scholarly journals Dirac–Bergmann constraints in physics: Singular Lagrangians, Hamiltonian constraints and the second Noether theorem

2018 ◽  
Vol 15 (10) ◽  
pp. 1830004 ◽  
Author(s):  
Luca Lusanna
Keyword(s):  
Canonical Transformation ◽  
Hessian Matrix ◽  
Theory Of Constraints ◽  
Noether Theorem ◽  
Lagrange Equations ◽  
Gauge Transformations ◽  
Mathematical Properties ◽  
Singular Lagrangians

There is a review of the main mathematical properties of system described by singular Lagrangians and requiring Dirac–Bergmann theory of constraints at the Hamiltonian level. The following aspects are discussed:(i)the connection of the rank and eigenvalues of the Hessian matrix in the Euler–Lagrange equations with the chains of first- and second-class constraints;(ii)the connection of the Noether identities of the second Noether theorem with the Hamiltonian constraints;(iii)the Shanmugadhasan canonical transformation for the identification of the gauge variables and for the search of the Dirac observables, i.e. the quantities invariant under Hamiltonian gauge transformations.

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 ROUTH REDUCTION FOR SINGULAR LAGRANGIANS

2010 ◽  
Vol 07 (08) ◽  
pp. 1451-1489 ◽  
Author(s):  
BAVO LANGEROCK ◽  
MARCO CASTRILLÓN LÓPEZ
Keyword(s):  
Kinetic Energy ◽  
Original System ◽  
Regularity Conditions ◽  
Lagrangian Systems ◽  
Lagrange Equations ◽  
Reduction Procedure ◽  
Momentum Map ◽  
Routh Reduction ◽  
Regular Value ◽  
Singular Lagrangians

This paper concerns the Routh reduction procedure for Lagrangians systems with symmetry. It differs from the existing results on geometric Routh reduction in the fact that no regularity conditions on either the Lagrangian L or the momentum map JL are required apart from the momentum being a regular value of JL. The main results of this paper are: the description of a general Routh reduction procedure that preserves the Euler–Lagrange nature of the original system and the presentation of a presymplectic framework for Routh reduced systems. In addition, we provide a detailed description and interpretation of the Euler–Lagrange equations for the reduced system. The proposed procedure includes Lagrangian systems with a non-positively definite kinetic energy metric.

THREE ELABORATIONS ON BERRY’S CONNECTION, CURVATURE AND PHASE

1988 ◽  
Vol 03 (02) ◽  
pp. 285-297 ◽  
Author(s):  
R. JACKIW
Keyword(s):  
Conservation Laws ◽  
Quantum System ◽  
Canonical Transformation ◽  
Time Dependent ◽  
Berry's Phase ◽  
Berry’S Phase ◽  
Gauge Transformations ◽  
Constants Of Motion ◽  
Symmetry Transformations ◽  
Higher Order Corrections

We discuss how symmetries and conservation laws are affected when Berry’s phase occurs in a quantum system: symmetry transformations of coordinates have to be supplemented by gauge transformations of Berry’s connection, and consequently constants of motion acquire terms beyond the familiar kinematical ones. We show how symmetries of a problem determine Berry’s connection, curvature and, once a specific path is chosen, the phase as well. Moreover, higher order corrections are also fixed. We demonstrate that in some instances Berry’s curvature and phase can be removed by a globally well-defined, time-dependent canonical transformation. Finally, we describe how field theoretic anomalies may be viewed as manifestations of Berry’s phase.

scholarly journals Noether symmetry approach in Eddington-inspired Born–Infeld gravity

2021 ◽  
Vol 81 (4) ◽  
Author(s):  
Thanyagamon Kanesom ◽  
Phongpichit Channuie ◽  
Narakorn Kaewkhao
Keyword(s):  
Gravity Model ◽  
Hessian Matrix ◽  
Noether Symmetry ◽  
De Sitter ◽  
Lagrange Equations ◽  
Underlying Theory ◽  
Sitter Solution ◽  
Symmetry Approach ◽  
Gauge Symmetries ◽  
Formal Framework

AbstractIn this work, we take a short recap of a formal framework of the Eddington-inspired Born–Infeld (EiBI) theory of gravity and derive the point-like Lagrangian for underlying theory based on the use of Noether gauge symmetries (NGS). We study a Hessian matrix and quantify Euler–Lagrange equations of EiBI universe. We discuss the NGS approach for the Eddington-inspired Born–Infeld theory and show that there exists the de Sitter solution in this gravity model.

