Axiomatics of general Hamiltonian theories including classical and quantum ones as particular cases

The set of axioms of the hamiltonian theory is constructed in the form which is valid for classical and quantum theories as well. These axioms define the algebra with two multiplication operations called the free hamiltonian algebra. Classical and quantum theories are derived as factor algebras corresponding to different equivalence relations. The axiomatics admits essentially new hamiltonian theories with the nonassociative multiplication of the observables.