  We consider Feynman's path integral approach to quantum mechanics with a
noncommutativity in position and momentum sectors of the phase space. We show
that a quantum-mechanical system with this kind of noncommutativity is
equivalent to the another one with usual commutative coordinates and momenta.
We found connection between quadratic classical Hamiltonians, as well as
Lagrangians, in their commutative and noncommutative regimes. The general
procedure to compute Feynman's path integral on this noncommutative phase space
with quadratic Lagrangians (Hamiltonians) is presented. Using this approach, a
particle in a constant field, ordinary and inverted harmonic oscillators are
elaborated in detail.
