Adiabatic Invariants and the “Third” Integral

In a general case of Hamiltonian systems of n degrees of freedom, depending periodically on time, n formal “third” integrals of motion are found. Their application in finding boundaries for the orbits is illustrated in a special case. Then a comparison is made between these integrals and the adiabatic invariants. Both are series expansions but the small parameter used is of different character in each case. This is shown explicitly in a simple example and the relative accuracy of the two expansions is discussed.