deletions | additions       diff --git a/First_step_in_Velocity_Verlet__.tex b/First_step_in_Velocity_Verlet__.tex  index 5971193..e9b9eda 100644  --- a/First_step_in_Velocity_Verlet__.tex  +++ b/First_step_in_Velocity_Verlet__.tex 
...
is a forward finite-difference approximation of $\vec{v}(\Delta t)$, i.e.  $$ \frac{1}{\Delta t} (\vec{x}(2 \vec{x}(2  \Delta t)- \vec{x}(\Delta t))  = \vec{\hat{v}}(\Delta \vec{x}(\Delta  t) + O(\Delta \vec{v}(\Delta  t) \Delta t + \frac{1}{2} \vec{a} (\Delta t) \Delta t^2 + O(\Delta t^3)  $$ Euler's iteration has $O(\Delta t^2)$ error. $$ \frac{\vec{x}(2 \Delta t) - \vec{x}(\Delta t)}{\Delta t} = \vec{v}(\Delta t) + \frac{1}{2} \vec{a} (\Delta t) \Delta t + O(\Delta t^2) $$  $$ \vec{v}(0) \frac{1}{\Delta t} (\vec{x}(2 \Delta t) - \vec{x}(\Delta t)) = \vec{\hat{v}}(\Delta t)  + \frac{1}{2}\vec{a}(0)\Delta \frac{1}{2} \vec{a} (\Delta t) \Delta  t= \vec{\hat{v}}(\frac{\Delta t}{2})  + O(\Delta t^2) $$ So we get  $$\vec{v}(\Delta $$2 \vec{v}(\Delta  t) = \frac{\vec{\hat{v}}(\Delta \vec{v}(0) + \frac{1}{2}\vec{a}(0)\Delta t + \vec{\hat{v}}(\Delta  t) + \vec{\hat{v}}(\frac{\Delta t}{2})}{2} \frac{1}{2} \vec{a} (\Delta t) \Delta t  + O(\Delta t^2)$$  $$2 \vec{v}(\Delta t) = \vec{v}(\Delta t) + \vec{\hat{v}}(\Delta  t) + O(\Delta t^2)$$  $$ \vec{v}(\Delta t) = \vec{\hat{v}}(\Delta t) + O(\Delta t^2)$$  where $\vec{\hat{v}}$ is a precise value.