Knowledge Bank

We reported $previously1$ that an algebraic minimization procedure could be used to simplify the derivation of centrifugal distortion constants of high order, but the relation between this procedure and the conventional use of successive contact transformations had not been clearly understood at that time. As we see it now, the minimization procedure is equivalent to a contact transformation in closed from where the transformation function S($pi$,$qi$) is linear in the conjugate variables $pi$, $qi$, and is used to eliminate linear terms in these variables in the Hamiltonian. Such a transformation represents a translation in more-dimensional phase-space. It appears not to have been noticed previously that exact formulas can also be derived when the transformation function is of second order in $pi$, $qi$. The transformations thus generated can be interpreted as rotations in phase-space, and can be combined with the translations. The transformations themselves can be easily performed in closed form, but the exact determination of the parameters needed in the transformation functions presents a more difficult though solvable problem.