On the change in the adiabatic invariant on crossing a separatrix in systems with two degrees of freedom

Abstract Hamiltonian systems with two degrees of freedom are studied. One degree of freedom corresponds to rapid motion, and the other to slow motion. The phase point intersects the separatrix of the rapid motion. Formulas are obtained for the change in the adiabatic invariant during this crossing. An example is solved, dealing with the change in the adiabatic invariant, of an asteroid near the 3:l resonance with Jupiter.