  This paper concerns Floer homology for periodic orbits and for a Lagrangian
intersection problem on the cotangent bundle of a compact orientable manifold
M. The first result is a new uniform estimate for the solutions of the Floer
equation, which allows to deal with a larger - and more natural - class of
Hamiltonians. The second and main result is a new construction of the
isomorphism between the Floer homology and the singular homology of the free
loop space of M, in the periodic case, or of the based loop space of M, in the
Lagrangian intersection problem. The idea for the construction of such an
isomorphism is to consider a Hamiltonian which is the Legendre transform of a
Lagrangian on TM, and to construct an isomorphism between the Floer complex and
the Morse complex of the classical Lagrangian action functional on the space of
free or based loops on M of Sobolev class W(1,2).
