User:Tohline/Appendix/Ramblings/T2Integrals
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Integrals of Motion in T2 Coordinates
Motivated by the HNM82 derivation, in an accompanying chapter we have introduced a new T2 Coordinate System and have outlined a few of its properties. Here we explore whether an analytic prescription of the <math>3^\mathrm{rd}</math> integral of motion can be formulated in a potential that conforms to this set of coordinates.
Review
We begin by summarizing properties of the T2 Coordinate System that we have derived earlier. By defining the dimensionless angle,
<math> \Zeta \equiv \sinh^{-1}\biggl( \frac{qz}{\varpi} \biggr) , </math>
the two key "T2" coordinates can be written as,
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<math> \chi_1 </math> |
<math>\equiv</math> |
<math>B \varpi \cosh\Zeta</math> |
and |
<math> \chi_2 </math> |
<math>\equiv</math> |
<math>\frac{A \sinh\Zeta}{ \varpi^{q^2-1}}</math> |
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<math>=</math> |
<math> B ( \varpi^2 + q^2z^2 )^{1/2} = qB \xi_1 </math> |
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<math>=</math> |
<math> \frac{Aqz}{\varpi^{q^2}} = Aq \biggl[\frac{1}{\tan\xi_2} \biggr]^{q^2} </math> |
Here are a variety of relevant partial derivatives:
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<math> \frac{\partial}{\partial x} </math> |
<math> \frac{\partial}{\partial y} </math> |
<math> \frac{\partial}{\partial z} </math> |
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<math>\chi_1</math> |
<math> \biggl(\frac{B^2}{\chi_1}\biggr) x </math> |
<math> \biggl(\frac{B^2}{\chi_1}\biggr) y </math> |
<math> \biggl(\frac{B^2}{\chi_1}\biggr) q^2 z </math> |
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<math>\chi_2</math> |
<math> - \biggl( \frac{q^3 A z}{\varpi^{q^2+2}} \biggr) x </math> |
<math> - \biggl( \frac{q^3 A z}{\varpi^{q^2+2}} \biggr) y </math> |
<math> \frac{qA}{\varpi^{q^2}} </math> |
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<math>\chi_3</math> |
<math> - \biggl( \frac{1}{\varpi^{2}} \biggr) y </math> |
<math> + \biggl( \frac{1}{\varpi^{2}} \biggr) x </math> |
<math> 0 </math> |
The scale factors are,
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<math>h_1^2</math> |
<math>=</math> |
<math> \biggl[ \biggl( \frac{\partial\chi_1}{\partial x} \biggr)^2 + \biggl( \frac{\partial\chi_1}{\partial y} \biggr)^2 + \biggl( \frac{\partial\chi_1}{\partial z} \biggr)^2 \biggr]^{-1} </math> |
<math>=</math> |
<math> \frac{\chi_1^2}{B^4 (\varpi^2 + q^4 z^2)} </math> |
<math>=</math> |
<math> \frac{\chi_1^2 \ell^2}{B^4} </math> |
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<math>h_2^2</math> |
<math>=</math> |
<math> \biggl[ \biggl( \frac{\partial\chi_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial\chi_2}{\partial y} \biggr)^2 + \biggl( \frac{\partial\chi_2}{\partial z} \biggr)^2 \biggr]^{-1} </math> |
<math>=</math> |
<math> \frac{z^2 \varpi^2 }{\chi_2^2 (\varpi^2 + q^4 z^2)} </math> |
<math>=</math> |
<math> \frac{z^2 \varpi^2 \ell^2}{\chi_2^2} </math> |
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<math>h_3^2</math> |
<math>=</math> |
<math> \biggl[ \biggl( \frac{\partial\chi_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial\chi_3}{\partial y} \biggr)^2 + \biggl( \frac{\partial\chi_3}{\partial z} \biggr)^2 \biggr]^{-1} </math> |
<math>=</math> |
<math> \varpi^2 </math> |
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where, <math>\ell \equiv (\varpi^2 + q^2 z^2)^{-1/2}</math>. |
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The position vector is,
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<math>\vec{x}</math> |
<math>=</math> |
<math> \hat{i}x + \hat{j}y + \hat{k} </math> |
<math>=</math> |
<math> \hat{e}_1 (h_1 \chi_1) + (1 - q^2) \hat{e}_2 (h_2 \chi_2) . </math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |