scholarly journals POISSON GEOMETRY IN CONSTRAINED SYSTEMS

2003 ◽  
Vol 15 (07) ◽  
pp. 663-703 ◽  
Author(s):  
MARTIN BOJOWALD ◽  
THOMAS STROBL
Keyword(s):  
Phase Space ◽  
Symplectic Manifold ◽  
Sufficient Conditions ◽  
Constrained Systems ◽  
Poisson Manifolds ◽  
Poisson Geometry ◽  
Dirac Bracket ◽  
Semisimple Lie Algebra ◽  
Constrained System ◽  
Symplectic Embedding

Associated to a constrained system with closed constraint algebra there are two Poisson manifolds P and Q forming a symplectic dual pair with respect to the original, unconstrained phase space: P is the image of the constraint map (equipped with the algebra of constraints) and Q the Poisson quotient with respect to the orbits generated by the constraints (the orbit space is assumed to be a manifold). We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf of Q. By these methods, a second class constrained system with closed algebra is reformulated as an abelian first class system in an extended phase space. While any Poisson manifold (P,Π) has a symplectic realization (Karasev, Weinstein 87), it does not always permit a leafwise symplectic embedding into a symplectic manifold (M,ω). For regular P, it is seen that such an embedding exists, iff the characteristic form-class of Π, a certain element of the third relative cohomology of P, vanishes. A tubular neighborhood of the constraint surface of a general second class constrained system equipped with the Dirac bracket provides a physical example for such an embedding into the original symplectic manifold. In contrast, a leafwise symplectic embedding of e.g. (the maximal regular part of) a Poisson Lie manifold associated to a compact, semisimple Lie algebra does not exist.

scholarly journals DEFORMATION QUANTIZATION OF RELATIVISTIC PARTICLES IN ELECTROMAGNETIC FIELDS

2008 ◽  
Vol 23 (11) ◽  
pp. 1757-1790 ◽  
Author(s):  
LAURA SÁNCHEZ ◽  
IMELDA GALAVIZ ◽  
HUGO GARCÍA-COMPEÁN
Keyword(s):  
Charged Particle ◽  
Phase Space ◽  
Deformation Quantization ◽  
Free Particle ◽  
Constrained Systems ◽  
Dirac Bracket ◽  
Moyal Product ◽  
Weyl Correspondence ◽  
Particles In Electromagnetic Fields ◽  
Relativistic Particles

The Weyl–Wigner–Moyal formalism for Dirac second-class constrained systems has been proposed recently as the deformation quantization of Dirac bracket. In this paper, after a brief review of this formalism, it is applied to the case of the relativistic free particle. Within this context, the Stratonovich–Weyl quantizer, Weyl correspondence, Moyal ⋆-product and Wigner function in the constrained phase-space are obtained. The recent Hamiltonian treatment for constrained systems, whose constraints depend explicitly on time, are used to perform the deformation quantization of the relativistic free charged particle in an arbitrary electromagnetic background. Finally, the system consisting of a charged particle interacting with a dynamical Maxwell field is quantized in this context.

Fractional Constrained Systems and Caputo Derivatives

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Dumitru Baleanu
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Variational Principles ◽  
Constrained Systems ◽  
Control Problems ◽  
Fractional Dynamics ◽  
Constrained System ◽  
Caputo Derivatives ◽  
Fractional Dynamical System ◽  
Made In

During the last few years, remarkable developments have been made in the theory of the fractional variational principles and their applications to control problems and fractional quantization issue. The variational principles have been used in physics to construct the phase space of a fractional dynamical system. Based on the Caputo derivatives, the fractional dynamics of discrete constrained systems is presented and the notion of the reduced phase space is discussed. Two examples of discrete constrained system are analyzed in detail.

