Trajectory calculations initial value formulation

An algorithm to compute classical trajectories using boundary value formulation is presented and discussed. It is based on an optimization of a functional of the complete trajectory. This functional can be the usual classical action, and is approximated by discrete and sequential sets of coordinates. In contrast to initial value formulation, the pre-specified end points of the trajectories are useful for computing rare trajectories. Each of the boundary-value trajectories ends at desired products. A difficulty in applying boundary value formulation is the high computational cost of optimizing the whole trajectory in contrast to the calculation of one temporal frame at a time in initial value formulation. 

A complementary approach to the initial value formulation is the calculation of trajectories as a solution of a boundary value problem. In this approach the two end points are given as input to the calculations. It is therefore obvious that the trajectories will end at the pre-set interesting configurations. This simple construction solves the second limitation on initial value calculations that we mentioned above. Of course if the end points are not known and only the beginning configuration is available (e.g. the protein folding problem), then the initial value approach is the only viable option. 

There are important differences between the initial value formulation and the boundary value approach. Initial value solutions are based on interpolation forward in time one coordinate set after another. The boundary value approach is based on minimization of a target function of the whole trajectory. Minimization (and the study of a larger system) is more expensive in the boundary value formulation compared to initial value solver. However, the calculations of state to state trajectories and the abilities to use approximations (next section), make it a useful alternative for a large number of problems. 

The last hmitrng expression is the Gauss action, Sc, for classical mechanics. It clearly has a minimum that satisfies the equations of motion (when the action is zero). The action is non-negative, which makes it easier to identify the trae minimum. The non-negativity is an important difference from the classical action formulation that we introduced at the beginning and makes the calculations with the Ssdet and Sg significantly more stable. The approximate trajectories that are produced by optimization of Ssdet are stable. An exponential solution of the type expjmf] (co positive) cannot be obtained in a reasonable formulation of a boundary value problem if the boundaries are not explosive . However, an explosive solution can be obtained with initial value formulation. 

In support of the philosophical appeal is the fact that Eq. (4) is an integral while in Eqs. (2) we use derivatives. Numerical estimates of integrals are, in general, more accurate and more stable compared to estimates of derivatives. On the other hand, computations of the whole path are more expensive than the calculation of one temporal slice of the trajectory at a time. The computational effort is larger in the boundary value formulation by at least a factor of N, where N is the number of time shces, compared to the calculation of a step in the initial value approach. To make the global approach computationally attractive (assuming that it does work), the gain in step size must be substantial. 

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