  We use path integral methods and topological quantum field theory techniques
to investigate a generic classical Hamiltonian system. In particular, we show
that Floer's instanton equation is related to a functional Euler character in
the quantum cohomology defined by the topological nonlinear $\sigma$--model.
This relation is an infinite dimensional analog of the relation between
Poincar\'e--Hopf and Gauss--Bonnet--Chern formul\ae$~$ in classical Morse
theory, and can also be viewed as a loop space generalization of the Lefschetz
fixed point theorem. By applying localization techniques to path integrals we
then show that for a K\"ahler manifold our functional Euler character coincides
with the Euler character determined by the finite dimensional de Rham
cohomology of the phase space. Our results are consistent with the Arnold
conjecture which estimates periodic solutions to classical Hamilton's equations
in terms of de Rham cohomology of the phase space.
