Editing Jaycall/KillingVectorApproach (section)

===Introduce Characteristic Vector===
The trick is to introduce an arbitrary vector field, which will be used to build a location-dependent weighted linear combination of the equations of motion.  Since the equations of motion associated with each of the coordinates collectively form a single vector equation, we will just form an inner product of our characteristic vector with each side of the vector equation of motion.  The result is

<span id="CV.01"><table align="right" border="1" cellpadding="10" width="10%">
<tr><th><font color="darkblue">CV.01</font></th></tr>
</table></span>
<div align="center">
<math>
C_i \frac{d}{dt} \left( m \ {h_i}^2 \dot{\lambda}_i \right) = m \ {h_k}^2 \Gamma^k_{ij} \dot{\lambda}_j \dot{\lambda}_k C_i - m \ C_i \ \partial_i \Phi
</math>
</div>

Next, we bring <math>C_i</math> inside the total time-derivative on the left-hand side (LHS).  This produces an additional term, which we promptly move over to the RHS and include as part of what is commonly referred to as the ''source'' (since it's the source of any change in the quantity in parentheses).

<div align="center">
<math>
\frac{d}{dt} \left( m \ {h_i}^2 \dot{\lambda}_i C_i \right) = m \ {h_i}^2 \dot{\lambda}_i \dot{C}_i + m \ {h_k}^2 \Gamma^k_{ij} \dot{\lambda}_j \dot{\lambda}_k C_i - m \ C_i \ \partial_i \Phi
</math>
</div>

By doing this, we have formed a new ''conservative'' quantity; that is, a new quantity in the parentheses which will be conserved if and only if the source is zero.  The utility is in the fact that we should be able to force the source associated with this new conservative quantity to go to zero by choosing the right characteristic vector.  Once we find the right characteristic vector, we'll be able to use it directly to build the conserved quantity <math>m \ {h_i}^2 \dot{\lambda}_i C_i</math>.