Dynamics of helium trapped in fullerene cages

Lee-Dadswell, Geoffrey R.
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University of Guelph

The unconstrained reformulated Maupertuis principle and Percival's variational principle are used to predict the perturbed frequencies of the system with potential V=(ω\sbspx(0))\sp2ρ\sp2/2+(ω\sbspz(0))\sp2z\sp2/2+Cρ\sp2z\sp2. It is found that these variational principles make equivalent predictions when a simple harmonic oscillator trial solution is used. Predictions of the radial and z frequencies are quite accurate and can be calculated very easily in this way. However, this trial solution is unable to account for precession in the system. A more complicated trial solution is shown to be able to predict the precessional frequency at the cost of some accuracy in the predictions of the other frequencies. The revised trial solution also greatly increases the difficulty of the calculation. An approximate constant of the motion for the system is generated using the method of Birkhoff-Gustavson normal forms. It is found to be divergent and only good at low energies. The Birkhoff-Gustavson normal form is compared with expressions for the energy of the system in terms of approximate actions obtained from the variational results. Agreement at low degrees is revealed. The structure of phase space for the system is examined in detail and it is found that the results of the variational calculations can be used to make useful predictions about the phase space; in particular, their allow the prediction of resonances in the system to a good degree of accuracy.

dynamics, helium, fullerene cages, Maupertuis principle, Percival's variational principle