Hamiltonian function Hamilton-Jacobi

It may happen that the Hamiltonian function does not consist o a sum of terms depending on only one pair of variables qkpk, but that the Hamilton-Jacobi equation may be solved by separation of the variables, i.e. on the assumption that... 

It is the Hamiltonian function of a system in which all co-ordinates but one are cyclic. The motion may be found in the usual way by solving a Hamilton-Jacobi differential equation for one degree of freedom. Since and ij (like ° and rj°) must vanish with A, we need only consider small motions, that is, those belonging to a system whose Hamiltonian function is... 

After calculating the unperturbed motion of the inner electron, we can find the secular motions of the remaining variables by introducing a new Hamiltonian function, the mean value of Hx taken over the unperturbed motion of the inner electron. The integration of the corresponding Hamilton-Jacobi equation is again performed by the methods of the theory of perturbations. 

Equation (5.2) is a combination of the two two-dimensional Hamiltonians (2.39) and (3.15) which describe the vibrational and rotational excitations of BC separately. The Jacobi coordinates R, r, and 7 are defined in Figures 2.1 and 3.1 and P and p denote the linear momenta corresponding to R and r, respectively, j is the classical angular momentum vector of BC and 1 stands for the classical orbital angular momentum vector describing the rotation of A with respect to BC. For zero total angular momentum J=j+l = 0we have 1 = — j and the Hamilton function reduces to... 

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