THE THEORY OF DIRAC; USE OF POISSON BRACKETS; THE ENERGY LAW AND BOHR'S FREQUENCY CONDITION

2011 ◽  
pp. 69-77
Author(s):  
George Birtwistle
Keyword(s):  
Poisson Brackets ◽  
Frequency Condition ◽  
Energy Law

Energy Law Principles and COVID-19

2020 ◽  
Vol 1 (2) ◽  
pp. 245-247
Author(s):  
Vicente Lopez-Ibor Mayor ◽  
Raphael J. Heffron
Keyword(s):  
Energy Sector ◽  
Economic Recovery ◽  
Low Carbon ◽  
Left Behind ◽  
Core Energy ◽  
Low Carbon Energy ◽  
Energy Law

It is advanced here that a principle-based approach is needed to develop the energy sector during and after COVID-19. The economic recovery that is needed needs to revolve around ensuring that no one is left behind, and it should be an inclusive transition to a secure and stable low-carbon energy future. There are seven core energy law principles that if applied to the energy sector could enable this to be achieved.

Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator

2014 ◽  
Vol 4 (1) ◽  
pp. 404-426
Author(s):  
Vincze Gy. Szasz A.
Keyword(s):  
Harmonic Oscillator ◽  
Classical Mechanics ◽  
Universal Time ◽  
Variation Theory ◽  
Quantum Oscillator ◽  
Variation Principle ◽  
Poisson Brackets ◽  
Mathematical Meaning ◽  
Damped Harmonic Oscillator ◽  
Damped Oscillator

Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

Innovation in Energy Law and Technology

2018 ◽  
Keyword(s):  
Technological Innovation ◽  
Technology Development ◽  
Energy Sources ◽  
Market Demand ◽  
Energy Technologies ◽  
Law And Technology ◽  
Low Emission ◽  
Consumer Access ◽  
Legal Innovation ◽  
Energy Law

Technological and legal innovation have been central to energy development for centuries. Today’s era of accelerating change is transforming energy law. Disruption and change to established energy sources, supply, distribution, and energy consumer access is driven by legal innovations that, in turn, prompt or respond to technology. Interaction between legal and technological innovation is advancing the growing global effort to transition from high-carbon energy to low-energy or no-carbon energy—evidenced by the 2015 Paris Agreement on climate change and the growing market demand for carbon-free electricity. This global transition to low-emission energy sources allows nations to take advantage of emerging economic opportunities and facilitates new forms of energy technology development, energy distribution, and governance. But progress is uneven and concerns such as energy security are initiating technological innovation in many existing energy technologies. These authors from twenty-one nations examine relevant developments in global energy law triggered by these innovations.

Reduced forms in the nominal morphology of the Lindisfarne Gospel Gloss. A case of accusative/dative syncretism?

2020 ◽  
Vol 54 (s41) ◽  
pp. 37-65
Author(s):  
Julia Fernández-Cuesta ◽  
Nieves Rodríguez-Ledesma
Keyword(s):  
Statistical Analysis ◽  
Old English ◽  
Frequency Condition ◽  
Facsimile Edition ◽  
Noun Class ◽  
Characteristic Features ◽  
Nominal Morphology ◽  
Reduced Forms ◽  
Syntactic Context

Abstract One of the most characteristic features of the grammar of the Lindisfarne Gospel gloss is the absence of the etymological -e inflection in the dative singular in the paradigm of the strong masculine and neuter declension (a-stems). Ross (1960: 38) already noted that endingless forms of the nominative/accusative cases were quite frequent in contexts where a dative singular in -e would be expected, to the extent that he labeled the forms in -e ‘rudimentary dative.’ The aim of this article is to assess to what extent the dative singular is still found as a separate case in the paradigms of the masculine and neuter a-stems and root nouns. To this end a quantitative/statistical analysis of nouns belonging to these classes has been carried out in contexts where the Latin lemma is either accusative or dative. We have tried to determine whether variables such as syntactic context, noun class, and frequency condition the presence or absence of the -e inflection, and whether the distribution of the inflected and uninflected forms is different in the various demarcations that have been identified in the gloss. The data have been retrieved using the Dictionary of Old English Corpus. All tokens have been checked against the facsimile edition and the digitised manuscript in order to detect possible errors.

scholarly journals PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

2021 ◽  
Author(s):  
DMITRI I. PANYUSHEV ◽  
OKSANA S. YAKIMOVA
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Invariant Theory ◽  
Cyclic Permutation ◽  
Poisson Brackets ◽  
Enveloping Algebra ◽  
Order Automorphism ◽  
Finite Dimensional ◽  
Compatible Poisson Brackets ◽  
Commutative Subalgebras

AbstractLet 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).

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