  We study the ions dynamics inside an Electrostatic Ion Beam Trap (EIBT) and
show that the stability of the trapping is ruled by a Hill's equation. This
unexpectedly demonstrates that an EIBT, in the reference frame of the ions
works very similar to a quadrupole trap. The parallelism between these two
kinds of traps is illustrated by comparing experimental and theoretical
stability diagrams of the EIBT. The main difference with quadrupole traps is
that the stability depends only on the ratio of the acceleration and trapping
electrostatic potentials, not on the mass nor the charge of the ions. All kinds
of ions can be trapped simultaneously and since parametric resonances are
proportional to the square root of the charge/mass ratio the EIBT can be used
as a mass spectrometer of infinite mass range.
