Derivation of the Verlet Method

Now let us consider the discrete version of minimizing the action . We work on the time interval [0, r]. Consider the v + 1 points qa to qv in configuration space 

We must first define the action for such a discrete path. One perspective is that we first formulate a discrete version of the Lagrangian in terms of only the point set qo,qi, qv, but this somehow requires that we define velocities. Noting that 

At first glance, it looks like this is a crude approximation, since we have employed a one-sided difference, however, let us proceed anyway to see the implications of this choice. In the case of a mechanical system with Lagrangian L q, ) = v Mvfl + U q), we then have 

We should think of. A as a function of all the positions qo,qi.qv defining the discrete path. Critical points of this function satisfy 

The method (2.4) is commonly referred to as StOrmer s rule. It was used by the mathematician Stormer for calculations in the first decade of the 1900s. In molecular dynamics this method is referred to as the Verlet method since it was used by Verlet in his important 1967 paper [387]. 

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