Alilouche A.

asked • 10/26/13

Could you help me to find a solution for this integral?

int {1/(x^2 Sqrt[1 - b^2/x^2 + (k/E) (exp[-2 a (x - c)] - 2 exp[-a (x - c)])])} dx
a,b,c,E,k are parameters

1 Expert Answer


Andre W. answered • 10/26/13

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Alilouche A.

thank you for your reply,
effectively U (x) has a form of a central potential, it is known as the name of the Morse potential. But the idea that you mention resolve not exactly the integral and it remains very complicated to solve.
Taylor series are an approximations , and even if we use this series, the integral remains difficult to solve (I think!)


Andre W.

This is indeed the Morse potential for diatomic molecules, which is usually treated quantum mechanically, but formulated here as a classical central motion problem in polar coordinates, with the integral representing φ(r). I doubt there is an exact (analytic) solution. I did a literature search and didn't find one.
The 2-dim isotropic harmonic oscillator is a very good approximation near x=c, and this integral is easy to solve: you basically get combinations of inverse trigonometric functions leading to elliptical orbits. For x=c, you get circular motion.


Alilouche A.

This is true, but I'm looking for an analytical solution, probably based on combination of arcsin and log functions.
We can have a solutions with numerical codes, but an analytical solution is more appreciated.


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