  For each representative $\mathfrak{B}$ of a bordism class in the free loop
space of a manifold, we associate a moduli space of finite length Floer
cylinders in the cotangent bundle. The left end of the Floer cylinder is
required to be a lift of one of the loops in $\mathfrak{B}$, and the right end
is required to lie on the zero section. Under certain assumptions on the
Hamiltonian functions, the length of the Floer cylinder is a smooth proper
function, and evaluating the level sets at the right end produces a family of
loops cobordant to $\mathfrak{B}$. The argument produces arbitrarily long Floer
cylinders with certain properties. We apply this to prove an existence result
for 1-periodic orbits of certain Hamiltonian systems in cotangent bundles, and
also to estimate the relative Gromov width of starshaped domains in certain
cotangent bundles. The moduli space is similar to moduli spaces considered by
Abbondandolo-Schwarz and Abouzaid for Tonelli Hamiltonians. The Hamiltonians we
consider are not Tonelli, but rather of ``contact-type'' in the symplectization
end.
