scholarly journals Theory of Generalized Canonical Transformations for Birkhoff Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yi Zhang
Keyword(s):  
Canonical Transformation ◽  
Natural Generalization ◽  
Analytical Mechanics ◽  
Hamiltonian Mechanics ◽  
Canonical Transformations ◽  
Transformation Theory ◽  
Special Cases ◽  
Criterion Equation ◽  
Important Means ◽  
Transformation Formulas

Transformation is an important means to study problems in analytical mechanics. It is often difficult to solve dynamic equations, and the use of variable transformation can make the equations easier to solve. The theory of canonical transformations plays an important role in solving Hamilton’s canonical equations. Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. This paper attempts to extend the canonical transformation theory of Hamilton systems to Birkhoff systems and establish the generalized canonical transformation of Birkhoff systems. First, the definition and criterion of the generalized canonical transformation for the Birkhoff system are established. Secondly, based on the criterion equation and considering the generating functions of different forms, six generalized canonical transformation formulas are derived. As special cases, the canonical transformation formulas of classical Hamilton’s equations are given. At the end of the paper, two examples are given to illustrate the application of the results.

scholarly journals COVARIANT HAMILTONIAN FIELD THEORY

2008 ◽  
Vol 17 (03) ◽  
pp. 435-491 ◽  
Author(s):  
JÜRGEN STRUCKMEIER ◽  
ANDREAS REDELBACH
Keyword(s):  
Field Theory ◽  
Canonical Transformation ◽  
Generating Functions ◽  
Lagrangian Description ◽  
Poisson Brackets ◽  
Canonical Transformations ◽  
Transformation Theory ◽  
Field Equations ◽  
Time Step ◽  
Transformation Rules

A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. Whereas the covariant canonical field equations are equivalent to the Euler–Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proven that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinitesimal canonical transformation furnishes the most general form of Noether's theorem. Furthermore, we specify the generating function of an infinitesimal space-time step that conforms to the field equations.

Hamiltonian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Angular Momentum ◽  
Jacobi Equation ◽  
Modern Theory ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Light Waves ◽  
Hamilton Jacobi Equation ◽  
Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

scholarly journals Flexible models for overdispersed and underdispersed count data

2021 ◽  
Author(s):  
Dexter Cahoy ◽  
Elvira Di Nardo ◽  
Federico Polito
Keyword(s):  
Poisson Distribution ◽  
Count Data ◽  
Hypergeometric Functions ◽  
Natural Generalization ◽  
Model Parameters ◽  
Probability Models ◽  
Limiting Behavior ◽  
Poisson Models ◽  
Special Cases ◽  
Flexible Models

AbstractWithin the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD’s) to allow both overdispersion and underdispersion. Similarly to Kemp’s generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD’s related to a generalization of Mittag–Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.

q-Canonical transformations for many q-bosons and new q–k symmetry

1994 ◽  
Vol 72 (7-8) ◽  
pp. 321-325
Author(s):  
Joseph L. Birman
Keyword(s):  
Plane Wave ◽  
Wave Vector ◽  
Canonical Transformation ◽  
Bose System ◽  
Canonical Transformations ◽  
Negative Wave ◽  
The Many ◽  
Wave Modes

A new symmetry is identified for the many particle q-Bose system, which permits carrying out a linear q-canonical transformation, i.e., one which preserves the q-deformed commutation rules. The symmetry is exhibited when the set of q-modes is partitioned into "positive" and "negative," each subset having its own type of deformation. For plane-wave modes, labelled by wave vector k, this means separate positive and negative wave vectors. Then, identifying the new symmetry: (q → q−1, k → −k) the transformation can be implemented. The new aspect is the simultaneous transformation of wave vector k and parameter q.

Anatomy of the canonical transformation

1993 ◽  
Vol 345 (1677) ◽  
pp. 577-598 ◽  
Keyword(s):  
Fluid Dynamics ◽  
Canonical Transformation ◽  
Generating Functions ◽  
Hamiltonian Structure ◽  
Rigid Bodies ◽  
Symplectic Integrators ◽  
Classical Dynamics ◽  
Canonical Transformations ◽  
Geophysical Fluid Dynamics ◽  
Dynamics Of Particles

A concise account of the structure of the canonical transformation is given, in the lowest dimensional case. This case is chosen because it offers a special clarity in several respects. In particular, the diversity of possible generating functions is illustrated by m any examples which are not available elsewhere. Many of these are of physical interest, and some of them are multivalued. These examples are used to inform a comparative study of the several different definitions of a canonical transformation to be found in the literature. The paper is pertinent to all those branches of mechanics which can be given a hamiltonian representation. These include not only the classical dynamics of particles and rigid bodies, but also some more recent studies in continuum mechanics, including geophysical fluid dynamics. An area of particular modern interest is that of symplectic integrators. These are numerical integrating algorithms which generate a solution to Hamilton’s equations via a sequence of canonical transformations, which preserve the hamiltonian structure in the numerical solution.

