scholarly journals The first integrals and their Lie algebra of the most general autonomous Hamiltonian of the form H = T + V possessing a Laplace-Runge-Lenz vector

1993 ◽  
Vol 34 (4) ◽  
pp. 511-522 ◽  
Author(s):  
V. M. Gorringe ◽  
P. G. L. Leach
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Kepler Problem ◽  
Three Dimensional ◽  
First Integral ◽  
Negative Energy ◽  
First Integrals ◽  
Two Dimensions ◽  
Positive Energy ◽  
Zero Energy

AbstractIn two dimensions it is found that the most general autonomous Hamiltonian possessing a Laplace-Runge-Lenz vector is The Poisson bracket of the two components of this vector leads to a third first-integral, cubic in the momenta. The Lie algebra of the three integrals under the operation of the Poisson bracket closes, and is shown to be so(3) for negative energy and so(2, 1) for positive energy. In the case of zero energy, the algebra is W(3, 1). The result does not have a three-dimensional analogue, apart from the usual Kepler problem.

scholarly journals Hamiltonian Formulation of Two-Dimensional Gyroviscous MHD

1984 ◽  
Vol 39 (11) ◽  
pp. 1023-1027 ◽  
Author(s):  
Philip J. Morrison ◽  
I. L. Caldas ◽  
H. Tasso
Keyword(s):  
Field Theory ◽  
Poisson Bracket ◽  
Lie Algebra ◽  
Hamiltonian Formulation ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Poisson Type

Gyroviscous MHD in two dimensions is shown to be a Hamiltonian field theory in terms of a non-canonical Poisson bracket. This bracket is of the Lie-Poisson type, but possesses an unfamiliar inner Lie algebra. Analysis of this algebra motivates a transformation that enables a Clebsch-like potential decomposition that makes Lagrangian and canonical Hamiltonian formulations possible.

A model of matter, mind, and consciousness

2020 ◽  
Vol 33 (4) ◽  
pp. 453-459
Author(s):  
Olav Drageset
Keyword(s):  
Dark Matter ◽  
Dark Energy ◽  
String Theory ◽  
Energy Levels ◽  
Three Dimensional ◽  
Negative Energy ◽  
Positive Energy ◽  
Physical Universe ◽  
Parallel Universes ◽  
The Mind

This article shows how string theory is able to model nonphysical particles and how three-dimensional string theory “branes” (parallel universes) could hold dark matter and dark energy. Introspective experience from scientifically oriented groups gives us some clues of how the mind and consciousness could be described. The resulting synthesis from science and direct introspection, for understanding mind and consciousness, are presented. It shows a cosmos with: (1) A parallel nonphysical universe containing dreams, thoughts, emotions, and memories. This universe, called the psychological universe, is probably based on dark matter; (2) A parallel nonphysical universe where intuitive nonphenomenal thinking takes place and where personality and worldview are stored. This universe is called the intuitive universe and is probably based on dark energy and seems to have quantum mechanical qualities. These two universes together make up the mind such as it is defined in this article; and (3) A third nonphysical universe filled with negative energy could make up consciousness. All four universes (including the physical universe) have different vacua, dimensions, and energy levels, so they are all around us but separated. I propose that biological beings consist of a physical body in the physical universe plus entangled bodies in the three nonphysical universes. Entanglement is established by signals going both ways between the different bodies. String theory shows how the interaction between branes/universes can take place. Such a worldview seems to match the requirements from string theory so that it becomes a theory that includes the physical universe and the mind (all kinds of positive energy), and the connection to consciousness. Consciousness itself is based on negative energy, according to mathematician Luigi Fantappiè. The physical base for negative energy is still an open question.

scholarly journals Freely falling observer and black hole radiation

2014 ◽  
Vol 29 (11) ◽  
pp. 1450052 ◽  
Author(s):  
Wontae Kim ◽  
Edwin J. Son
Keyword(s):  
Black Hole ◽  
Energy Density ◽  
Event Horizon ◽  
Negative Energy ◽  
Positive Energy ◽  
Energy Curve ◽  
Zero Energy ◽  
Negative Energy Density ◽  
Black Hole Radiation

We find radiation in an infalling frame and present an explicit analytic evidence of the failure of no drama condition by showing that an infalling observer finds an infinite negative energy density at the event horizon. The negative and positive energy density regions are divided by the newly defined zero-energy curve (ZEC). The evaporating black hole is surrounded by the negative energy which can also be observed in the infalling frame.

scholarly journals Three dimensional similarity solutions of the nonlinear diffusion equation from optimization and first integrals

1982 ◽  
Vol 23 (3) ◽  
pp. 297-309 ◽  
Author(s):  
J.-Y. Parlange ◽  
R. D. Braddock ◽  
G. Sander ◽  
F. Stagnitti
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Spherical Geometry ◽  
First Integral ◽  
Similarity Solutions ◽  
First Integrals ◽  
Nonlinear Diffusion Equation ◽  
Exponential Law ◽  
Optimization Principle ◽  
Diffusion Problems

AbstractFor diffusion problems, the boundary conditions are specified at two distinct points, yielding a two end-point boundary value problem which normally requires iterative techniques. For spherical geometry, it is possible to specify the boundary conditions at the same points, approximately, by using an optimization principle for arbitrary diffusivity. When the diffusivity obeys a power or an exponential law, a first integral exists and iteration can be avoided. For those two exact cases, it is shown that the general optimization result is extremely accurate when diffusivity increases rapidly with concentration.

scholarly journals Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
K. S. Mahomed ◽  
E. Momoniat
Keyword(s):  
Lie Algebra ◽  
First Integral ◽  
First Integrals ◽  
Order Equation ◽  
Third Order ◽  
Maximal Symmetry ◽  
Symmetry Properties ◽  
Group Methods ◽  
Algebraic Properties ◽  
The Relationship

By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs) and their point symmetries. It is well known that there are three classes of linear third-order ODEs for maximal cases of point symmetries which are 4, 5, and 7. The simplest scalar linear third-order equation has seven-point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equationy′′′=0is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals.

