  Different methods have been recently put forward and implemented
experimentally to inverse engineer the time dependent Hamiltonian of a quantum
system and accelerate slow adiabatic processes via non-adiabatic shortcuts. In
the "transitionless tracking algorithm" proposed by Berry, shortcut
Hamiltonians are designed so that the system follows exactly, in an arbitrarily
short time, the approximate adiabatic path defined by a reference Hamiltonian.
A different approach is based on designing first a Lewis-Riesenfeld invariant
to carry the eigenstates of a Hamiltonian from specified initial to final
configurations, again in an arbitrary time, and then constructing from the
invariant the transient Hamiltonian connecting these boundary configurations.
We show that the two approaches, apparently quite different in form and so far
in results, are in fact strongly related and potentially equivalent, so that
the inverse-engineering operations in one of them can be reinterpreted and
understood in terms of the concepts and operations of the other one. We study
as explicit examples the expansions of time-dependent harmonic traps and state
preparation of two level systems.
