deletions | additions       diff --git a/Trajectory spawning.tex b/Trajectory spawning.tex  index b4cc16d..01d8022 100644  --- a/Trajectory spawning.tex  +++ b/Trajectory spawning.tex 
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\end{equation}  Where $H$ is the Hessian. I suppose that $J(x, t=0)$ is simply set equal to 1.   For a trajectory starting from $x(t=0)$ a small surface element of area $dS$ sweeps out a volume, $dS \int J(x(t)) dt$ (should check this). The extra term in the definition of $\rho$ above is for the projection onto the reference surface. This means that If $dt$ has units length (measuring the length of the trajectory), then  $dS J(x(t),t))$ is the area of the surface element at time $t$, or $J(x(t),t)$ is the factor by which the surface element has increased or decreased by time $t$.  The idea was to spawn new trajectories when the density of trajectories becomes very low. This can be implemented by spawning new trajectories when alpha is larger than a factor $\alpha > 1$. At this point the local density of trajectories has decreased by a factor of $\alpha$.