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===Speculation===
Building on our experience developing [[User:Tohline/Appendix/Ramblings/T3Integrals#Integrals_of_Motion_in_T3_Coordinates|T3 Coordinates]] and, more recently, [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5 Coordinates]], let's define the two "angles,"
Building on our experience developing [[User:Tohline/Appendix/Ramblings/T3Integrals#Integrals_of_Motion_in_T3_Coordinates|T3 Coordinates]] and, more recently, [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5 Coordinates]], let's define the two "angles,"

Revision as of 12:14, 29 October 2020

Contents

Concentric Ellipsoidal (T6) Coordinates

Whitworth's (1981) Isothermal Free-Energy Surface
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Background

Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, and on our previous development of T3 (concentric oblate-spheroidal) and T5 (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T6) coordinate system. This is motivated by our desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids.

Orthogonal Coordinates

Primary (radial-like) Coordinate

We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,

~\lambda_1

~\equiv

~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, .

When ~\lambda_1 = a, we obtain the standard definition of an ellipsoidal surface, it being understood that, ~q^2 = a^2/b^2 and ~p^2 = a^2/c^2. (We will assume that ~a > b > c, that is, ~p^2 > q^2 > 1.)

A vector, ~\bold{\hat{n}}, that is normal to the ~\lambda_1 = constant surface is given by the gradient of the function,

~F(x, y, z)

~\equiv

~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} - \lambda_1 \, .

In Cartesian coordinates, this means,

~\bold{\hat{n}}(x, y, z)

~=

~
\hat\imath \biggl( \frac{\partial F}{\partial x} \biggr)
+ \hat\jmath \biggl( \frac{\partial F}{\partial y} \biggr)
+ \hat{k} \biggl( \frac{\partial F}{\partial z} \biggr)

 

~=

~
\hat\imath \biggl[ x(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr]
+ \hat\jmath \biggl[ q^2y(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr]
+ \hat{k}\biggl[ p^2 z(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr]

 

~=

~
\hat\imath \biggl( \frac{x}{\lambda_1} \biggr)
+ \hat\jmath \biggl( \frac{q^2y}{\lambda_1} \biggr)
+ \hat{k}\biggl(\frac{p^2 z}{\lambda_1} \biggr) \, ,

where it is understood that this expression is only to be evaluated at points, ~(x, y, z), that lie on the selected ~\lambda_1 surface — that is, at points for which the function, ~F(x,y,z) = 0. The length of this normal vector is given by the expression,

~[ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2}

~=

~
\biggl[ \biggl( \frac{\partial F}{\partial x} \biggr)^2 + \biggl( \frac{\partial F}{\partial y} \biggr)^2 + \biggl( \frac{\partial F}{\partial z} \biggr)^2 \biggr]^{1 / 2}

 

~=

~
\biggl[ \biggl( \frac{x}{\lambda_1} \biggr)^2
+ \biggl( \frac{q^2y}{\lambda_1} \biggr)^2
+ \biggl(\frac{p^2 z}{\lambda_1} \biggr)^2 \biggr]^{1 / 2}

 

~=

~
\frac{1}{\lambda_1 \ell_{3D}}

where,

~\ell_{3D}

~\equiv

~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, .

It is therefore clear that the properly normalized normal unit vector that should be associated with any ~\lambda_1 = constant ellipsoidal surface is,

~\hat{e}_1

~\equiv

~
\frac{ \bold\hat{n} }{ [ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2} }
=
\hat\imath (x \ell_{3D}) + \hat\jmath (q^2y \ell_{3D}) + \hat\jmath (p^2 z \ell_{3D}) \, .

From our accompanying discussion of direction cosines, it is clear, as well, that the scale factor associated with the ~\lambda_1 coordinate is,

~h_1^2

~=

~\lambda_1^2 \ell_{3D}^2 \, .

