Jacobi vectors dynamics

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

Fig. 10.7. The bond coordinates used to describe the photodissociation of H2O2.

For AB + CD systems the most straightforward choice of coordinates to describe the dynamical system is the Jacobi coordinates corresponding to the diatom-diatom arrangement. As shown in Fig. 1, the three vectors (R,ri,r2) denote, respectively, the vector R from the center of mass (CM) of diatom AB to that of CD, the AB diatomic vector F, and the CD diatomic vector F2. The full Hamiltonian expressed in this set of coordinates is written as... 

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