  Extension of Feynman's path integral to quantum mechanics of noncommuting
spatial coordinates is considered. The corresponding formalism for
noncommutative classical dynamics related to quadratic Lagrangians
(Hamiltonians) is formulated. Our approach is based on the fact that a
quantum-mechanical system with a noncommutative configuration space may be
regarded as another effective system with commuting spatial coordinates. Since
path integral for quadratic Lagrangians is exactly solvable and a general
formula for probability amplitude exists, we restricted our research to this
class of Lagrangians. We found general relation between quadratic Lagrangians
in their commutative and noncommutative regimes. The corresponding
noncommutative path integral is presented. This method is illustrated by two
quantum-mechanical systems in the noncommutative plane: a particle in a
constant field and a harmonic oscillator.
