of the Verlet iteration at step , time , computing , one already needs the position vector at time . At first sight, this could give problems, because the initial conditions are known only at the initial time . However, from these the acceleration is known, and a suitable approximation for the position at the first time step can be obtained using the Taylor polynomial of degree two: The error on the first time step then is of order . This is not considered a problem because on a simulation over a large number of time steps, the error on the first time step is only a negligibly small amount of the total error, which at time is of the order , both for the distance of the position vectors to as for the distance of the divided differences to . Moreover, to obtain this second-order global error, the initial error needs to be of at least third order. Non-constant time differences A disadvantage of the Störmer–Verlet method is that if the time step () changes, the method does not approximate the solution to the differential equation. This can be corrected using the formula A more exact derivation uses the Taylor series (to second order) at for times and to obtain after elimination of so that the iteration formula becomes Computing velocities – Störmer–Verlet method The velocities are not explicitly given in the basic Störmer equation, but often they are necessary for the calculation of certain physical quantities like the kinetic energy. This can create technical challenges in molecular dynamics simulations, because kinetic energy and instantaneous temperatures at time cannot be calculated for a system until the positions are known at time . This deficiency can either be dealt with using the velocity Verlet algorithm or by estimating the velocity using the position terms and the mean value theorem: Note that this velocity term is a step behind the position term, since this is for the velocity at time , not , meaning that is a second-order approximation to . With the same argument, but halving the time step, is a second-order approximation to , with . One can shorten the interval to approximate the velocity at time at the cost of accuracy: Velocity Verlet A related, and more commonly used, algorithm is the velocity Verlet algorithm, similar to the leapfrog method, except that the velocity and position are calculated