THE SHANMUGADHASAN CANONICAL TRANSFORMATION, FUNCTION GROUPS AND THE EXTENDED SECOND NOETHER THEOREM

1993 ◽  
Vol 08 (24) ◽  
pp. 4193-4233 ◽  
Author(s):  
LUCA LUSANNA
Keyword(s):  
Phase Space ◽  
Vector Fields ◽  
Continuous Group ◽  
Canonical Transformations ◽  
Noether Theorem ◽  
First Order ◽  
Global Formulation ◽  
Function Group ◽  
Definition Of ◽  
Singular Lagrangians

After the definition of a class of well-behaved singular Lagrangians, an analysis of all the consequences of the extended second Noether theorem in the second-order formalism is made. The phase-space reformulation contains arbitrary first- and second-class constraints. An answer to the problem of the Dirac conjecture is given for this class of singular Lagrangians. By using the concepts of function groups and of the associated Shanmugadhasan canonical transformations, an attempt is made to arrive at a global formulation of the theorem, in which the original invariance under an “infinite continuous group” of transformations is replaced by weak quasi-invariance under an “infinite continuous group [Formula: see text],” whose algebra is an involutive distribution of Lie-Bäcklund vector fields generating the Noether transformations. Its phase-space counterpart is the involutive distribution associated with a special function group Ḡpm, which contains a function subgroup Ḡp connected (when in canonical form) to the Shanmugadhasan canonical transformations. Also, the various possible first-order formalisms are analyzed.

To Construct Gauge Transformations from Singular Lagrangians

1986 ◽  
Vol 2 (3) ◽  
pp. 187-194 ◽  
Author(s):  
J Gomis ◽  
K Kamimura ◽  
J. M Pons
Keyword(s):  
Gauge Transformations ◽  
Singular Lagrangians

Foundations of higher-order variational theory on Grassmann fibrations

2014 ◽  
Vol 11 (07) ◽  
pp. 1460023 ◽  
Author(s):  
Zbyněk Urban ◽  
Demeter Krupka
Keyword(s):  
Vector Fields ◽  
Variational Principles ◽  
Variational Calculus ◽  
Higher Order ◽  
Noether Theorem ◽  
Variational Theory ◽  
Lagrange Equations ◽  
Variation Formula ◽  
One Dimensional ◽  
The First Variation

A setting for higher-order global variational analysis on Grassmann fibrations is presented. The integral variational principles for one-dimensional immersed submanifolds are introduced by means of differential 1-forms with specific properties, similar to the Lepage forms from the variational calculus on fibred manifolds. Prolongations of immersions and vector fields to the Grassmann fibrations are defined as a geometric tool for the variations of immersions, and the first variation formula in the infinitesimal form is derived. Its consequences, the Euler–Lagrange equations for submanifolds and the Noether theorem on invariant variational functionals are proved. Examples clarifying the meaning of the Noether theorem in the context of variational principles for submanifolds are given.

scholarly journals General variational principle, general Noether theorem, their classical and quantum new physics and solution to crisis deducing all physics laws

2020 ◽  
Author(s):  
C. Huang ◽  
Yong-Chang Huang
Keyword(s):  
Variational Principle ◽  
Extreme Values ◽  
New Physics ◽  
Noether Theorem ◽  
Lagrange Equations ◽  
Famous Theorem ◽  
The Real ◽  
Equivalent Relation

This paper discovers that current variational principle and Noether theorem for different physics systems with (in)finite freedom systems have missed the double extremum processes of the general extremum functional that both is deduced by variational principle and is necessarily taken in deducing all the physics laws, but these have not been corrected for over a century since Noether's proposing her famous theorem, which result in the crisis deducing relevant mathematical laws and all physics laws. This paper discovers there is the hidden logic cycle that one assumes Euler-Lagrange equations, and then he finally deduces Euler-Lagrange equations via the equivalent relation in the whole processes in all relevant current references. This paper corrects the current key mistakes that when physics systems choose the variational extreme values, the appearing processes of the physics systems are real physics processes, otherwise, are virtual processes in all current articles, reviews and (text)books. The real physics should be after choosing the variational extreme values of physics systems, the general extremum functional of the physics systems needs to further choose the minimum absolute extremum zero of the general extremum functional, otherwise, the appearing processes of physics systems are still virtual processes. Using the double extremum processes of the general extremum functionals, the crisis and the hidden logic cycle in current variational principle and current Noether theorem are solved. Furthermore, the new mathematical and physical double extremum processes and their new mathematical pictures and physics for (in)finite freedom systems are discovered. This paper gives both general variational principle and general Noether theorem as well as their classical and quantum new physics, which would rewrite all relevant current different branches of science, as key tools of studying and processing them.

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