A theory of libration

It is shown here that many problems of libration in celestial mechanics can be reduced to a perturbation of anintermediary defined by the Hamiltonian $$F = Bleft( y right) + 2mu ^2 Aleft( y right)fleft( x right).$$ This generalization of the Ideal Resonance Problem, with a periodic functionf(x) replacing sin² x, is solved here toO(μ ²) by an algorithm that is essentially the same as the one used in the original formulation. The solution is of the formx=x(u), u=u(t), y=y(x), with the functionx(u) commonly involving the inversion of a hyperelliptic integralu(x), evaluated by quadrature. Libration may be simple or multiple, depending on the nature of the functionf(x) and on the initial conditions. Double libration is illustrated here by the horseshoe-shaped orbits enclosing two libration centers.