In consideration of the correspondence between

Poisson bracket and quantum commutator, noncommutative extended dynamics is also interesting even at the classical level [46-49], particularly in the fluid dynamics [50-53].

Considering a smooth function C on P, a Poisson structure on P is generated by the

Poisson bracketIn a fusion of a research monograph on function theory on symplectic manifolds and an introductory survey of symplectic topology, Polterovich and Rosen discuss such topics as three wonders of symplectic geometry, quasi-morphisms, sub-additive spectral invariants, symplectic quasi-states and quasi-measures, applications of partial symplectic quasi-states, a

Poisson bracket invariant of quadruples, symplectic approximation theory, the geometry of covers and quantum noise, an overview of Floer theory, and constructing sub-additive spectral invariants.

A

Poisson bracket on Mis a bilinear map from [C.sup.[infinity]](M) x [C.sup.[infinity]](M) into [C.sup.[infinity]](M), denoted as

As an added note, we can mention here, that the aforementioned Feynman's derivation of Maxwell equations is based on commutator relation which has classical analogue in the form of

Poisson bracket. Then there can be a plausible way to extend directly this "classical" dynamics to quaternion extension of

Poisson bracket, by assuming the dynamics as element of the type: r [member of] H [conjunction] H of the type: r = ai [conjunction] j + bi [conjunction] k + cj [conjunction] k, from which we can define

Poisson bracket on H.

be a

Poisson bracket. It is well-known that the association df[[right arrow]][X.sub.f] (the hamiltonian vector field which correspond to a real function f[member of]F(M) is given by [X.sub.f] (g) = {f, g}) extends to an anchor D: [tau] *M[right arrow][tau]M.

In order to show the structure of extremals it is useful to recall the

Poisson bracket. If F and H are any functions on the cotangent bundle [T.sup.*]M of an arbitrary manifold M, then {F, H} denotes their

Poisson bracket.

where [A(x), B(y)} is the

Poisson bracket between the functionals A, B; the matrix [C.sub.[alpha][beta]] [equivalent to] {[[phi].sup.[alpha]], [phi].sup.[beta]]}; here, {[[phi].sup.[alpha]]} is the complete set of second-class constraints found above; and [C.sup.-1] represents the inverse of C.

A novel star product deformations of (super) p-brane actions based on the noncommutative spacetime Yang's algebra where the deformation parameter is [[??].sub.eff] = [??][L.sub.P] / R, for nonzero values of [??], was obtained in [15] The modified (noncommutative)

Poisson bracket is now given by

Since [Psi] = H, and the system evolves according to [Sigma] = {[Sigma], H} + [Lambda]{[Sigma], [Phi]} + [Mu]{[Sigma], [Psi]} = (1 + [Mu]){[Sigma], H} + [Lambda]{[Sigma], [Phi]}, for Lagrange multipliers [Lambda], [Mu] and

Poisson bracket {,}, it suffices, by setting [Mu] = 0, to consider the Hamiltonian system (M, [Omega], H) constrained by

The bracket [{.,.}.sub.1] verifies the Jacobi identity; thus it is a

Poisson bracket. We have proven the following result.

This leads one to the conclusion that the only other needed deformation is in the

Poisson bracket between two boosts,