MOMENTUM MAPS, INDEPENDENT FIRST INTEGRALS AND INTEGRABILITY FOR CENTRAL POTENTIALS

2009 ◽  
Vol 06 (08) ◽  
pp. 1323-1341 ◽  
Author(s):  
EMANUELE FIORANI
Keyword(s):  
Angular Momentum ◽  
First Integral ◽  
Functional Independence ◽  
Negative Energy ◽  
Momentum Tensor ◽  
First Integrals ◽  
Central Potential ◽  
Momentum Maps ◽  
Kepler System ◽  
Natural Symmetry

It is shown that all the motions of a natural Hamiltonian H(q, p) = ½‖p‖2+V(q) lie on planes through 0 ∈ Rn if and only if V is a central potential, i.e. H admits SO(n) symmetry. Then, using the momentum maps associated to their natural symmetry groups, we study in detail the functional independence of first integrals of a general central potential, of the isotropic harmonic potential and of the Kepler potential. We show that all the smooth first integral of the isotropic harmonic oscillator are dependent of the angular momentum tensor L and of the Fradkin tensor H, and that all the smooth first integrals of the Kepler system on the region of negative energy are dependent of the angular momentum tensor L and of the Laplace–Runge–Lenz vector B.

Laser Scanning Confocal Microscopy and Three-Dimesnsional Reconstruction of the Mitotic Spindles of Unequally Dividing Embryonic Cells

1990 ◽  
Vol 48 (3) ◽  
pp. 166-167
Author(s):  
J. Holy ◽  
G. Schatten
Keyword(s):  
Confocal Microscopy ◽  
Mitotic Spindle ◽  
Laser Scanning ◽  
Three Dimensional ◽  
Laser Scanning Confocal Microscopy ◽  
Two Dimensions ◽  
Embryonic Cells ◽  
Mitotic Spindles ◽  
Cleavage Division ◽  
Scanning Confocal Microscopy

One of the classic limitations of light microscopy has been the fact that three dimensional biological events could only be visualized in two dimensions. Recently, this shortcoming has been overcome by combining the technologies of laser scanning confocal microscopy (LSCM) and computer processing of microscopical data by volume rendering methods. We have employed these techniques to examine morphogenetic events characterizing early development of sea urchin embryos. Specifically, the fourth cleavage division was examined because it is at this point that the first morphological signs of cell differentiation appear, manifested in the production of macromeres and micromeres by unequally dividing vegetal blastomeres.The mitotic spindle within vegetal blastomeres undergoing unequal cleavage are highly polarized and develop specialized, flattened asters toward the micromere pole. In order to reconstruct the three-dimensional features of these spindles, both isolated spindles and intact, extracted embryos were fluorescently labeled with antibodies directed against either centrosomes or tubulin.

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

Turbulent three-dimensional dielectric electrohydrodynamic convection between two plates

2012 ◽  
Vol 696 ◽  
pp. 228-262 ◽  
Author(s):  
A. Kourmatzis ◽  
J. S. Shrimpton
Keyword(s):  
Spatial Data ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Energy Budgets ◽  
Turbulent Regime ◽  
Scalar Flux ◽  
Wide Range ◽  
Scalar Variance

AbstractThe fundamental mechanisms responsible for the creation of electrohydrodynamically driven roll structures in free electroconvection between two plates are analysed with reference to traditional Rayleigh–Bénard convection (RBC). Previously available knowledge limited to two dimensions is extended to three-dimensions, and a wide range of electric Reynolds numbers is analysed, extending into a fully inherently three-dimensional turbulent regime. Results reveal that structures appearing in three-dimensional electrohydrodynamics (EHD) are similar to those observed for RBC, and while two-dimensional EHD results bear some similarities with the three-dimensional results there are distinct differences. Analysis of two-point correlations and integral length scales show that full three-dimensional electroconvection is more chaotic than in two dimensions and this is also noted by qualitatively observing the roll structures that arise for both low (${\mathit{Re}}_{E} = 1$) and high electric Reynolds numbers (up to ${\mathit{Re}}_{E} = 120$). Furthermore, calculations of mean profiles and second-order moments along with energy budgets and spectra have examined the validity of neglecting the fluctuating electric field ${ E}_{i}^{\ensuremath{\prime} } $ in the Reynolds-averaged EHD equations and provide insight into the generation and transport mechanisms of turbulent EHD. Spectral and spatial data clearly indicate how fluctuating energy is transferred from electrical to hydrodynamic forms, on moving through the domain away from the charging electrode. It is shown that ${ E}_{i}^{\ensuremath{\prime} } $ is not negligible close to the walls and terms acting as sources and sinks in the turbulent kinetic energy, turbulent scalar flux and turbulent scalar variance equations are examined. Profiles of hydrodynamic terms in the budgets resemble those in the literature for RBC; however there are terms specific to EHD that are significant, indicating that the transfer of energy in EHD is also attributed to further electrodynamic terms and a strong coupling exists between the charge flux and variance, due to the ionic drift term.

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