- #1

Xyius

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1. Homework Statement

*I only really need help evaluating the contour integral. I added more detail for completeness.*1. Homework Statement

A particle of mass ##m## is constrained to move on the ##x## axis subject to the following potential.

[tex]V(x)=a \sec^2\left( \frac{x}{l} \right)[/tex]

Find the action angle variables ##(\phi,J)##

## Homework Equations

[tex]J=\frac{1}{2\pi}\oint p dq[/tex]

## The Attempt at a Solution

So I really only need help with the evaluation of the contour integral for J. My background in complex analysis is limited and I believe I must use the residue theorem. I will walk you through what I have.

So first find the lagrangian

##L=T-V=\frac{1}{2}m\dot{x}^2-a\sec^2\left( \frac{x}{l} \right)##

Then find the Hamiltonian using the energy function definition and writing everything in terms of momentum.

##H=\frac{p^2}{2m}+a\sec^2\left( \frac{x}{l} \right)##

Next, use the Hamilton-Jacobi equation to find the generating function (I won't go through this as it is not the focus of my question). From here You can find that ##p=\sqrt{2m(X_0-a\sec^2\left( \frac{x}{l} \right))}## where ##X_0## is the space canonical transformation (acting as a constant for the purposes of the integral next).

So using the equation in section 2 from above, I have the following.

[tex]J=\frac{\sqrt{2m}}{2\pi}\oint \sqrt{X_0-a\sec^2\left( \frac{x}{l} \right)}dx[/tex]

And this is where I am stuck. I believe I need to use the residue theorem, but again my background in complex analysis is limited. So one thing I notice is that because the secant squared is inside the square root, it cannot go to infinity, otherwise the square root becomes complex. ##p## will equal zero when ##x=l \arccos{\sqrt{\frac{a}{x}}}##. So if I plot ##p## I can sort of see it's motion but I am not sure. The curve starts on the Y (or p) axis and then dips down and touches the x axis, then it goes imaginary is is periodic from here. I Believe the path in phase space is such that this dip is mirrored about the y-axis and forms an upside-down parabolic looking shape. From this I can clearly see a closed loop path. Two curves segments making the parabolic-looking shape, and a straight segment connection the two when p=0.

If anyone can offer any help I would appreciate it!

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