deletions | additions       diff --git a/Let_us_derive_basic_Verlet__.tex b/Let_us_derive_basic_Verlet__.tex  index 475b679..9c5abf2 100644  --- a/Let_us_derive_basic_Verlet__.tex  +++ b/Let_us_derive_basic_Verlet__.tex 
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Let us derive basic Verlet from velocity formulation: If we assume  $$\vec{v}(t) = \vec{v}(t - $$ \vec{v}(t)  \Delta t) + \frac{\vec{a}(t t = (\vec{x}(t)  - \Delta t) + \vec{a}(t)}{2} \Delta t$$ \vec{x}(t-\Delta t)) $$  $$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t - \Delta t) \Delta t + \frac{\vec{a}(t - \Delta t) + \vec{a}(t)}{2} \Delta t^2 + \frac{1}{2} \vec{a}(t) \Delta t^2$$ then  $$\vec{x}(t) = \vec{x}(t - \Delta $$ \vec{x}(t+\Delta  t) = \vec{x}(t)  + \vec{v}(t - \Delta t) \vec{v}(t)  \Delta t + \frac{1}{2} \vec{a}(t - \Delta t) \Delta t^2$$ \vec{a}(t)\Delta t^2 $$  which is \textit{exactly} Velocity Verlet for $\vec{x}$.