  While it is well-known that every nearly-periodic Hamiltonian system
possesses an adiabatic invariant, extant methods for computing terms in the
adiabatic invariant series are inefficient. The most popular method involves
the heavy intermediate calculation of a non-unique near-identity coordinate
transformation, even though the adiabatic invariant itself is a
uniquely-defined scalar. A less well-known method, developed by S. Omohundro,
avoids calculating intermediate sequences of coordinate transformations but is
also inefficient as it involves its own sequence of complex intermediate
calculations. In order to improve the efficiency of future calculations of
adiabatic invariants, we derive generally-applicable, readily computable
formulas for the first several terms in the adiabatic invariant series. To
demonstrate the utility of these formulas, we apply them to charged particle
dynamics in a strong magnetic field and magnetic field-line dynamics when the
field lines are nearly closed.
