Time-independence of original coordinates in canonical transform

[mjordan2nd]

I am going through my professors notes on generating functions. I come across the following equation:

$$\frac{\partial}{\partial t} \frac{\partial F}{\partial \xi^k} = \frac{\partial}{\partial t} \left( \gamma_{il} \frac{\partial \eta^i}{\partial \xi^k}\eta^l - \gamma_{kl}\xi^l \right ).$$

Here $\xi$ are the original coordinates, $\eta$ are the transformed coordinates, and $\gamma_{il}$ are the components of the block matrix

$$\left( \begin{array}{ccc} 0 & -I \\ 0 & 0 \end{array} \right ).$$

When the time derivative is taken on the right hand side, the $\xi^l$ are said to be independent of time. Why is this. Certainly (q,p) depend on time, right?

 

[Valerie Park]

I'm confused as to why the \xi^l are independent of time.The reason that the \xi^l are said to be independent of time is because they are the original coordinates, and the transformation from the original coordinates to the transformed coordinates is a static transformation - it does not change over time. Therefore, the \xi^l remain constant throughout the transformation, and thus are not dependent on time.