Integrals of Motion in T4 Coordinates

In an accompanying Wiki document, we have derived the properties of an orthogonal, axisymmetric, T3 coordinate system in which the first coordinate, <math>\lambda_1</math>, defines a family of concentric oblate-spheroidal surfaces whose (uniform) flattening is defined by a parameter <math>q \equiv R_\mathrm{eq}/Z_\mathrm{pole}</math>. In a separate, but related, Wiki document, we attempt to derive the <math>3^\mathrm{rd}</math> isolating integral of motion for massless particles that move inside a flattened, axisymmetric potential whose equipotential surfaces align with <math>\lambda_1 = \mathrm{constant}</math> surfaces in the special (quadratic) case when <math>q^2 = 2</math>. While examining this special case, we noticed that, in T3 Coordinates, the <math>h_1</math> and <math>h_2</math> scale factors are only a function of the coordinate ratio <math>\lambda_1/\lambda_2</math>. This has led us to wonder whether it might be more fruitful to search for the <math>3^\mathrm{rd}</math> isolating integral using a coordinate system in which one of the coordinates is defined by this T3-coordinate ratio. It is with this in mind that we explore the development of a new T4 coordinate system.

Definition

In what follows, the coordinates <math>(\lambda_1,\lambda_2,\lambda_3)</math> refer to T3 Coordinates. Let's define a set of orthogonal T4 Coordinates such that,

<math> \xi_1 </math>	<math> \equiv </math>	<math> (\lambda_1^2 + \lambda_2^2)^{1/2} ; </math>
<math> \tan\xi_2 </math>	<math> \equiv </math>	<math> \frac{\lambda_2}{\lambda_1} ; </math>
<math> \tan\xi_3 </math>	<math> \equiv </math>	<math> \frac{y}{x} . </math>

The coordinate inversion — from <math>(\xi_1,\xi_2,\xi_3)</math> back to <math>(\lambda_1,\lambda_2,\lambda_3)</math> — is straightforward. Specifically,

<math> \lambda_1 </math>	<math> = </math>	<math> \xi_1 \cos\xi_2 ; </math>
<math> \lambda_2 </math>	<math> = </math>	<math> \xi_1 \sin\xi_2 ; </math>
<math> \lambda_3 </math>	<math> = </math>	<math> \xi_3 . </math>

Here are some relevant partial derivatives:

	<math> \frac{\partial}{\partial x} </math>	<math> \frac{\partial}{\partial y} </math>	<math> \frac{\partial}{\partial z} </math>
<math>\xi_1</math>	<math> ~~ </math>	<math> ~~ </math>	<math> ~~ </math>
<math>\xi_2</math>	<math> ~~ </math>	<math> ~~ </math>	<math> ~~ </math>
<math>\xi_3</math>	<math> ~~ </math>	<math> ~~ </math>	<math> ~~ </math>

Ghe inverted partials are

	<math> \frac{\partial}{\partial \xi_1} </math>	<math> \frac{\partial}{\partial \xi_2} </math>	<math> \frac{\partial}{\partial \xi_3} </math>
<math>x</math>	<math> ~~ </math>	<math> ~~ </math>	<math> ~~ </math>
<math> y </math>	<math> ~~ </math>	<math> ~~ </math>	<math> ~~ </math>
<math> z </math>	<math> ~~ </math>	<math> ~~ </math>	<math> ~~ </math>

The scale factors are,

<math>h_1^2</math>	<math>=</math>	<math> \biggl[ \biggl( \frac{\partial\xi_1}{\partial x} \biggr)^2 + \biggl( \frac{\partial\xi_1}{\partial y} \biggr)^2 + \biggl( \frac{\partial\xi_1}{\partial z} \biggr)^2 \biggr]^{-1} </math>	<math>=</math>	<math> ~~ </math>
<math>h_2^2</math>	<math>=</math>	<math> \biggl[ \biggl( \frac{\partial\xi_2}{\partial x} \biggr)^2 + \biggl( \frac{\partial\xi_2}{\partial y} \biggr)^2 + \biggl( \frac{\partial\xi_2}{\partial z} \biggr)^2 \biggr]^{-1} </math>	<math>=</math>	<math> ~~ </math>
<math>h_3^2</math>	<math>=</math>	<math> \biggl[ \biggl( \frac{\partial\xi_3}{\partial x} \biggr)^2 + \biggl( \frac{\partial\xi_3}{\partial y} \biggr)^2 + \biggl( \frac{\partial\xi_3}{\partial z} \biggr)^2 \biggr]^{-1} </math>	<math>=</math>	<math> ~~ </math>
where, <math>~~</math>.

The position vector is,

<math> \hat\imath x + \hat\jmath y + \hat{k}z </math>

User:Tohline/Appendix/Ramblings/T4Integrals

Integrals of Motion in T4 Coordinates

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