User:Tohline/Appendix/Ramblings/T2Integrals
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Integrals of Motion in T2 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 explore whether an analytic prescription of the 3^{rd} integral of motion can be formulated in a potential that conforms to this set of coordinates.
Review
We begin by summarizing properties of the T2 Coordinate System that we have derived earlier. By defining the dimensionless angle,
the two key "T2" coordinates can be written as,
χ_{1} 


and 
χ_{2} 



= 



= 

Here are a variety of relevant partial derivatives:




χ_{1} 



χ_{2} 



χ_{3} 


0 
The scale factors are,

= 

= 

= 


= 

= 

= 


= 

= 



where, . 
The position vector is,

= 

= 

Other Potentially Useful Differential Relations
In examining the equation of motion and searching for analytic representations of the 3^{rd} integral of motion, we will need to know how each of the scale factors varies along each of the coordinate directions. Since our T2 coordinate system is an orthogonal system of coordinates, in general we can write,
where x_{i} are the three Cartesian coordinates. Analytic expressions for the first partial derivative in each term on the RHS, , can be obtained from the table shown above. To derive expressions for the second partial derivative in each RHS term, the following differential relations also will be useful:







0 




Putting all of these expressions together, we derive the following:




h_{1} 


0 
h_{2} 


0 
Time Derivatives
Assuming no mistakes have been made in our derivation of the above expressions, the timederivative of the two key scale factors become,

= 


= 


= 


= 

If the potential is only a function of the first T2 coordinate, χ_{1}, then the 2^{nd} component of the equation of motion states,
In other words,
And the time derivative of a quantity that resembles the traditional angular momentum is,
First Special Case (quadratic)
As has been discussed in an accompanying chapter, when q^{2} = 2 the product of χ_{1} and χ_{2} shows up as a key quantity when inverting the coordinate definitions. In particular, by defining,
and its companion,
we can write,

= 

= 

= 

= 

z^{2} 
= 

= 

= 

= 


= 

= 

= 

= 

Hence, potentially useful expressions for the scale factors are,

= 

= 


= 

= 

Notice that, because it is expressible entirely in terms of Χ_{q}, the variable Ζ is a function only of the product of the two key coordinates. Hence, the scale factor h_{1} is only a function of the product of the coordinates while h_{2} depends on the ratio, as well as the product of the two coordinates. In this context, note that Ζ can be derived from the product of the two key coordinates via one of the following two relations:

= 


= 

See Also
© 2014  2020 by Joel E. Tohline 