We can also fill in the top line of our direction-cosines table, namely,

Direction Cosines for T6 Coordinates
~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)

~n ~i = x, y, z
~1  

~x\ell_{3D}
 

~q^2 y \ell_{3D} ~p^2 z \ell_{3D}
~2

 
---
 

 
---
 

 
---
 

~3

 
---
 

 
---
 

 
---
 

Other Coordinate Pair in the Tangent Plane

Let's focus on a particular point on the ~\lambda_1 = constant surface, ~(x_0, y_0, z_0), that necessarily satisfies the function, ~F(x_0, y_0, z_0) = 0. We have already derived the expression for the unit vector that is normal to the ellipsoidal surface at this point, namely,

~\hat{e}_1

~\equiv

~
\hat\imath (x_0 \ell_{3D}) + \hat\jmath (q^2y_0 \ell_{3D}) + \hat\jmath (p^2 z_0 \ell_{3D}) \, ,

where, for this specific point on the surface,

~\ell_{3D}

~=

~\biggl[ x_0^2 + q^4y_0^2 + p^4 z_0^2 \biggr]^{- 1 / 2} \, .


Tangent Plane

The two-dimensional plane that is tangent to the ~\lambda_1 = constant surface at this point is given by the expression,

~0

~=

~
(x - x_0) \biggl[ \frac{\partial \lambda_1}{\partial x} \biggr]_0 
+ (y - y_0) \biggl[\frac{\partial \lambda_1}{\partial y} \biggr]_0  
+ (z - z_0) \biggl[\frac{\partial \lambda_1}{\partial z} \biggr]_0

 

~=

~
(x - x_0) \biggl[ \frac{\partial F}{\partial x} \biggr]_0  
+ (y - y_0) \biggl[\frac{\partial F}{\partial y} \biggr]_0  
+ (z - z_0) \biggl[ \frac{\partial F}{\partial z} \biggr]_0

 

~=

~
(x - x_0) \biggl( \frac{x}{\lambda_1}\biggr)_0 + (y - y_0)\biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + (z - z_0)\biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0

~\Rightarrow~~~
x \biggl( \frac{x}{\lambda_1}\biggr)_0 + y \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0

~=

~
x_0 \biggl( \frac{x}{\lambda_1}\biggr)_0 + y_0 \biggl( \frac{q^2 y  }{ \lambda_1 } \biggr)_0 + z_0 \biggl( \frac{ p^2z }{ \lambda_1 } \biggr)_0

~\Rightarrow~~~
x x_0  + q^2 y y_0  + p^2 z z_0

~=

~
x_0^2  +  q^2 y_0^2 + p^2 z_0^2

~\Rightarrow~~~
x x_0  + q^2 y y_0  + p^2 z z_0

~=

~
(\lambda_1^2)_0 \, .

Speculation

Building on our experience developing T3 Coordinates and, more recently, T5 Coordinates, let's define the two "angles,"

~\Zeta

~\equiv

~\sinh^{-1}\biggl(\frac{qy}{x} \biggr)

      and,      

~\Upsilon

~\equiv

~\sinh^{-1}\biggl(\frac{pz}{x} \biggr) \, ,

in which case we can write,

~\lambda_1^2

~=

~x^2(\cosh^2\Zeta + \sinh^2\Upsilon)\, .

We speculate that the other two orthogonal coordinates may be defined by the expressions,

~\lambda_2

~\equiv

~x \biggl[ \sinh\Zeta \biggr]^{1/(1-q^2)} 
=
x \biggl[ \frac{qy}{x}\biggr]^{1/(1-q^2)}
=
x \biggl[ \frac{x}{qy}\biggr]^{1/(q^2-1)}
=
\biggl[ \frac{x^{q^2}}{qy}\biggr]^{1/(q^2-1)}
\, ,

~\lambda_3

~\equiv

~x \biggl[ \sinh\Upsilon \biggr]^{1/(1-p^2)} 
=
x \biggl[ \frac{pz}{x}\biggr]^{1/(1-p^2)} 
=
x \biggl[ \frac{x}{pz}\biggr]^{1/(p^2-1)} 
=
\biggl[ \frac{x^{p^2}}{pz}\biggr]^{1/(p^2-1)} 
\, .

