AV-differential geometry: Poisson and Jacobi structures

Grabowska, Katarzyna; Grabowski, Janusz; Urbański, Paweł

doi:10.1016/j.geomphys.2004.04.004

Cited by 39 publications

(127 citation statements)

References 19 publications

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“…On the other hand, it became clear nowadays that an intrinsic, i.e., a frame-independent formulation of the Newtonian dynamics requires affine and not vectorial objects. We refer here to our earlier work [5,6,10,24], to recent papers by Janyška and Modugno [13], and Mangiarotti and Sardanashvily [16].…”

Section: Introductionmentioning

confidence: 99%

The Schrödinger operator as a generalized Laplacian

Grabowska¹,

Grabowski²,

Urbański³

2008

J. Phys. A: Math. Theor.

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The Schrödinger operators on the Newtonian space-time are defined in a way which make them independent on the class of inertial observers. In this picture the Schrödinger operators act not on functions on the space-time but on sections of certain one-dimensional complex vector bundle -the Schrödinger line bundle. This line bundle has trivializations indexed by inertial observers and is associated with an U (1)-principal bundle with an analogous list of trivializations -the Schrödinger principal bundle. If an inertial frame is fixed, the Schrödinger bundle can be identified with the trivial bundle over space-time, but as there is no canonical trivialization (inertial frame), these sections interpreted as 'wave-functions' cannot be viewed as actual functions on the space-time. In this approach the change of an observer results not only in the change of actual coordinates in the space-time but also in a change of the phase of wave functions. For the Schrödinger principal bundle a natural differential calculus for 'wave forms' is developed that leads to a natural generalization of the concept of Laplace-Beltrami operator associated with a pseudoRiemannian metric. The free Schrödinger operator turns out to be the Laplace-Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schrödinger principal bundle. The presented framework does not involve any ad hoc or axiomatically introduced geometrical structures. It is based on the traditional understanding of the Schrödinger operator in a given reference frame -which is supported by producing right physics predictions -and it is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton-Jacobi equations, that makes a bridge between the classical and quantum theory. MSC 2000: 35J10, 70G45.

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Section: Introductionmentioning

confidence: 99%

The Schrödinger operator as a generalized Laplacian

Grabowska¹,

Grabowski²,

Urbański³

2008

J. Phys. A: Math. Theor.

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“…Using (11)(12) we can proceed to construct Jacobi structures using the well-known language of vector calculus in R 3 . Before we do that, it is interesting to interpret geometrically equation (11).…”

Section: Jacobi Structures In Rmentioning

confidence: 99%

“…Before we do that, it is interesting to interpret geometrically equation (11). Consider (M, Λ, E) a given Jacobi structure and let Λ ♯ : T * M → T M be the vector bundle map associated with Λ.…”

Section: Jacobi Structures In Rmentioning

confidence: 99%

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Jacobi structures in R3

Haas¹

2005

Journal of Mathematical Physics

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The most general Jacobi brackets in R 3 are constructed after solving the equations imposed by the Jacobi identity. Two classes of Jacobi brackets were identified, according to the rank of the Jacobi structures. The associated Hamiltonian vector fields are also constructed. I. INTRODUCTIONJacobi brackets have all properties of Poisson brackets, except from the fact that they are not necessarily derivations. A manifold endowed with a Jacobi bracket is called a Jacobi manifold. In this context, Jacobi manifolds are natural generalizations of Poisson, contact and locally conformal symplectic manifolds. Jacobi manifolds were introduced by Lichnerowicz 1 and, in a local Lie algebra setting, by Kirillov 2 The general properties of Jacobi manifolds are discussed, for instance, in references 3 and 4 below. Some recent advances on the study of Jacobi manifolds can be found in references from 5 to 16.The present work is devoted to the explicit construction of Jacobi structures. Although contact and locally conformal symplectic manifolds are general concrete examples of Jacobi manifolds, there is a lack of knowledge of other possible classes of Jacobi brackets even for low dimensional manifolds. An exception in this regard is given by linear Jacobi structures on vector bundles 17 . This is to be compared with the Poisson manifolds case, where, in recent years, there has been much work for the explicit construction of Poisson structures, generalizing the well-known case of Lie-Poisson structures 18 -22 . In particular, there have been * Electronic address: ferhaas@unisinos.br

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“…Lagrangian systems on so-called 'affine' Lie algebroids motivate our interest in such systems, as we will explain in the next section (see also [6,7]). Dynamical systems of the form (1) are called 'pseudo-SODEs', where SODE stands for 'second-order ordinary differential equations'.…”

Section: Introductionmentioning

confidence: 99%