We describe the classical Schwinger model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classied by 2 (S 2 ), we construct hermitian connections with values in the universal dierential envelope which leads us to the Schwinger model on the sphere. The Connes-Lott program is then applied using the Hilbert space of complexied inhomogeneous forms with its Atiyah-K ahler structure. This Hilbert space splits in two minimal left ideals of the Cliord algebra preserved by the Dirac-K ahler operator D = i (d ). The induced representation of the universal dierential envelope, in order to recover its dierential structure, is divided by the unwanted dierential ideal and the obtained quotient is the usual complexied de Rham exterior algebra over the sphere with Cliord action on the "spinors" of the Hilbert space. The subsequent steps of the Connes-Lott program allow to dene a matter action, and the eld action is obtained using the Dixmier trace which reduces to the integral of the curvature squared.
In this work we examine noncommutativity of position coordinates in classical symplectic mechanics and its quantisation. In coordinates {q i , p k } the canonical symplectic two-form is ω 0 = dq i ∧ d p i . It is well known in symplectic mechanics [5,6,9] that the interaction of a charged particle with a magnetic field can be described in a Hamiltonian formalism without a choice of a potential. This is done by means of a modified symplectic twoform ω = ω 0 −eF, where e is the charge and the (time-independent) magnetic field F is closed: dF = 0. With this symplectic structure, the canonical momentum variables acquire non-vanishing Poisson brackets: {p k , p l } = e F kl (q). Similarly a closed two-form in p-space G may be introduced. Such a dual magnetic field G interacts with the particle's dual charge r. A new modified symplectic two-form ω = ω 0 − eF + rG is then defined. Now, both p-and q-variables will cease to Poisson commute and upon quantisation they become noncommuting operators. In the particular case of a linear phase space R 2N , it makes sense to consider constant F and G fields. It is then possible to define, by a linear transformation, global Darboux coordinates: {ξ i , π k } = δ i k . These can then be quantised in the usual way [ ξ i , π k ] = i δ i k . The case of a quadratic potential is examined with some detail when N equals 2 and 3.
In this paper we deal with the problem of controlling some Chaplygin systems in the framework of the vakonomic approach for nonholonomic systems. Equations of motion for these systems are obtained which contain a free parameter that permits to control the system. It is show that given a prescribed path it is possible to determine the parameter of control which inserted in the equations of motion compel the trajectory of the system to follow the input function.
In earlier work [1], we studied an extension of the canonical symplectic structure in the cotangent bundle of an affine space Q = R N , by additional terms implying the Poisson non-commutativity of both configuration and momentum variables. In this article, we claim that such an extension can be done consistently when Q is a Lie group G.
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