scholarly journals Formulation of singular theories in a partial Hamiltonian formalism using a new bracket and multi-time dynamics

2014 ◽  
Vol 12 (01) ◽  
pp. 1550001 ◽  
Author(s):  
Steven Duplij
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Hessian Matrix ◽  
Hamiltonian Formalism ◽  
Dirac Bracket ◽  
Algebraic Equations ◽  
Time Dynamics ◽  
Linear Algebraic Equations ◽  
Singular Theory

A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller than the number of velocities) is proposed. The equations of motion become first-order differential equations, and they coincide with those of multi-time dynamics, if a certain condition is imposed. In a singular theory, this condition is fulfilled in the case of the coincidence of the number of generalized momenta with the rank of the Hessian matrix. The noncanonical generalized velocities satisfy a system of linear algebraic equations, which allows an appropriate classification of singular theories (gauge and nongauge). A new antisymmetric bracket (similar to the Poisson bracket) is introduced, which describes the time evolution of physical quantities in a singular theory. The origin of constraints is shown to be a consequence of the (unneeded in our formulation) extension of the phase space, when the new bracket transforms into the Dirac bracket. Quantization is briefly discussed.

scholarly journals HARMONIC BALANCE THEORY FOR SCHEME TECHNICAL DESIGN

T-Comm ◽  
2020 ◽  
Vol 14 (11) ◽  
pp. 21-32
Author(s):  
Svetlana F. Gorgadze ◽  
◽  
Anton A. Maximov ◽  
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Projection Methods ◽  
Algebraic Equations ◽  
Electronic Circuits ◽  
Large Signal ◽  
Linear Algebraic Equations ◽  
Methods Of Synthesis ◽  
Signal Mode

The analysis and generalization of the main publications on the methods of synthesis and analysis of non-linear active microwave circuits based on the use of the harmonic balance method are presented. As a result of some classification of mathematical approaches and techniques used in the context of this method, a selection and review of basic algorithms was made, the sequential application of which makes it possible to obtain the final result for a scheme of any complexity. The principles of drawing up the initial system of differential equations for electronic circuits and reducing it to a system of linear algebraic equations are considered. A detailed and, at the same time, simplified interpretation of the approaches involving the use of projection methods and Krylov subspaces is given in order to make them easier to understand. Both the complete and the restart generalized method of minimal residuals are considered, in which the desired solution is obtained in the course of an iterative process, at each stage of which subspaces of lower dimension are constructed. The possibilities of simulators and application packages intended for circuit design of electronic circuits are considered. The problem of matching a power amplifier in large signal mode using the APLAC simulator, which is NI AWR technology for designing high-frequency circuits, is discussed.

scholarly journals FEATURES OF ELECTROMECHANICAL ACOUSTIC ENERGY CONVERSION BY CYLINDRICAL PIEZOCERAMIC TRANSDUCERS WITH INTERNAL SCREENS IN COMPOSITION OF FLAT SYSTEMS

2018 ◽  
Vol 9 (1) ◽  
pp. 85-95 ◽  
Author(s):  
A. G. Leiko ◽  
I. V. Kandrachuk ◽  
A. O. Sviatnenko
Keyword(s):  
Energy Conversion ◽  
Equations Of Motion ◽  
Longitudinal Axis ◽  
Radial Symmetry ◽  
Acoustic Energy ◽  
Acoustic Fields ◽  
Algebraic Equations ◽  
Acoustic Interaction ◽  
Flat Systems ◽  
Linear Algebraic Equations

The problem of sound emission is considered by a system formed from cylindrical piezoceramic radiators with internal acoustically soft screens. Longitudinal axis of emitters lie in one plane. This system is characterized by the interaction of electric, mechanical and acoustic fields in the process of conversion electrical energy to acoustical energy and acoustic fields in the process of forming them in the environments. The purpose of the work is to determine the peculiarities of the electromechanical acoustic transformation of energy by cylindrical piezoceramic radiators with internal screens in the composition of flat systems, taking into account all types of interaction.The research was carried out by the method of bound fields in multiply connected domains with the use of addition theorems for the cylindrical wave functions. The physical fields arising from the emission of sound by such a system are determined by the joint solution of the system of differential equations: the wave equation; equations of motion of thin piezoceramic shells with circular polarization in displacements; the equations of forced electrostatics for piezoceramics at given boundary conditions, the conditions of conjugation of fields at the boundaries of the division of domains and electric conditions.The solution of the problem is reduced to the solution of an infinite system of linear algebraic equations with respect to unknown coefficients of field expansions.An analysis of the results of numerical calculations, performed on the basis of the obtained analytical relations, called to establish a number of features in the electromechanical acoustic transformation of energy by emitters in the composition of flat systems. They include: the role of acoustic interaction in the process of energy conversion; determination of the mechanism of quantitative assessment of the influence of interaction on these processes; the dependence of the degree of violation of the radial symmetry of the acoustic loading of the emitters on the amount of acoustic interaction; the appearance of multimodality of the mechanical field of emitters in the structure of the plane system and the dependence of the redistribution of energy between all modes on the degree of disturbance of the radial symmetry of the acoustic loading of the emitters.

