Shooting system dynamics

To calculate the partiais we define shooting system dynamics for each unknown initial condition. These equations come from taking partiais with respect to the missing initial conditions (a and 6), of the describing state differential equations (7.2.1) to (7.2.4). 

Develop the generalized shooting technique for solving this problem. Write down the shooting system dynamics for this problem and the updating equations for unknown initial conditions. Discuss a computer implementation. 

Figure 7.5b Dynamic model and Shooting System Equations,...

For some applications, the distribution of initial conditions is distinct from the stationary distribution preserved by a system s dynamics. For example, one might be interested in the relaxation of a system that has been driven away from equilibrium. One might even be interested in an unstable dynamics that does not preserve a stationary distribution. In both cases, it is possible to sample trajectories using the same type of shooting move described above, provided the distribution of initial conditions p(x) is well defined. To derive an appropriate acceptance probability for this case, we return to Eq. (1.25). By using Eq. (1.32) and the chain rule for the Jacobian,... 

Our definition of a committor in Eq. (1.107) is applicable to both stochastic and deterministic dynamics. In the case of deterministic dynamics, care must be taken that fleeting trajectories are initiated with momenta drawn from the appropriate distribution. As discussed in Section III.A.2, global constraints on the system may complicate this distribution considerably. The techniques described in Section III.A.2 and in the Appendix of [10] for shooting moves may be simply generalized to draw initial momenta at random from the proper equilibrium distribution. 

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