Some relevant partial derivatives are,

~\frac{\partial \lambda_2}{\partial x}

~=

~\biggl[ \frac{1}{qy}\biggr]^{1/(q^2-1)} \biggl[ \frac{q^2}{q^2-1} \biggr]x^{1/(q^2-1)}
=
\biggl[ \frac{q^2}{q^2-1} \biggr]\biggl[ \frac{x}{qy}\biggr]^{1/(q^2-1)} 
=
\biggl[ \frac{q^2}{q^2-1} \biggr]\frac{\lambda_2}{x} 
\, ;

~\frac{\partial \lambda_2}{\partial y}

~=

~\biggl[ \frac{x^{q^2}}{q}\biggr]^{1/(q^2-1)} \biggl[ \frac{1}{1-q^2} \biggr] y^{q^2/(1-q^2)}
=
- \biggl[ \frac{1}{q^2-1} \biggr] \frac{\lambda_2}{y} 
\, ;

~\frac{\partial \lambda_3}{\partial x}

~=

~
\biggl[ \frac{p^2}{p^2-1} \biggr]\frac{\lambda_3}{x} 
\, ;

~\frac{\partial \lambda_3}{\partial z}

~=

~
- \biggl[ \frac{1}{p^2-1} \biggr] \frac{\lambda_3}{z} 
\, .

And the associated scale factors are,

~h_2^2

~=

~
\biggl\{ \biggl[ \biggl( \frac{q^2}{q^2-1} \biggr)\frac{\lambda_2}{x} \biggr]^2 + \biggl[ - \biggl( \frac{1}{q^2-1} \biggr) \frac{\lambda_2}{y} \biggr]^2 \biggr\}^{-1}

 

~=

~
\biggl\{ \biggl( \frac{q^2}{q^2-1} \biggr)^2 \frac{\lambda_2^2}{x^2}  + \biggl( \frac{1}{q^2-1} \biggr)^2 \frac{\lambda_2^2}{y^2} \biggr\}^{-1}

 

~=

~
\biggl\{x^2 + q^4 y^2 \biggr\}^{-1}
\biggl[ \frac{(q^2 - 1)^2x^2 y^2}{\lambda_2^2} \biggr] \, ;

~h_3^2

~=

~
\biggl\{x^2 + p^4 z^2 \biggr\}^{-1}
\biggl[ \frac{(p^2 - 1)^2x^2 z^2}{\lambda_3^2} \biggr] \, .

We can now fill in the rest of our direction-cosines table, namely,

Direction Cosines for T6 Coordinates
~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)

~n ~i = x, y, z
~1  

~x\ell_{3D}
 

~q^2 y \ell_{3D} ~p^2 z \ell_{3D}
~2

~q^2 y \ell_q

~-x\ell_q

~0

~3

~p^2 z \ell_p

~0

~-x\ell_p

Hence,

~\hat{e}_2

~=

~
\hat\imath \gamma_{21}
+ \hat\jmath \gamma_{22}
+\hat{k} \gamma_{23}
=
\hat\imath (q^2y\ell_q)
- \hat\jmath (x\ell_q) \, ;

~\hat{e}_3

~=

~
\hat\imath \gamma_{31}
+ \hat\jmath \gamma_{32}
+\hat{k} \gamma_{33}
=
\hat\imath (p^2z\ell_p)
-\hat{k} (x\ell_p) \, .

Check:

~\hat{e}_2 \cdot \hat{e}_2

~=

~
(q^2y\ell_q)^2
+ (x\ell_q)^2
=
1 \, ;

~\hat{e}_3 \cdot \hat{e}_3

~=

~
(p^2z\ell_p)^2
+ (x\ell_p)^2
=
1 \, ;

~\hat{e}_2 \cdot \hat{e}_3

~=

~
(q^2y\ell_q)(p^2z\ell_p) \ne 0 \, .

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2020 by Joel E. Tohline
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