deletions | additions       diff --git a/First_step_in_Velocity_Verlet__.tex b/First_step_in_Velocity_Verlet__.tex  index 758e444..5971193 100644  --- a/First_step_in_Velocity_Verlet__.tex  +++ b/First_step_in_Velocity_Verlet__.tex 
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Euler's iteration has $O(\Delta t^2)$ error.  $$ \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t = \vec{\hat{v}}(\Delta t) \vec{\hat{v}}(\frac{\Delta t}{2})  + O(\Delta t) t^2)  $$ So we get  $$\vec{v}(\Delta t) = \vec{\hat{v}}(\Delta \frac{\vec{\hat{v}}(\Delta  t) + \vec{\hat{v}}(\frac{\Delta t}{2})}{2} +  O(\Delta t) $$ where $\vec{\hat{v}}$ is a precise value.