User:Tohline/Appendix/Ramblings/T4Integrals
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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, λ_{1}, defines a family of concentric oblatespheroidal surfaces whose (uniform) flattening is defined by a parameter . In a separate, but related, Wiki document, we attempt to derive the 3^{rd} isolating integral of motion for massless particles that move inside a flattened, axisymmetric potential whose equipotential surfaces align with λ_{1} = constant surfaces in the special (quadratic) case when q^{2} = 2. While examining this special case, we noticed that, in T3 Coordinates, the h_{1} and h_{2} scale factors are only a function of the coordinate ratio λ_{1} / λ_{2}. This has led us to wonder whether it might be more fruitful to search for the 3^{rd} isolating integral using a coordinate system in which one of the coordinates is defined by this T3coordinate ratio.
It is with this in mind that we explore the development of a new T4 coordinate system. From the very beginning we will restrict the T4coordinate definition to the special case of q^{2} = 2 because, at present, we think that the coordinate T3coordinate ratio λ_{1} / λ_{2} is only interesting in the quadratic case. (See, for example, the polynomial root derived to complete the T1coordinate inversion for the cubic case q^{2} = 3; it is another combination of the T3 coordinates that appears to be relevant in the cubic case.)
STOP!
(7/06/2010)
As defined, below, this is not an orthogonal coordinate system.
Definition
In what follows, the coordinates (λ_{1},λ_{2},λ_{3}) refer to T3 Coordinates. Let's define a set of orthogonal T4 Coordinates for the special (quadratic) case q^{2} = 2 such that,
ξ_{1} 


= 



ξ_{2} 


= 



tanξ_{3} 






where,
The coordinate inversion — from (ξ_{1},ξ_{2},ξ_{3}) back to (λ_{1},λ_{2},λ_{3}) — is straightforward. Specifically,
λ_{1} 
= 

λ_{2} 
= 

λ_{3} 
= 
ξ_{3}. 
Here are some relevant partial derivatives:
Partial derivatives with respect to cylindrical coordinates are,




ξ_{1} 


0 
ξ_{2} 


0 
ξ_{3} 
0 
0 
1 
Hence, the partials with respect to Cartesian coordinates are,




ξ_{1} 



ξ_{2} 



ξ_{3} 


0 
The scale factors are,

= 

= 

= 


= 

= 

= 


= 

= 



where, . 
The position vector is,

= 

= 

See Also
© 2014  2019 by Joel E. Tohline 