Integrable and chaotic motions of four vortices. I. The case of identical vortices

1982 ◽  
Vol 380 (1779) ◽  
pp. 359-387 ◽  
Keyword(s):  
Degrees Of Freedom ◽  
Heteroclinic Orbit ◽  
Elliptic Functions ◽  
Explicit Construction ◽  
Resonant Interaction ◽  
Two Dimensions ◽  
Degree Of Freedom ◽  
Chaotic Motions ◽  
Interaction Terms ◽  
Coupled Nonlinear Oscillators

It is shown that the three-vortex problem in two-dimensional hydrodynamics is integrable, whereas the motion of four identical vortices is not. A sequence of canonical transformations is obtained that reduces the N -degree-of-freedom Hamiltonian, which describes the interaction of N identical vortices, to one with N — 2 degrees of freedom. For N = 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions. For N = 4 the reduction procedure leads to an effective Hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators. Resonant interaction terms in this Hamiltonian suggest non-integrable behaviour and this is verified by numerical experiments. Explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible. The relevance of the results obtained to fundamental problems in hydrodynamics, such as the question of integrability of Euler’s equation in two dimensions, is discussed. The paper also contains a general exposition of the Hamiltonian and Poisson-bracket formalism for point vortices.

Nonintegrability of an Infinite-Degree-of-Freedom Model for Unforced and Undamped, Straight Beams

2003 ◽  
Vol 70 (5) ◽  
pp. 732-738
Author(s):  
K. Yagasaki
Keyword(s):  
Differential Equation ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Degree Of Freedom ◽  
Differential Galois Theory ◽  
Finite Degree ◽  
Chaotic Motions ◽  
Infinite Degree ◽  
Simulation Results

We study a mathematical model for unforced and undamped, initially straight beams. This system is governed by an integro-partial differential equation, and its energy is conserved: It is an infinite-degree-of-freedom Hamiltonian system. We can derive “exact” finite-degree-of-freedom mode truncations for it. Using the differential Galois theory for Hamiltonian systems, we prove that any two or more modal truncations for the model are nonintegrable in the following sense: The Hamiltonian systems do not have the same number of “meromorphic” first complex integrals which are independent and in involution, as the number of their degrees of freedom, when they are regarded as Hamiltonian systems with complex time and coordinates. This also means the nonintegrability of the infinite-degree-of-freedom model for the beams. We present numerical simulation results and observe that chaotic motions occur as in typical nonintegrable Hamiltonian systems.

scholarly journals Design and Calibration of a Hall Effect System for Measurement of Six-Degree-of-Freedom Motion within a Stacked Column

Sensors ◽  
2021 ◽  
Vol 21 (11) ◽  
pp. 3740
Author(s):  
Olafur Oddbjornsson ◽  
Panos Kloukinas ◽  
Tansu Gokce ◽  
Kate Bourne ◽  
Tony Horseman ◽  
...  
Keyword(s):  
Hall Effect ◽  
Degrees Of Freedom ◽  
Reactor Core ◽  
Degree Of Freedom ◽  
Life Extension ◽  
Scale Model ◽  
Design Development ◽  
Seismic Safety ◽  
Six Degrees Of Freedom ◽  
Six Degree Of Freedom

This paper presents the design, development and evaluation of a unique non-contact instrumentation system that can accurately measure the interface displacement between two rigid components in six degrees of freedom. The system was developed to allow measurement of the relative displacements between interfaces within a stacked column of brick-like components, with an accuracy of 0.05 mm and 0.1 degrees. The columns comprised up to 14 components, with each component being a scale model of a graphite brick within an Advanced Gas-cooled Reactor core. A set of 585 of these columns makes up the Multi Layer Array, which was designed to investigate the response of the reactor core to seismic inputs, with excitation levels up to 1 g from 0 to 100 Hz. The nature of the application required a compact and robust design capable of accurately recording fully coupled motion in all six degrees of freedom during dynamic testing. The novel design implemented 12 Hall effect sensors with a calibration procedure based on system identification techniques. The measurement uncertainty was ±0.050 mm for displacement and ±0.052 degrees for rotation, and the system can tolerate loss of data from two sensors with the uncertainly increasing to only 0.061 mm in translation and 0.088 degrees in rotation. The system has been deployed in a research programme that has enabled EDF to present seismic safety cases to the Office for Nuclear Regulation, resulting in life extension approvals for several reactors. The measurement system developed could be readily applied to other situations where the imposed level of stress at the interface causes negligible material strain, and accurate non-contact six-degree-of-freedom interface measurement is required.