SYMPLECTIC QUANTIZATION OF CONSTRAINED SYSTEMS

1992 ◽  
Vol 07 (19) ◽  
pp. 1737-1747 ◽  
Author(s):  
J. BARCELOS-NETO ◽  
C. WOTZASEK
Keyword(s):  
Symplectic Form ◽  
Gauge Theories ◽  
Geometrical Structure ◽  
Constrained Systems ◽  
Dirac Bracket ◽  
Constrained System ◽  
Gauge Transformations ◽  
Symmetry Breaking Term ◽  
Singular Configuration ◽  
Breaking Term

It is shown that the symplectic two-form, which defines the geometrical structure of a constrained theory in the Faddeev-Jackiw approach, may be brought into a non-degenerated form, by an iterative implementation of the existing constraints. The resulting generalized brackets coincide with those obtained by the Dirac bracket approach, if the constrained system under investigation presents only second-class constraints. For gauge theories, a symmetry breaking term must be supplemented to bring the symplectic form into a non-singular configuration. At present, the singular symplectic two-form provides directly the generators of the time independent gauge transformations.

scholarly journals RIGID AND GAUGE NOETHER SYMMETRIES FOR CONSTRAINED SYSTEMS

2000 ◽  
Vol 15 (29) ◽  
pp. 4681-4721 ◽  
Author(s):  
J. ANTONIO GARCÍA ◽  
J. M. PONS
Keyword(s):  
Phase Space ◽  
Tangent Space ◽  
Constrained Systems ◽  
Simons Theory ◽  
Dirac Bracket ◽  
Noether Symmetries ◽  
Gauge Symmetries ◽  
Secondary Constraint

We develop the general theory of Noether symmetries for constrained systems, that is, systems that are described by singular Lagrangians. In our derivation, the Dirac bracket structure with respect to the primary constraints appears naturally and plays an important role in the characterization of the conserved quantities associated to these Noether symmetries. The issue of projectability of these symmetries from tangent space to phase space is fully analyzed, and we give a geometrical interpretation of the projectability conditions in terms of a relation between the Noether conserved quantity in tangent space and the presymplectic form defined on it. We also examine the enlarged formalism that results from taking the Lagrange multipliers as new dynamical variables; we find the equation that characterizes the Noether symmetries in this formalism, and we also prove that the standard formulation is a particular case of the enlarged one. The algebra of generators for Noether symmetries is discussed in both the Hamiltonian and Lagrangian formalisms. We find that a frequent source for the appearance of open algebras is the fact that the transformations of momenta in phase space and tangent space only coincide on shell. Our results apply with no distinction to rigid and gauge symmetries; for the latter case we give a general proof of the existence of Noether gauge symmetries for theories with first and second class constraints that do not exhibit tertiary constraints in the stabilization algorithm. Among some examples that illustrate our results, we study the Noether gauge symmetries of the Abelian Chern–Simons theory in 2n+1 dimensions. An interesting feature of this example is that its primary first class constraints can only be identified after the determination of the secondary constraint. The example is worked out retaining all the original set of variables.

Energy method for modelling delamination buckling in geometrically constrained systems

2003 ◽  
Vol 217 (9) ◽  
pp. 1015-1026 ◽  
Author(s):  
B J Hicks ◽  
G Mullineux ◽  
C Berry ◽  
C J McPherson ◽  
A J Medland
Keyword(s):  
Finite Element ◽  
Full Range ◽  
Constrained Systems ◽  
Third Party ◽  
Constrained System ◽  
Finite Element Approach ◽  
Delamination Buckling ◽  
Element Approach ◽  
Geometrically Constrained ◽  
Energy Formulation

Delamination buckling analysis of laminates is of considerable interest to the mechanical and materials engineering sectors, as well as having wider applications in geology and civil engineering. With advances in computing power, the ability to model ever increasingly complex problems at more detailed levels becomes more of a reality. However, many of the common finite element packages, with the exception of all but the most specialized, do not perform particularly well where complex non-linear problems are dealt with. In many cases, these packages can fail to determine the full range of solutions or accurately predict the properties and geometry of the final state. This is particularly the case where large deformations and buckling of laminates are considered. Because of this, many researchers prefer to use what they perceive to be more reliable techniques, such as the symbolic computation of the underlying differential equations, rather than finite element approaches. The use of finite element packages is further frustrated by the steep learning curve and implicit restrictions imposed by using third-party software. In this paper, a finite element approach and an energy formulation method are considered and used to model the delamination buckling in a geometrically constrained system. These methods are compared with experimental results and their relative merits are discussed. In particular, the accuracy and the ability to represent the geometry of the buckled system are discussed. Both the finite element approach and the energy formulation are described in detail and the numerical results are compared.

