Noncommutativity from the symplectic point of view

The great deal in noncommutative (NC) field theories started when it was noted that NC spaces naturally arise in string theory with a constant background magnetic field in the presence of D-branes. In this work we explore how NC geometry can be introduced into a commutative field theory besides the usual introduction of the Moyal product. We propose a nonperturbative systematic new way to introduce NC geometry into commutative systems, based mainly on the symplectic approach. Further, as example, this formalism describes precisely how to obtain a Lagrangian description for the NC version of some systems reproducing well-known theories.