scholarly journals Classification of primary constraints for new general relativity in the premetric approach

2021 ◽  
pp. 2140003
Author(s):  
María-José Guzmán ◽  
Shymaa Khaled Ibraheem
Keyword(s):  
General Relativity ◽  
Hessian Matrix ◽  
Hamiltonian Formalism ◽  
Temporal Part ◽  
Tetrad Field ◽  
Mathematical Properties ◽  
Constitutive Tensor ◽  
Hamiltonian Analysis

We introduce a novel procedure for studying the Hamiltonian formalism of new general relativity (NGR) based on the mathematical properties encoded in the constitutive tensor defined by the premetric approach. We derive the canonical momenta conjugate to the tetrad field and study the eigenvalues of the Hessian tensor, which is mapped to a Hessian matrix with the help of indexation formulas. The properties of the Hessian matrix heavily rely on the possible values of the free coefficients [Formula: see text] appearing in the NGR Lagrangian. We find four null eigenvalues associated with trivial primary constraints in the temporal part of the momenta. The remaining eigenvalues are grouped in four sets, which have multiplicity 3, 1, 5 and 3, and can be set to zero depending on different choices of the coefficients [Formula: see text]. There are nine possible different cases when one, two, or three sets of eigenvalues are imposed to vanish simultaneously. All cases lead to a different number of primary constraints, which are consistent with previous work on the Hamiltonian analysis of NGR by Blixt et al. (2018).

scholarly journals NON-ABELIAN PROCA MODEL BASED ON THE IMPROVED BFT FORMALISM

1998 ◽  
Vol 13 (13) ◽  
pp. 2179-2199 ◽  
Author(s):  
MU-IN PARK ◽  
YOUNG-JAI PARK
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Difficult Problem ◽  
Lagrangian Formulation ◽  
Exponential Form ◽  
Extended Phase ◽  
Physical Variables ◽  
Correction Terms ◽  
Insight Into

We present the newly improved Batalin–Fradkin–Tyutin (BFT) Hamiltonian formalism and the generalization to the Lagrangian formulation, which provide a much more simple and transparent insight into the usual BFT method, with application to the non-Abelian Proca model, which has been a difficult problem in the usual BFT method. The infinite terms of the effectively first class contraints can be made to be the regular power series forms by an ingenious choice of Xαβ and ωαβ matrices. In this new method, the first class Hamiltonian, which also needs infinite correction terms, is obtained simply by replacing the original variables in the original Hamiltonian with the BFT physical variables. Remarkably, all the infinite correction terms can be expressed in the compact exponential form. We also show that in our model the Poisson brackets of the BFT physical variables in the extended phase space have the same structure as the Dirac brackets of the original phase space variables. With the help of both our newly developed Lagrangian formulation and Hamilton's equations of motion, we obtain the desired classical Lagrangian corresponding to the first class Hamiltonian which can be reduced to the generalized Stückelberg Lagrangian which is a nontrivial conjecture in our infinitely many terms involved in the Hamiltonian and the Lagrangian.

scholarly journals FIRST-ORDER ACTIONS AND DUALITY

2007 ◽  
Vol 22 (04) ◽  
pp. 851-867 ◽  
Author(s):  
ALEJANDRO GAONA ◽  
J. ANTONIO GARCÍA
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
General Covariance ◽  
Dirac Bracket ◽  
Local Transformation ◽  
First Order ◽  
Duality Transformations ◽  
Reduced Space ◽  
Field Redefinition ◽  
Nonlocal Transformation

We consider some aspects of classical S-duality transformations in first-order actions taking into account the general covariance of the Dirac algorithm and the transformation properties of the Dirac bracket. By classical S-duality transformations we mean a field redefinition that interchanges the equations of motion and its associated Bianchi identities. By working from a first-order variational principle and performing the corresponding Dirac analysis we find that the standard electromagnetic duality can be reformulated as a canonical local transformation. The reduction from this phase space to the original phase space variables coincides with the well-known result about duality as a canonical nonlocal transformation. We have also applied our ideas to the bosonic string. These dualities are not canonical transformations for the Dirac bracket and relate actions with different kinetic terms in the reduced space.

scholarly journals Exploring Alternatives to the Hamiltonian Calculation of the Ashtekar-Olmedo-Singh Black Hole Solution

2021 ◽  
Vol 8 ◽  
Author(s):  
Alejandro García-Quismondo ◽  
Guillermo A. Mena Marugán
Keyword(s):  
Black Hole ◽  
Phase Space ◽  
Black Hole Solution ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Quantum Geometry ◽  
Dynamical Equations ◽  
Constants Of Motion ◽  
Geometry Effects ◽  
Hole Model

