CLASSICAL GEOMETRIC FORCES OF REACTION - AN EXACTLY SOLVABLE MODEL

We illustrate the effects of the classical 'magnetic' and 'electric' g eometric forces that enter into the adiabatic description of the slow motion of a heavy system coupled to a light one, beyond the Born Oppen heimer approximation of simple averaging. When the fast system is a sp in S and the slow system is a massive particle whose spatial position R is coupled to S with energy (fast hamiltonian) S . R, the magnetic f orce is that of a monopole of strength I (= adiabatic invariant S . R/ R) centred at R = 0, and the electric force is inverse-cube repulsion with strength S2 - I2. Confining the slow particle to the surface of a sphere eliminates the Born-Oppenheimer and electric forces, and gener ates motion with precession and nutation exactly equivalent to that of a heavy symmetrical top. In the adiabatic limit the nutation is small and the averaged precession is precisely reproduced by the magnetic f orce. Alternatively, choosing the exactly conserved total angular mome ntum to vanish eliminates the Born-Oppenheimer and magnetic forces, an d generates as exact orbits a one-parameter family of curly 'antelope horns' coiling in from infinity, reversing hand. and receding to infin ity. In the adiabatic limit the repulsion of the 'guiding centre' of t hese coils is exactly reproduced by the electric force. A by-product o f the 'antelope horn' analysis is a determination of the shape of a cu rve with a given curvature kappa and torsion tau in terms of the evolu tion of a quantum 2-spinor driven by a planar 'magnetic field' with co mponents kappa and tau.