Poisson brackets for densities of functionals

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.

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Kinetic theory

10.1093/oso/9780198786399.003.0010 ◽

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.

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Poisson Brackets & Angular Momentum

10.1093/oso/9780198822370.003.0017 ◽

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.

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PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Transformation Groups ◽

10.1007/s00031-021-09650-3 ◽

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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GAUGE INVARIANCE AND CONSTRAINTS: OPEN GAUGE ALGEBRAS

International Journal of Modern Physics A ◽

10.1142/s0217751x92002519 ◽

1992 ◽

Vol 07 (22) ◽

pp. 5549-5561 ◽

Cited By ~ 1

Author(s):

KH. S. NIROV ◽

P.N. PYATOV ◽

A.V. RAZUMOV

Keyword(s):

Wide Class ◽

Gauge Invariance ◽

Poisson Brackets ◽

Hamiltonian Description ◽

Gauge Invariant

For a wide class of gauge-invariant systems with open gauge algebras the Hamiltonian description is constructed and the Poisson brackets of the constraints are calculated. It is shown that in the case under consideration there arise only first class constraints.

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On third-order Poisson brackets of differential geometry

Russian Mathematical Surveys ◽

10.1070/rm1997v052n03abeh001817 ◽

1997 ◽

Vol 52 (3) ◽

pp. 617-618 ◽

Cited By ~ 8

Author(s):

G V Potëmin

Keyword(s):

Differential Geometry ◽

Poisson Brackets ◽

Third Order

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On the Dirac eigenvalues as observables of the on-shell N = 2D = 4 Euclidean supergravity

Open Physics ◽

10.2478/s11534-008-0116-z ◽

2008 ◽

Vol 6 (4) ◽

Author(s):

Ion Vancea

Keyword(s):

Phase Space ◽

Equations Of Motion ◽

Poisson Brackets ◽

Gauge Transformations

AbstractWe generalize previous works on the Dirac eigenvalues as dynamical variables of Euclidean gravity and N =1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their value for the Dirac eigenvalues.

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