  In this paper we extend the standard differential geometric theory of
Hamiltonian dynamics to noncommutative spaces, beginning with symplectic forms.
Derivations on the algebra are used instead of vector fields, and interior
products and Lie derivatives with respect to derivations are discussed. Then
the Poisson bracket of certain algebra elements can be defined by a choice of
closed 2-form. Examples are given using the noncommutative torus, the Cuntz
algebra, the algebra of matrices, and the algebra of matrix valued functions on
$\Bbb R^2$.