scholarly journals Fine structures of valley-polarized excitonic states in monolayer transitional metal dichalcogenides

Nanophotonics ◽  
2020 ◽  
Vol 9 (7) ◽  
pp. 1811-1829 ◽  
Author(s):  
Zhipeng Li ◽  
Tianmeng Wang ◽  
Shengnan Miao ◽  
Zhen Lian ◽  
Su-Fei Shi
Keyword(s):  
Optical Excitation ◽  
Coherence Time ◽  
Two Dimensions ◽  
Degree Of Freedom ◽  
Research Progress ◽  
Many Body ◽  
Transitional Metal ◽  
Metal Dichalcogenides ◽  
Fine Structures ◽  
Excitonic States

AbstractMonolayer transitional metal dichalcogenides (TMDCs), a new class of atomically thin semiconductor, respond to optical excitation strongly with robust excitons, which stem from the reduced screening in two dimensions. These excitons also possess a new quantum degree of freedom known as valley spin, which has inspired the field of valleytronics. The strongly enhanced Coulomb interaction allows the exciton to bind with other particles to form new excitonic states. However, despite the discovery of trions, most of the excitonic states in monolayer TMDCs remain elusive until recently, when new light was shed into the fascinating excitonic fine structures with drastically improved sample quality through boron nitride encapsulation. Here, we review the latest research progress on fine structures of excitonic states in monolayer TMDCs, with a focus on tungsten-based TMDCs and related alloy. Many of the new excitonic complexes inherit the valley degree of freedom, and the valley-polarized dark excitonic states are of particular interest because of their long lifetime and possible long valley coherence time. The capability of resolving the excitonic fine structures also enables the investigation of exciton–phonon interactions. The knowledge of the interlayer between excitons and other particles not only advances our understanding of many-body effects in the monolayer TMDCs but also provides guidance on future applications based on TMDCs.

scholarly journals Standard model from a supergravity model with a naturally small cosmological constant

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Shing Yan Li ◽  
Yu-Cheng Qiu ◽  
S.-H. Henry Tye
Keyword(s):  
String Theory ◽  
Standard Model ◽  
Cosmological Constant ◽  
Supersymmetry Breaking ◽  
Supersymmetric Standard Model ◽  
Degrees Of Freedom ◽  
Interaction Terms ◽  
Supersymmetric Version ◽  
The Standard Model ◽  
Linear Version

Abstract Guided by the naturalness criterion for an exponentially small cosmological constant, we present a string theory motivated 4-dimensional $$ \mathcal{N} $$ N = 1 non-linear supergravity model (or its linear version with a nilpotent superfield) with spontaneous supersymmetry breaking. The model encompasses the minimal supersymmetric standard model, the racetrack Kähler uplift, and the KKLT anti-D3-branes, and use the nilpotent superfield to project out the undesirable interaction terms as well as the unwanted degrees of freedom to end up with the standard model (not the supersymmetric version) of strong and electroweak interactions.

RELATIVISTIC NUCLEAR ENERGY DENSITY FUNCTIONALS

2010 ◽  
Vol 19 (04) ◽  
pp. 548-557 ◽  
Author(s):  
D. VRETENAR ◽  
T. NIKŠIĆ ◽  
P. RING
Keyword(s):  
Energy Density ◽  
Degrees Of Freedom ◽  
Nuclear Energy ◽  
Effective Interaction ◽  
Binding Energies ◽  
Large Set ◽  
Density Functionals ◽  
Heavy Nuclei ◽  
Interaction Terms ◽  
Range Dynamics

A class of relativistic nuclear energy density functionals is explored, in which only nucleon degrees of freedom are explicitly used in the construction of effective interaction terms. Short-distance correlations, as well as intermediate and long-range dynamics, are encoded in the nucleon-density dependence of the strength functionals of an effective interaction Lagrangian. The resulting phenomenological effective interaction, adjusted to experimental binding energies of a large set of axially deformed nuclei, together with a new separable pairing interaction adjusted to reproduce the pairing gap in nuclear matter calculated with the Gogny force, is applied in triaxial relativistic Hartree-Bogoliubov calculations of sequences of heavy nuclei: Th , U , Pu , Cm , Cf , Fm , and No .

