scholarly journals ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS

1995 ◽  
Vol 10 (04) ◽  
pp. 579-610 ◽  
Author(s):  
V. MUKHANOV ◽  
A. WIPF
Keyword(s):  
String Theory ◽  
Hamiltonian Systems ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Linear Constraints ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Diffeomorphism Invariance ◽  
Symmetry Transformations ◽  
Local Symmetries

In this paper we show how the well-known local symmetries of Lagrangian systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangian system. The non-linear constraints (which we have, for instance, in gravity, supergravity and string theory) generate the dynamics of the corresponding Lagrangian system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We show the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems, in particular those which are diffeomorphism-invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian and Lagrangian formalisms is found. The possible applications of our results are discussed.

scholarly journals LAGRANGIAN APPROACH OF THE FIRST-CLASS CONSTRAINED SYSTEMS

1998 ◽  
Vol 13 (33) ◽  
pp. 2653-2663 ◽  
Author(s):  
YONG-WAN KIM ◽  
YOUNG-JAI PARK ◽  
SEUNG-KOOK KIM
Keyword(s):  
Hamiltonian Formulation ◽  
Constrained Systems ◽  
Lagrangian Approach ◽  
Lagrangian Formalism ◽  
Exact Form ◽  
Gauge Invariant ◽  
Symmetry Transformations ◽  
Local Symmetries

We show how to systematically derive the exact form of local symmetries for the Abelian Proca and CS models, which are converted into first-class constrained systems by the BFT formalism, in the Lagrangian formalism. As a result, without resorting to a Hamiltonian formulation we obtain the well-known U(1) symmetry for the gauge-invariant Proca model, while showing that for the CS model there exist novel symmetries as well as the usual symmetry transformations.

Quantization of fractional singular Lagrangian systems using WKB approximation

2018 ◽  
Vol 33 (36) ◽  
pp. 1850222 ◽  
Author(s):  
Eqab M. Rabei ◽  
Mohammed Al Horani
Keyword(s):  
Wave Function ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Singular Systems ◽  
Equations Of Motion ◽  
Singular System ◽  
Wkb Approximation ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Action Integral

In this paper, the fractional singular Lagrangian system is studied. The Hamilton–Jacobi treatment is developed to be applicable for fractional singular Lagrangian system. The equations of motion are obtained for the fractional singular systems and the Hamilton–Jacobi partial differential equations are obtained and solved to determine the action integral. Then the wave function for fractional singular system is obtained. Besides, to demonstrate this theory, the fractional Christ-Lee model is discussed and quantized using the WKB approximation.

scholarly journals VARIATIONAL PROBLEM FOR THE FRENKEL AND THE BARGMANN–MICHEL–TELEGDI (BMT) EQUATIONS

2013 ◽  
Vol 28 (01) ◽  
pp. 1250234 ◽  
Author(s):  
A. A. DERIGLAZOV
Keyword(s):  
Abelian Group ◽  
Variational Problem ◽  
Classical Theory ◽  
Equations Of Motion ◽  
Lagrangian Formulation ◽  
Gauge Invariant ◽  
Local Symmetries

We propose Lagrangian formulation for the particle with value of spin fixed within the classical theory. The Lagrangian is invariant under non-Abelian group of local symmetries. On this reason, all the initial spin variables turn out to be unobservable quantities. As the gauge-invariant variables for description of spin we can take either the Frenkel tensor or the Bargmann–Michel–Telegdi (BMT) vector. Fixation of spin within the classical theory implies O(ℏ)-corrections to the corresponding equations of motion.

INTEGRABILITY AND ACTION FUNCTION IN MULTI-HAMILTONIAN SYSTEMS

2004 ◽  
Vol 19 (11) ◽  
pp. 863-870 ◽  
Author(s):  
S. I. MUSLIH
Keyword(s):  
Differential Equations ◽  
Hamiltonian Systems ◽  
Equations Of Motion ◽  
Integral Function ◽  
Jacobi Method ◽  
Action Function ◽  
Action Integral ◽  
Method Integration ◽  
Total Differential

Multi-Hamiltonian systems are investigated by using the Hamilton–Jacobi method. Integration of a set of total differential equations which includes the equations of motion and the action integral function is discussed. It is shown that this set is integrable if and only if the total variations of the Hamiltonians vanish. Two examples are studied.

scholarly journals Symmetries, pseudosymmetries and conservation laws in Lagrangian and Hamiltonian k-symplectic formalisms

2016 ◽  
Vol 13 (03) ◽  
pp. 1650026
Author(s):  
Florian Munteanu
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Systems ◽  
Conservation Law ◽  
Type Theorem ◽  
Natural Extension ◽  
Lagrangian Systems ◽  
Symmetry And Conservation Law

In this paper, we will present Lagrangian and Hamiltonian [Formula: see text]-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will obtain new kinds of conservation laws for [Formula: see text]-symplectic Hamiltonian systems and [Formula: see text]-symplectic Lagrangian systems.

scholarly journals Gauge-independent Theory of Symmetry. I.

