  We consider Carroll-invariant limits of Lorentz-invariant field theories. We
show that just as in the case of electromagnetism, there are two inequivalent
limits, one "electric" and the other "magnetic". Each can be obtained from the
corresponding Lorentz-invariant theory written in Hamiltonian form through the
same "contraction" procedure of taking the ultrarelativistic limit $c
\rightarrow 0$ where $c$ is the speed of light, but with two different
consistent rescalings of the canonical variables. This procedure can be applied
to general Lorentz-invariant theories ($p$-form gauge fields, higher spin free
theories etc) and has the advantage of providing explicitly an action principle
from which the electrically-contracted or magnetically-contracted dynamics
follow (and not just the equations of motion). Even though not manifestly so,
this Hamiltonian action principle is shown to be Carroll invariant. In the case
of $p$-forms, we construct explicitly an equivalent manifestly
Carroll-invariant action principle for each Carroll contraction. While the
manifestly covariant variational description of the electric contraction is
rather direct, the one for the magnetic contraction is more subtle and involves
an additional pure gauge field, whose elimination modifies the Carroll
transformations of the fields. We also treat gravity, which constitutes one of
the main motivations of our study, and for which we provide the two different
contractions in Hamiltonian form.
