deletions | additions       diff --git a/First_step_in_Velocity_Verlet__.tex b/First_step_in_Velocity_Verlet__.tex  index 615cc98..b00c635 100644  --- a/First_step_in_Velocity_Verlet__.tex  +++ b/First_step_in_Velocity_Verlet__.tex 
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$$\vec{v}(\Delta t) = \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t + \frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t) - \vec{v}(\Delta t) \Delta t)$$  $$2 \vec{v}(\Delta t) = \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t + \frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t))$$ Now  $$ \frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t)) $$  is a forward finite-difference approximation of $\vec{v}(\Delta t)$, i.e.  $$ \frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t)) = \vec{\hat{v}}(\Delta t) + O(\Delta t^2) $$  Again, Euler's iteration has $O(\Delta t^2)$ error.  $$ \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t = \vec{\hat{v}}(\Delta t) + O(\Delta t^2) $$  So we get  $$\vec{v}(\Delta t) = \vec{\hat{v}}(\Delta t) + O(\Delta t^2) $$  where $\vec{\hat{v}}$ is a precise value.