Harmonic Oscillator with Additional Repulsive Cubic Force: Solutions and Study

[Miesvama]

Hi all,

this is my first time on PF.

I do not know English but I have a problem of a harmonic oscillator.
I have rather large head, help me please , I do not know what else to do ...
I have this problem:

Consider the harmonic oscillator with an additional repulsive
cubic force, whose potential is U(q1)=$\frac{k}{2}$*$q1^{2}$ - k'$q1^{4}$, (k, k > 0), and study all
possible solutions, periodic and non-periodic.

I do know the Hamiltonian and the equation solution of the system, giving

q1*=$\sqrt{1/2}$$\int(dq1/\sqrt{(E/m)-U(q1)})$

I tried to do it by trigonometric substitution but does not work, i do not know if anyone could give me some idea of how I can solve, I'll be very grateful.

 

[stevendaryl]

Hi all,

this is my first time on PF.

I do not know English but I have a problem of a harmonic oscillator.
I have rather large head, help me please , I do not know what else to do ...
I have this problem:

Consider the harmonic oscillator with an additional repulsive
cubic force, whose potential is U(q1)=$\frac{k}{2}$*$q1^{2}$ - k'$q1^{4}$, (k, k > 0), and study all
possible solutions, periodic and non-periodic.

I do know the Hamiltonian and the equation solution of the system, giving

q1*=$\sqrt{1/2}$$\int(dq1/\sqrt{(E/m)-U(q1)})$

I tried to do it by trigonometric substitution but does not work, i do not know if anyone could give me some idea of how I can solve, I'll be very grateful.

Hi. Nobody else has responded to your question, so I guess I'll give it a try.

The equation that I think you meant is this:

$t = \sqrt{m/2}\int(dq1/\sqrt{E-U(q1)})$

If we change to new variables $x$ and $s$ where
$x = A q1$ and $s = B t$, where $A$ and
$B$ are constants, we can choose the constants to make the
equation look like this:

$t = \int(dx/\sqrt{(1-x^2)(1- \lambda^2 x^2)}$

where $\lambda$ is yet another constant.

That's called the "Jacobi form of the incomplete elliptic integral of the first kind", $F(s;\lambda)$.

http://en.wikipedia.org/wiki/Jacobi..._of_nonlinear_ordinary_differential_equations

 

[Miesvama]

Thank you.. :D