  We consider classical and quantum mechanics related to an additional
noncommutativity, symmetric in position and momentum coordinates. We show that
such mechanical system can be transformed to the corresponding one which allows
employment of the usual formalism. In particular, we found explicit connections
between quadratic Hamiltonians and Lagrangians, in their commutative and
noncommutative regimes. In the quantum case we give general procedure how to
compute Feynman's path integral in this noncommutative phase space with
quadratic Lagrangians (Hamiltonians). This approach is applied to a charged
particle in the noncommutative plane exposed to constant homogeneous electric
and magnetic fields.
