Geometry of Hamiltonian Chaos

Horwitz, L. P.; Zion, Yossi Ben; Lewkowicz, M.; Schiffer, Marcelo; Levitan, Jacob

doi:10.1103/physrevlett.98.234301

Cited by 46 publications

(97 citation statements)

References 11 publications

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“…In the author's work torsion geometry appears as the natural geometry for unification of spacetime structures, Brownian motion, quantum mechanics and fluid-dynamics[77][78][79][80]. This geometry has appeared recently in the work of Horwitz et al[41] on the geometrical structure of Hamiltonian chaos.…”

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confidence: 99%

Surmounting the Cartesian Cut Through Philosophy, Physics, Logic, Cybernetics, and Geometry: Self-reference, Torsion, the Klein Bottle, the Time Operator, Multivalued Logics and Quantum Mechanics

Rapoport

2009

Found Phys

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In this transdisciplinary article which stems from philosophical considerations (that depart from phenomenology-after Merleau-Ponty, Heidegger and Rosen-and Hegelian dialectics), we develop a conception based on topological (the Moebius surface and the Klein bottle) and geometrical considerations (based on torsion and non-orientability of manifolds), and multivalued logics which we develop into a unified world conception that surmounts the Cartesian cut and Aristotelian logic. The role of torsion appears in a self-referential construction of space and time, which will be further related to the commutator of the True and False operators of matrix logic, still with a quantum superposed state related to a Moebius surface, and as the physical field at the basis of Spencer-Brown's primitive distinction in the protologic of the calculus of distinction. In this setting, paradox, self-reference, depth, time and space, higher-order non-dual logic, perception, spin and a time operator, the Klein bottle, hypernumbers due to Musès which include non-trivial square roots of ±1 and in particular non-trivial nilpotents, quantum field operators, the transformation of cognition to spin for two-state quantum systems, are found to be keenly interwoven in a world conception compatible with the philosophical approach taken for basis of this article. The Klein bottle is found not only to be the topological information for self-reference and paradox whose logical counterpart in the calculus of indications are the paradoxical imaginary time waves, but also a classical-quantum transformer (Hadamard's gate in quantum computation) which is indispensable to be able to obtain a complete multivalued logical system, and still to generate the matrix extension of classical connective Boolean logic. We further find that the multivalued logic that stems from considering the paradoxical equation in the calculus of distinctions, and in particular, the imaginary solutions to this equation, generates the matrix logic which supersedes the classical logic of connectives and which has for D.L. Rapoport ( ) 34 Found Phys (2011) 41: 33-76 particular subtheories fuzzy and quantum logics. Thus, from a primitive distinction in the vacuum plane and the axioms of the calculus of distinction, we can derive by incorporating paradox, the world conception succinctly described above.

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confidence: 99%

Surmounting the Cartesian Cut Through Philosophy, Physics, Logic, Cybernetics, and Geometry: Self-reference, Torsion, the Klein Bottle, the Time Operator, Multivalued Logics and Quantum Mechanics

Rapoport

2009

Found Phys

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“…The approach provides an explanation of the origin of Hamiltonian chaos, and has been successfully applied to the study for Hamiltonian systems. In addition, a new Riemannian geometric criterion [8] for chaotic motion in Hamiltonian systems has recently been developed by Horwitz, Ben Zion, Lewkowicz, Schiffer and Levitan (HBLSL), based on the idea that the orbits are determined as geodesics on a dynamically induced surface. The new geometrical criterion is directly effective for Hamiltonians which are dominated by oscillator-like potentials at small distances [9].…”

Section: Introductionmentioning

confidence: 99%

The potential energy surface and chaos in 2D Hamiltonian systems

Zhang

2011

Physics Letters A

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“…We then follow the method of Ref. [10] to show that there is a corresponding Hamiltonian K with a conformally modified metric, and no explicit additive scalar field, which has the form of the construction of Bekenstein and Milgrom [11,12] for the realization of Milgrom's MOND program (modified Newtonian dynamics) [13][14][15][16] for achieving the observed galactic rotation curves. This simple form of Bekenstein's theory (called RAQUAL), which we discuss in some detail below for the sake of simplicity and clarity in the development of the mathematical method, does not properly account for causality and gravitational lensing; the theory has been further developed to include vector fields (which we shall call Bekenstein-Sanders fields) as well (TeVeS) [17][18][19], which has been relatively successful in accounting for these problems.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian Map to Conformal Modification of Spacetime Metric: Kaluza-Klein and TeVeS

2010

Self Cite

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It has been shown that the orbits of motion for a wide class of nonrelativistic Hamiltonian systems can be described as geodesic flows on a manifold and an associated dual by means of a conformal map. This method can be applied to a four dimensional manifold of orbits in spacetime associated with a relativistic system. We show that a relativistic Hamiltonian which generates Einstein geodesics, with the addition of a world scalar field, can be put into correspondence in this way with another Hamiltonian with conformally modified metric. Such a construction could account for part of the requirements of Bekenstein for achieving the MOND theory of Milgrom in the post-Newtonian limit. The constraints on the MOND theory imposed by the galactic rotation curves, through this correspondence, would then imply constraints on the structure of the world scalar field. We then use the fact that a Hamiltonian with vector gauge fields results, through such a conformal map, in a KaluzaKlein type theory, and indicate how the TeVeS structure of Bekenstein and Saunders can be put into this framework. We exhibit a class of infinitesimal gauge transformations on the gauge fields U μ (x) which preserve the Bekenstein-Sanders condition U μ U μ = −1. The underlying quantum structure giving rise to these gauge fields is a Hilbert bundle, and the gauge transformations induce a non-commutative behavior to the fields, i.e. they become of Yang-Mills type. Working in the infinitesimal gauge neighborhood of the initial Abelian theory we show that in the Abelian limit the Yang-Mills field equations provide residual nonlinear terms which may avoid the caustic singularity found by Contaldi et al.

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Geometry of Hamiltonian Chaos

Cited by 46 publications

References 11 publications

Surmounting the Cartesian Cut Through Philosophy, Physics, Logic, Cybernetics, and Geometry: Self-reference, Torsion, the Klein Bottle, the Time Operator, Multivalued Logics and Quantum Mechanics

Surmounting the Cartesian Cut Through Philosophy, Physics, Logic, Cybernetics, and Geometry: Self-reference, Torsion, the Klein Bottle, the Time Operator, Multivalued Logics and Quantum Mechanics

The potential energy surface and chaos in 2D Hamiltonian systems

Hamiltonian Map to Conformal Modification of Spacetime Metric: Kaluza-Klein and TeVeS

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