User:Tohline/Appendix/Ramblings/T3Integrals

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 will be written as,

<math> \lambda_1 </math>

<math>\equiv</math>

<math>\varpi \cosh\Zeta = ( \varpi^2 + q^2z^2 )^{1/2}</math>

and

<math> \lambda_2 </math>

<math>\equiv</math>

<math>\varpi [\sinh\Zeta ]^{1/(1-q^2)} = \biggl[\frac{\varpi^{q^2}}{qz}\biggr]^{1/(q^2-1)}</math>

Here are some relevant partial derivatives (there may be a mistake in the derivation of the partials of <math>\lambda_2</math>):

	<math> \frac{\partial}{\partial x} </math>	<math> \frac{\partial}{\partial y} </math>	<math> \frac{\partial}{\partial z} </math>
<math>\lambda_1</math>	<math> \frac{x}{\lambda_1} </math>	<math> \frac{y}{\lambda_1} </math>	<math> \frac{q^2}{\lambda_1} </math>
<math>\lambda_2</math>	<math> \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\Zeta} \biggr]^{q^2/(q^2-1)} \biggl( \frac{q^3 z}{\varpi^{q^2+2}} \biggr) x </math> <math> =\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{x}{\varpi} \biggr) </math>	<math> \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\Zeta} \biggr]^{q^2/(q^2-1)} \biggl( \frac{q^3 z}{\varpi^{q^2+2}} \biggr) y </math> <math> =\frac{q^2}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{1/(q^2-1)} \biggl( \frac{y}{\varpi} \biggr) </math>	<math> - \frac{1}{(q^2-1)} \biggl[ \frac{\varpi^{q^2-1}}{\sinh\zeta} \biggr]^{q^2/(q^2-1)} \frac{q}{\varpi^{q^2}} </math> <math> =- \frac{q}{(q^2-1)} \biggl[ \frac{\varpi}{qz} \biggr]^{q^2/(q^2-1)} </math>
<math>\lambda_3</math>	<math> - \frac{y}{\varpi^{2}} </math>	<math> + \frac{x}{\varpi^{2}} </math>	<math> 0 </math>

The scale factors are (there is a mistake in the derivation of <math>h_2</math>),

<math>h_1^2</math>	<math>=</math>	<math> \biggl[ \biggl( \frac{\partial\lambda_1}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_1}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_1}{\partial z} \biggr)^2 \biggr]^{-1} </math>	<math>=</math>	<math> \frac{\lambda_1^2}{(\varpi^2 + q^4 z^2)} </math>	<math>=</math>	<math> \lambda_1^2 \ell^2 </math>
<math>h_2^2</math>	<math>=</math>	<math> \biggl[ \biggl( \frac{\partial\lambda_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_2}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_2}{\partial z} \biggr)^2 \biggr]^{-1} </math>	<math>=</math>	<math> \frac{(q^2-1)}{q^2} [\sinh\Zeta]^{2q^2/(q^2-1)} \frac{\varpi^2 }{(\varpi^2 + q^4 z^2)} </math>	<math>=</math>	<math> \frac{(q^2-1)}{q^2} \biggl[\frac{\varpi}{\lambda_2} \biggr]^{2q^2} \varpi^2 \ell^2 </math>
<math>h_3^2</math>	<math>=</math>	<math> \biggl[ \biggl( \frac{\partial\lambda_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial\lambda_3}{\partial y} \biggr)^2 + \biggl( \frac{\partial\lambda_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,

See Also

• Old discussion of Integrals of Motion

 

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation