Editing Appendix/Ramblings/T3CharacteristicVector (section)

==Thoughts on Integrating This Conserved Quantity==

The quantity appearing inside the parentheses has an interesting symmetry.  Each variable appearing without a dot in the first term appears in the same place with a dot in the second term, and vice versa.  Certainly there must be some differentiation rule that will allow us to express this quantity as a total time derivative.

On the other hand, the factor of <math>\dot{\lambda_2}</math> appearing in the denominator of the first term is troublesome.  I can't think of any differentiation rule that puts a derivative in the denominator.  Product rule, quotient rule, and chain rule all end up ''multiplying'' by derivatives.  So I wonder if there's some way to eliminate the <math>\dot{\lambda_2}</math> in favor of undotted variables.  This would require transforming the equation of motion for the <math>\dot{\lambda_2}</math> coordinate into a first-order equation.  Right now, the second-order equation reads

<div align="center">
<math>
\ddot{\lambda_2} + \frac{\dot{h_2}}{h_2} \dot{\lambda_2} - \frac{\dot{\lambda_1} \dot{h_2}}{\lambda_1 h_2} \lambda_2 = 0 .
</math>
</div>

The first step in reducing this to a first-order equation is to perform a transformation of variables that eliminates that <math>\dot{\lambda_2}</math> term.  I have successfully accomplished this.  By defining <math>b \equiv {h_2}^{1/2} \lambda_2</math>, the equation can be written:

<div align="center">
<math>
\ddot{b} + \left( \tfrac{1}{4} \frac{{\dot{h_2}}^2}{h_2} - \tfrac{1}{2} \frac{\ddot{h_2}}{h_2} - \frac{\dot{\lambda_1} \dot{h_2}}{\lambda_1 h_2} \right) b = 0 .
</math>
</div>

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