VARIATIONAL-PRINCIPLES AND 2ND-ORDER LAGRANGIAN DENSITIES

The functionals of the Kohn and Sil variational principles of atomic c ollision theory contain second-order spatial derivatives of the wave-f unction. Ipso facto the 'second-order'' generalised Euler-Lagrange dif ferential equation is a necessary condition for these functionals to y ield a stationary value. To illustrate this we provide a classical pre scription for the time-dependent and time-independent quantum potentia l scattering problems, generalising the latter to the three-body probl em. As practical examples where the presence of second-order derivativ es in the Lagrangian density makes a non-trivial contribution to the e quations of motion, we consider the optimisation of translation factor s in a trial wave-function using the Sil variational principle and the optimisation of a variable charge using the Kohn variational principl e. As further examples of Lagrangian densities where second-order deri vatives may occur we briefly mention electromagnetism and relativistic quantum mechanics.