In this article, we reexamine the derivation of the dynamical equations of the Ashtekar-Olmedo-Singh black hole model in order to determine whether it is possible to construct a Hamiltonian formalism where the parameters that regulate the introduction of quantum geometry effects are treated as true constants of motion. After arguing that these parameters should capture contributions from two distinct sectors of the phase space that had been considered independent in previous analyses in the literature, we proceed to obtain the corresponding equations of motion and analyze the consequences of this more general choice. We restrict our discussion exclusively to these dynamical issues. We also investigate whether the proposed procedure can be reconciled with the results of Ashtekar, Olmedo, and Singh, at least in some appropriate limit.

Diagnosing Knowledge States in Algebra Using the Rule-Space Model

1993 ◽  
Vol 24 (5) ◽  
pp. 442-459
Author(s):  
Menucha Birenbaum ◽  
Anthony E. Kelly ◽  
Kikumi K. Tatsuoka
Keyword(s):  
Ninth Graders ◽  
Algebraic Equations ◽  
Space Model ◽  
Rule Space ◽  
Knowledge States ◽  
Linear Algebraic Equations ◽  
Rule Space Model ◽  
Mathematical Behavior ◽  
Space Approach

This paper illustrates the use of rule space as a tool to support cognitive analyses of students' mathematical behavior. The rule-space approach is explained and is then used to classify students into one of two methods for solving linear algebraic equations in one unknown and to diagnose their knowledge states in this topic. A 32-item test with open-ended questions was administered to 231 eighth and ninth graders. Two outcomes of the rule-space model are presented: (a) a classification of examinees into knowledge states resulting from the two solution approaches at the group level along with individual examples and (b) tree diagrams of the transitional relationships among the states for each strategy. Implications for using the feedback provided by the rule-space model in the context of instruction and assessment are discussed.

scholarly journals Covariant phase space, constraints, gauge and the Peierls formula

2014 ◽  
Vol 29 (05) ◽  
pp. 1430009 ◽  
Author(s):  
Igor Khavkine
Keyword(s):  
Phase Space ◽  
Gauge Theories ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Hamiltonian Formalism ◽  
Classical Field Theory ◽  
The Novel ◽  
Poisson Structures ◽  
Gauge Symmetries ◽  
Space Formalism

It is well known that both the symplectic structure and the Poisson brackets of classical field theory can be constructed directly from the Lagrangian in a covariant way, without passing through the noncovariant canonical Hamiltonian formalism. This is true even in the presence of constraints and gauge symmetries. These constructions go under the names of the covariant phase space formalism and the Peierls bracket. We review both of them, paying more careful attention, than usual, to the precise mathematical hypotheses that they require, illustrating them in examples. Also an extensive historical overview of the development of these constructions is provided. The novel aspect of our presentation is a significant expansion and generalization of an elegant and quite recent argument by Forger and Romero showing the equivalence between the resulting symplectic and Poisson structures without passing through the canonical Hamiltonian formalism as an intermediary. We generalize it to cover theories with constraints and gauge symmetries and formulate precise sufficient conditions under which the argument holds. These conditions include a local condition on the equations of motion that we call hyperbolizability, and some global conditions of cohomological nature. The details of our presentation may shed some light on subtle questions related to the Poisson structure of gauge theories and their quantization.

The Generalized Transmission Error of Parallel-Axis Gears

1989 ◽  
Vol 111 (3) ◽  
pp. 414-423 ◽  
Author(s):  
W. D. Mark
Keyword(s):  
Equations Of Motion ◽  
Transmission Error ◽  
Inertial Forces ◽  
Algebraic Equations ◽  
Elastic Deformations ◽  
Small Deviations ◽  
Helical Gears ◽  
Linear Algebraic Equations ◽  
Static Transmission Error ◽  
Parallel Axis

The traditional one-component transmission error of parallel-axis helical gears is generalized to a three-component transmission error which characterizes the composite displacement in the plane-of-contact resulting from arbitrary small deviations in the positions of both gears of a meshing pair from the positions of their rigid perfect involute counterparts. A set of linear algebraic equations is derived for the contribution to the three generalized transmission error components arising from elastic deformations of the teeth and gear bodies and deviations of the tooth running surfaces from equispaced perfect involute surfaces. It is shown how to combine this set of equations with the generalized transmission error definition and the equations of motion of a gear system to predict the dynamic response of gear elements in the system. For the case of negligible gearbody and bearing/bearing support inertial forces, an additional set of algebraic equations that includes the effects of bearing flexibility and misalignment is derived. Combining the solution of this set of equations with the above-mentioned generalized transmission error equations yields the three-component generalized static transmission error.

Sign in / Sign up

Export Citation Format

Share Document