Verlet leap-frog algorithm

$r(t+\delta t)=r(t)+\delta tv\left(t+{\frac {1}{2}}\delta t\right)$

$v\left(t+{\frac {1}{2}}\delta t\right)=v\left(t-{\frac {1}{2}}\delta t\right)+\delta ta(t)$

References

1. Loup Verlet "Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules", Physical Review 159 pp. 98 - 103 (1967)
2. R. W. Hockney, Methods in Computational Physics vol. 9, Academic Press, New York pp. 135–211 (1970)