The thermodynamics of intracrystalline sorption I. Models of the sorbed state

1956 ◽  
Vol 234 (1196) ◽  
pp. 24-34 ◽  
Keyword(s):  
Mobile Phase ◽  
Degrees Of Freedom ◽  
Degree Of Freedom ◽  
Translational Degree ◽  
Sorbate Concentration ◽  
The Ideal

Three basic models of the intracrystalline sorbed state are discussed: a localized phase, a mobile phase possessing two translational degrees of freedom, and a mobile phase with one translational degree of freedom. The isotherm and entropy of each of these models have been investigated for the ideal phase, and where possible the influence of sorbate-sorbate interactions has been considered. Expressions for the molal and differential entropies of each model are given as a function of sorbate concentration. The method of comparing theoretical isotherms and entropies with experimental observations is outlined.

Research on Simulation Customers Participate Type in Open Innovation

2014 ◽  
Vol 926-930 ◽  
pp. 2054-2057
Author(s):  
Jun Hui He
Keyword(s):  
Open Innovation ◽  
Degrees Of Freedom ◽  
New Products ◽  
Innovation Process ◽  
Degree Of Freedom ◽  
Three Dimensions ◽  
Dynamic Evolution ◽  
Evolution Simulation ◽  
High Degree

This paper proposed customers to participate typology based on three dimensions, which are the customers’ autonomy in the process, the nature of the firm‐customer collaboration, and the stage of the innovation process. Then proposed customers to participate in the type of open innovation framework. Through the static comparative and dynamic evolution simulation found: customers tend to be open to participate in the development of new products pre innovation, the tendency to begin to choose the low participation of degrees of freedom, and ultimately tend to opt for a high degree of freedom to participate.

The Dynamics of a Nonharmonically Excited System Having Rigid Amplitude Constraints

1992 ◽  
Vol 59 (3) ◽  
pp. 693-695 ◽  
Author(s):  
Pi-Cheng Tung
Keyword(s):  
Dynamic Response ◽  
Bifurcation Analysis ◽  
Piecewise Linear ◽  
Coefficient Of Restitution ◽  
Degree Of Freedom ◽  
Linear Oscillator ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Excited System

We consider the dynamic response of a single-degree-of-freedom system having two-sided amplitude constraints. The model consists of a piecewise-linear oscillator subjected to nonharmonic excitation. A simple impact rule employing a coefficient of restitution is used to characterize the almost instantaneous behavior of impact at the constraints. In this paper periodic and chaotic motions are found. The amplitude and stability of the periodic responses are determined and bifurcation analysis for these motions is carried out. Chaotic motions are found to exist over ranges of forcing periods.

Analysis of Five-Degree-of-Freedom Robot Arms

1983 ◽  
Vol 105 (1) ◽  
pp. 23-27 ◽  
Author(s):  
K. Sugimoto ◽  
J. Duffy
Keyword(s):  
Arc Welding ◽  
Degrees Of Freedom ◽  
Screw Theory ◽  
Industrial Robot ◽  
Degree Of Freedom ◽  
Numerical Example ◽  
Robot Arms ◽  
Special Cases ◽  
Computer Controlled ◽  
Five Axes

Many kinds of robot arms with five degrees of freedom are widely used in industry for arc welding, spray painting, assembling etc. It is necessary to be able to compute joint displacements when such devices are computer controlled. A solution to this problem is presented and the analysis is illustrated by a numerical example using the most common industrial robot with five axes. Further, special cases are discussed using screw theory.

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