}$, what would the maximum permissible velocity be?

(I think there is some margin in the given conditions: I am getting some dimensionless multipliers on the order of 1 for the limits on displacement and velocity.)