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In the Wikipedia article on Verlet method, under the Velocity Verlet method, the first algorithm proposed uses half steps:

Calculate $\vec{v}\left(t + \tfrac12\,\Delta t\right) = \vec{v}(t) + \tfrac12\,\vec{a}(t)\,\Delta t$.
Calculate $\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}\left(t + \tfrac12\,\Delta t\right)\, \Delta t$.
Derive $\vec{a}(t + \Delta t)$ from the interaction potential using $\vec{x}(t + \Delta t)$.
Calculate $\vec{v}(t + \Delta t) = \vec{v}\left(t + \tfrac12\,\Delta t\right) + \tfrac12\,\vec{a}(t + \Delta t)\Delta t$.

One sentence later it is mentioned that you may also eliminate the half steps.

I don't really get why one should use the half steps in the first place when the solution without half steps is trivially derived from the base equations:

$$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t)\, \Delta t + \frac{1}{2} \,\vec{a}(t) \Delta t^2$$ $$\vec{v}(t + \Delta t) = \vec{v}(t) + \frac{\vec{a}(t) + \vec{a}(t + \Delta t)}{2} \Delta t$$

Kyle Kanos
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Alexander Haas
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1 Answers1

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Your answer is found in the sentence you omitted that immediately follows the shortened form:

Note, however, that this algorithm assumes that acceleration $\vec{a}\left(t+\Delta t\right)$ only depends on position $\vec{x}\left(t+\Delta t\right)$ and does not depend on velocity $\vec{v}\left(t+\Delta t\right)$.

For gravitationally interacting systems, the force depends only on the position so that the shortened form can be used easily enough. But if you are dealing with a velocity dependent force (e.g., drag; cf also this answer of mine, among others), then the shortened form cannot be used because $\vec{a}\left(t\right)=f\left(t,\,x\left(t\right),\,v\left(t\right)\right)$ and incrementing to $t+\Delta t$ leads to some complications that are more easily resolved using the split step mentioned first.

Kyle Kanos
  • 29,127