A generalization of the Whitney rank generating function

1993 ◽  
Vol 113 (2) ◽  
pp. 267-280 ◽  
Author(s):  
G. E. Farr
Keyword(s):  
Generating Function ◽  
Linear Code ◽  
Tutte Polynomial ◽  
Natural Generalization ◽  
Chromatic Polynomial ◽  
Weight Enumerator ◽  
Hadamard Transform ◽  
Percolation Probability ◽  
Special Cases

AbstractThe Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.

POTENTIALS OF ARBITRARY FORCES WITH FRACTIONAL DERIVATIVES

2004 ◽  
Vol 19 (17n18) ◽  
pp. 3083-3092 ◽  
Author(s):  
EQAB M. RABEI ◽  
TAREQ S. ALHALHOLY ◽  
AKRAM ROUSAN
Keyword(s):  
Laplace Transform ◽  
General Formula ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Hamiltonian Mechanics ◽  
Dissipative Effects ◽  
Special Cases ◽  
Exact Agreement ◽  
The Laplace Transform ◽  
Lagrangian And Hamiltonian Mechanics

The Laplace transform of fractional integrals and fractional derivatives is used to develop a general formula for determining the potentials of arbitrary forces: conservative and nonconservative in order to introduce dissipative effects (such as friction) into Lagrangian and Hamiltonian mechanics. The results are found to be in exact agreement with Riewe's results of special cases. Illustrative examples are given.

scholarly journals Canonical transformation theory from extended normal ordering

2007 ◽  
Vol 127 (10) ◽  
pp. 104107 ◽  
Author(s):  
Takeshi Yanai ◽  
Garnet Kin-Lic Chan
Keyword(s):  
Canonical Transformation ◽  
Transformation Theory ◽  
Normal Ordering

scholarly journals LOCAL EXPONENTS AND INFINITESIMAL GENERATORS OF CANONICAL TRANSFORMATIONS ON BOSON FOCK SPACES

2004 ◽  
Vol 07 (04) ◽  
pp. 547-571 ◽  
Author(s):  
F. HIROSHIMA ◽  
K. R. ITO
Keyword(s):  
Canonical Transformation ◽  
Symplectic Group ◽  
Unitary Operator ◽  
Fock Space ◽  
Adjoint Operator ◽  
The Self ◽  
Canonical Transformations ◽  
Fock Spaces ◽  
Infinitesimal Generators ◽  
Boson Fock Space

A one-parameter symplectic group {etÂ}t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator Ut implementing such a proper canonical transformation gives a projective unitary representation of {etÂ}t∈ℝ on the Boson–Fock space and that Ut can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent [Formula: see text] with a real-valued function τÂ(·) such that [Formula: see text].

scholarly journals CANONICAL TRANSFORMATIONS IN THREE-DIMENSIONAL PHASE-SPACE

2009 ◽  
Vol 24 (25n26) ◽  
pp. 4769-4788 ◽  
Author(s):  
TEKİN DERELİ ◽  
ADNAN TEĞMEN ◽  
TUĞRUL HAKİOĞLU
Keyword(s):  
Phase Space ◽  
Canonical Transformation ◽  
Generating Functions ◽  
General Framework ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Canonical Transformations ◽  
Dimensional Phase Space ◽  
Nambu Bracket ◽  
Definition Of

Canonical transformation in a three-dimensional phase-space endowed with Nambu bracket is discussed in a general framework. Definition of the canonical transformations is constructed based on canonoid transformations. It is shown that generating functions, transformed Hamilton functions and the transformation itself for given generating functions can be determined by solving Pfaffian differential equations corresponding to that quantities. Types of the generating functions are introduced and all of them are listed. Infinitesimal canonical transformations are also discussed. Finally, we show that the decomposition of canonical transformations is also possible in three-dimensional phase space as in the usual two-dimensional one.

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