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Verlet integrator method [1]
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- Leapfrog integrator [2]
Verlet integrator method [1] Let us suppose that we already know the position vectors and their necessary derivatives at the time of tk. Using the Taylor expansion of the position vector we can calculate its value at tk+Δ t and tk-Δt
where O( Δ t4) is an error term of order t4. Adding these two series, the terms with odd orders will cancel and we obtain
After rearrangement
Using the Newton’s equation we can substitute the acceleration with the force calculated from the position vectors at tk:
In this method we need to know the values of the position vectors at tk and tk - Δ t (green points in Figure 6.3.). The forces at tk can be calculated from ri(tk)-s (blue spot in Figure 6.3.). Finally, from the positions and the forces we can calculate the new positions (red spot in Figure 6.3.). Moving the reference time one step further we can repeat the procedure until reaching the desired simulation time. The Verlet method does not require the explicit calculation of the velocities but we may need it during the evaluation of the results (e.g. calculation of kinetic energy). Formally we can obtain it by subtracting the second equation of (6.2) from the first:
The error of this approximate value is second order in Δt, considerably larger that in the case of position (fourth order in Δt) . 
Figure 6.3. Steps of the Verlet integration algorithm Leapfrog integrator [2] The other widespread method for the numerical solution of the Newton's equations is the leapfrog integration. It can be easily derived by rearranging the central formula of Verlet’s procedure [6.4] and dividing it by Δt:
The first term in the left hand side and right hand side of the equation is an approximation of v i(t+½Δt) and v i(t-½Δt), respectively. This gives the first equation of the leapfrog method.
The second equation comes from the approximate expression of the velocity used in the derivation. The schematic representation of this method is given in Figure 6.4. 
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