deletions | additions       diff --git a/If_we_assume_2_vec__.tex b/If_we_assume_2_vec__.tex  index 2ec1a35..2130ade 100644  --- a/If_we_assume_2_vec__.tex  +++ b/If_we_assume_2_vec__.tex 
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then  $$\vec x(t+\Delta t) - \vec{x}(t-\Delta t) = 2\vec x(t)-2 \vec t)=2\vec x(t)-\vec  x(t-\Delta t)+\vec a(t)\,\Delta t^2$$ $$2 \vec{v}(t) \Delta t = 2\vec x(t)-2 \vec $$\vec x(t+\Delta t)=\vec x(t) + (\vec x(t)-\vec  x(t-\Delta t)+\vec a(t)\,\Delta t^2$$  $$\vec{v}(t) \Delta t = t)+\frac{1}{2}  \vec x(t)- a(t) \Delta t^2) + \frac{1}{2}  \vec x(t-\Delta t)+\frac{1}{2}\vec a(t)\,\Delta a(t) \Delta  t^2$$ which is \textit{exactly} Velocity Verlet for $\vec{x}$.