User:Tohline/Appendix/Ramblings/T3Integrals

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Whitworth's (1981) Isothermal Free-Energy Surface
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Integrals of Motion in T3 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 offer a modest redefinition of the second radial coordinate in an effort to bring even more symmetry to the definition of the position vector, <math>\vec{x}</math>.


Definition

By defining the dimensionless angle,

<math> \Zeta \equiv \sinh^{-1}\biggl( \frac{qz}{\varpi} \biggr) , </math>

the two key "T3" coordinates can be written as,

<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>

 

<math>=</math>

<math> B ( \varpi^2 + q^2z^2 )^{1/2} = qB \xi_1 </math>

 

 

<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:

 

<math> \frac{\partial}{\partial x} </math>

<math> \frac{\partial}{\partial y} </math>

<math> \frac{\partial}{\partial z} </math>

<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>

<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>

<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,

<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>

<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>

<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>

 

 

where,        <math>\ell \equiv (\varpi^2 + q^2 z^2)^{-1/2}</math>.


The position vector is,

<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

 

Whitworth's (1981) Isothermal Free-Energy Surface

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