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In this paper we use the gradient flow equation introduced in [10] toconstruct a Morse complex for the Hamiltonian action $\mathbb A_H$ on a mixedregularity space of loops in the cotangent bundle $T^*M$ of a closed manifold$M$. Connections between pairs of critical points are realized as genuineintersections between unstable and stable manifolds, which (despite beinginfinite dimensional objects) turn out to have finite dimensional intersectionwith good compactness properties. This follows from the existence of anadditional structure, namely a strongly integrable (0)-essential subbundle,which behaves nicely under the negative gradient flow of the Hamiltonian actionand which is needed to make comparisons. Transversality is achieved bygenerically perturbing the negative gradient vector field $-\nabla \mathbb A_H$of the Hamiltonian action within a class of pseudo-gradient vector fieldspreserving all good compactness properties of $-\nabla \mathbb A_H$. Thisfollows from an abstract transversality result of independent interest forvector fields on a Hilbert manifold for which stable and unstable manifolds ofrest points are infinite dimensional. The resulting Morse homology isindependent of the choice of the Hamiltonian (and of all other choices but thechoice of the (0)-essential subbundle, which however only changes theMorse-complex by a shift of the indices) and is isomorphic to the Floerhomology of $T^*M$ as well as to the singular homology of the free loop spaceof $M$.

Morse homology for the Hamiltonian action in cotangent bundles