Editing Appendix/Ramblings/T3CharacteristicVector (section)

==Conserved Quantity==

Let's cut to the chase.  As shown on the page describing the characteristic vector approach, I can write down the third conserved quantity right now--just not in closed form.  Assuming there's no potential variation in the direction of <math>\lambda_2</math>, it is

<div align="center">
<math>
m{h_2}^2 \dot{\lambda_2} \exp \left\{ - \int \frac{{h_k}^2}{{h_2}^2} \Gamma^k_{2j} \frac{\dot{\lambda_j} \dot{\lambda_k}}{\dot{\lambda_2}} \ dt \right\} .
</math>
</div>

In the case of T3 coordinates, this becomes more specific.

<div align="center">
<math>
m {h_2}^2 \dot{\lambda_2} \exp \left\{ - \int 2 {\lambda_1}^2 \ell^4 \left( \frac{\lambda_2 \dot{\lambda_1}^2}{\dot{\lambda_2}} - \frac{\dot{\lambda_2}{\lambda_1}^2}{\lambda_2} \right) dt \right\}
</math>
</div>

Although this is not all that useful in an analytic sense until we can integrate it, I wonder if it can be a guide to building a more accurate numerical model.  Certainly this function can be integrated numerically, and that's got to be useful somehow...