deletions | additions       diff --git a/First_step_in_Velocity_Verlet__.tex b/First_step_in_Velocity_Verlet__.tex  index e9b9eda..4e0ff5d 100644  --- a/First_step_in_Velocity_Verlet__.tex  +++ b/First_step_in_Velocity_Verlet__.tex 
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is a forward finite-difference approximation of $\vec{v}(\Delta t)$, i.e.  $$ \vec{x}(2 \Delta t) = \vec{x}(\Delta t) + \vec{v}(\Delta \vec{\hat{v}}(\Delta  t) \Delta t + \frac{1}{2} \vec{a} (\Delta t) \Delta t^2 + O(\Delta t^3) $$ $$ \frac{\vec{x}(2 \Delta t) - \vec{x}(\Delta t)}{\Delta t} = \vec{v}(\Delta \vec{\hat{v}}(\Delta  t) + \frac{1}{2} \vec{a} (\Delta t) \Delta t + O(\Delta t^2) $$ $$ \frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t)) = \vec{\hat{v}}(\Delta t) + \frac{1}{2} \vec{a} (\Delta t) \Delta t + O(\Delta t^2) $$