  BRST formulation of cohomological Hamiltonian mechanics is presented. In the
path integral approach, we use the BRST gauge fixing procedure for the
partition function with trivial underlying Lagrangian to fix symplectic
diffeomorphism invariance. Resulting Lagrangian is BRST and anti-BRST exact and
the Liouvillian of classical mechanics is reproduced in the ghost-free sector.
The theory can be thought of as a topological phase of Hamiltonian mechanics
and is considered as one-dimensional cohomological field theory with the target
space a symplectic manifold. Twisted (anti-)BRST symmetry is related to global
N=2 supersymmetry, which is identified with an exterior algebra.
Landau-Ginzburg formulation of the associated $d=1$, N=2 model is presented and
Slavnov identity is analyzed. We study deformations and perturbations of the
theory. Physical states of the theory and correlation functions of the BRST
invariant observables are studied. This approach provides a powerful tool to
investigate the properties of Hamiltonian systems.