1964 ◽  
Vol 17 (4) ◽  
pp. 431 ◽  
Author(s):  
LJ Tassie ◽  
HA Buchdahl
Keyword(s):  
Electromagnetic Field ◽  
Single Particle ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Conserved Quantities ◽  
Continuous Symmetry ◽  
Symmetry Properties ◽  
Symmetry Transformations ◽  
Hamiltonian Forms

The invariance of a system under a given transformation of coordinates is usually taken to mean that its Lagrangian is invariant under that transformation. Consequently, whether or not the system is invariant will depend on the gauge used in describing the system. By defining invariance of a system to mean the invariance of its equations of motion, a gauge-independent theory of symmetry properties is obtained for classical mechanics in both the Lagrangian and Hamiltonian forms. The conserved quantities associated with continuous symmetry transformations are obtained. The system of a single particle moving in a given electromagnetic field is considered in detail for various symmetries of the electromagnetic field, and the appropriate conserved quantities are found.

scholarly journals LOOP VARIABLES IN STRING THEORY

2003 ◽  
Vol 18 (05) ◽  
pp. 767-809 ◽  
Author(s):  
B. SATHIAPALAN
Keyword(s):  
Renormalization Group ◽  
String Theory ◽  
Equations Of Motion ◽  
Renormalization Group Equation ◽  
Group Method ◽  
Group Equation ◽  
World Sheet ◽  
Gauge Invariant ◽  
Variable Approach ◽  
Exact Renormalization Group

The loop variable approach is a proposal for a gauge-invariant generalization of the sigma-model renormalization group method of obtaining equations of motion in string theory. The basic guiding principle is space–time gauge invariance rather than world sheet properties. In essence it is a version of Wilson's exact renormalization group equation for the world sheet theory. It involves all the massive modes and is defined with a finite world sheet cutoff, which allows one to go off the mass-shell. On shell the tree amplitudes of string theory are reproduced. The equations are gauge-invariant off shell also. This paper is a self-contained discussion of the loop variable approach as well as its connection with the Wilsonian RG.

scholarly journals A NOTE ON THE (ANTI-)BRST INVARIANT LAGRANGIAN DENSITIES FOR THE FREE ABELIAN 2-FORM GAUGE THEORY

2010 ◽  
Vol 25 (28) ◽  
pp. 2457-2467
Author(s):  
SAURABH GUPTA ◽  
R. P. MALIK
Keyword(s):  
Gauge Theory ◽  
Vector Field ◽  
Lagrange Multiplier ◽  
Field Condition ◽  
Equations Of Motion ◽  
Present Theory ◽  
Lagrange Equations ◽  
Symmetry Transformations ◽  
The Absolute ◽  
Lagrangian Densities

We show that the previously known off-shell nilpotent [Formula: see text] and absolutely anticommuting (sb sab + sab sb = 0) Becchi–Rouet–Stora–Tyutin (BRST) transformations (sb) and anti-BRST transformations (sab) are the symmetry transformations of the appropriate Lagrangian densities of a four (3+1)-dimensional (4D) free Abelian 2-form gauge theory which do not explicitly incorporate a very specific constrained field condition through a Lagrange multiplier 4D vector field. The above condition, which is the analogue of the Curci–Ferrari restriction of the non-Abelian 1-form gauge theory, emerges from the Euler–Lagrange equations of motion of our present theory and ensures the absolute anticommutativity of the transformations s(a)b. Thus, the coupled Lagrangian densities, proposed in our present investigation, are aesthetically more appealing and more economical.

scholarly journals Is there a Connection between Local and Global (In-)Stability?

1986 ◽  
Vol 39 (3) ◽  
pp. 331 ◽  
Author(s):  
B Eckhardt ◽  
JA Louw ◽  
W-H Steeb
Keyword(s):  
Stability Analysis ◽  
Hamiltonian Systems ◽  
Local Stability ◽  
Large Scale ◽  
Equations Of Motion ◽  
Riemannian Curvature ◽  
Local Stability Analysis

We review two criteria which have been used to predict the onset of large scale stochasticity in Hamiltonian systems. We show that one of them, due to Toda and based on a local stability analysis of the equations of motion, is inconclusive. An approach based on the local Riemannian curvature K of trajectories correctly predicts chaos if K < 0 everywhere, but�no further conclusions can be drawn. New (counter-)examples are provided.

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