User:Jaycall/T3 Coordinates/Special Case
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Coordinate Transformations
If the special case q^{2} = 2 is considered, it is possible to invert the coordinate transformations in closed form. The coordinate transformations and their inversions become
λ_{1} 


and 
λ_{2} 


R^{2} 


and 
z 


where .
From this definition of Λ, we can compute both its partials with respect to the T3 coordinates, and its total time derivative.
Partials of the Coordinates
Partial derivatives of each of the T3 coordinates taken with respect to each of the cylindrical coordinates are:




λ_{1} 


0 
λ_{2} 


0 
λ_{3} 
0 
0 
1 
And partials of the cylindrical coordinates taken with respect to the T3 coordinates are:




R 


0 
z 


0 
φ 
0 
0 
1 
where .
Scale Factors
Furthermore, the scale factors become
h_{1} 
= 

h_{2} 
= 

h_{3} 
= 
R = λ_{3} 
Useful Relationships
In this special case, there are some additional useful relationships between various combinations of cylindrical variables and their T3 equivalents which can be written out.
R^{2} + 2z^{2} 
= 

R^{2} + 4z^{2} 
= 

R^{2} + 8z^{2} 
= 

R^{2} − 2z^{2} 
= 

Rz 
= 

Additional Partials
Partials of can be taken with respect to the coordinates of either system. They are:







0 







0 
Partials of the scale factors taken with respect to the T3 coordinates are:




h_{1} 


0 
h_{2} 


0 
h_{3} 


0 
Conserved Quantity
The conserved quantity associated with the λ_{2} coordinate is
The quantity in brackets needs to be integrated. In terms of Λ, it can be written
.
Notice that the thing in square brackets looks very closely related to . Could this be a hint? If only we could figure out what is, maybe we could factor out the , which appears in ...
If, by some miracle, it should turn out that , factorization would be possible and our integral would read
.
We ought to be able to integrate this, right...? Maybe we could handle the pesky with integration by parts...