PATH INTEGRAL QUANTIZATION OF SUPERSTRING

2010 ◽  
Vol 25 (02) ◽  
pp. 135-141
Author(s):  
H. A. ELEGLA ◽  
N. I. FARAHAT
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Path Integral ◽  
Equations Of Motion ◽  
Singular System ◽  
Constrained Systems ◽  
Classical Structure ◽  
Path Integral Quantization ◽  
Total Differential

Motivated by the Hamilton–Jacobi approach of constrained systems, we analyze the classical structure of a four-dimensional superstring. The equations of motion for a singular system are obtained as total differential equations in many variables. The path integral quantization based on Hamilton–Jacobi approach is applied to quantize the system, and the integration is taken over the canonical phase space coordinates.

Phase space geometry of constrained systems

1995 ◽  
Vol 105 (3) ◽  
pp. 1539-1545 ◽  
Author(s):  
V. P. Pavlov ◽  
A. O. Starinetz
Keyword(s):  
Phase Space ◽  
Constrained Systems ◽  
Space Geometry ◽  
Phase Space Geometry

scholarly journals Nonfragile Robust Model Predictive Control for Uncertain Constrained Systems with Time-Delay Compensation

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Wei Jiang ◽  
Hong-li Wang ◽  
Jing-hui Lu ◽  
Wei-wei Qin ◽  
Guang-bin Cai
Keyword(s):  
Time Delay ◽  
Model Predictive Control ◽  
Predictive Control ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Constrained Systems ◽  
Delay Compensation ◽  
Robust Model Predictive Control ◽  
Robust Model ◽  
Time Delay Compensation

This study investigates the problem of asymptotic stabilization for a class of discrete-time linear uncertain time-delayed systems with input constraints. Parametric uncertainty is assumed to be structured, and delay is assumed to be known. In Lyapunov stability theory framework, two synthesis schemes of designing nonfragile robust model predictive control (RMPC) with time-delay compensation are put forward, where the additive and the multiplicative gain perturbations are, respectively, considered. First, by designing appropriate Lyapunov-Krasovskii (L-K) functions, the robust performance index is defined as optimization problems that minimize upper bounds of infinite horizon cost function. Then, to guarantee closed-loop stability, the sufficient conditions for the existence of desired nonfragile RMPC are obtained in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approaches.

SCHRÖDINGER REPRESENTATION AND SYMPLECTIC EMBEDDING OF TOPOLOGICAL SOLITONS

2005 ◽  
Vol 20 (32) ◽  
pp. 2455-2465 ◽  
Author(s):  
SOON-TAE HONG
Keyword(s):  
Phase Space ◽  
Quantum Number ◽  
Lagrange Multiplier ◽  
Homotopy Class ◽  
Canonical Quantization ◽  
Topological Solitons ◽  
Symplectic Embedding ◽  
Class System ◽  
Collective Coordinates ◽  
Schrödinger Representation

Exploiting the SU(2) Skyrmion Lagrangian with second-class constraints associated with Lagrange multiplier and collective coordinates, we convert the second-class system into the first-class one in the Batalin–Fradkin–Tyutin embedding through the introduction of Stückelberg coordinates. In the enlarged phase space possessing the Stückelberg coordinates, we perform the "canonical" quantization to describe the Schrödinger representation of the SU(2) Skyrmion, so that we can assign via the homotopy class π4( SU (2))=Z2 half integers to the isospin quantum number for the solitons. The symplectic embedding and the Becchi–Rouet–Stora–Tyutin symmetries involved in the SU(2) Skyrmion are also investigated.

scholarly journals DUALITY THROUGH THE SYMPLECTIC EMBEDDING FORMALISM

2007 ◽  
Vol 22 (21) ◽  
pp. 3605-3620 ◽  
Author(s):  
E. M. C. ABREU ◽  
A. C. R. MENDES ◽  
C. NEVES ◽  
W. OLIVEIRA ◽  
F. I. TAKAKURA
Keyword(s):  
Phase Space ◽  
Gauge Invariance ◽  
Zero Mode ◽  
Mass Term ◽  
Gauge Invariant ◽  
Embedding Method ◽  
Symplectic Embedding ◽  
The One ◽  
Topological Term

In this work we show that we can obtain dual equivalent actions following the symplectic formalism with the introduction of extra variables which enlarge the phase space. We show that the results are equal as the one obtained with the recently developed gauging iterative Noether dualization method. We believe that, with the arbitrariness property of the zero mode, the symplectic embedding method is more profound since it can reveal a whole family of dual equivalent actions. We illustrate the method demonstrating that the gauge-invariance of the electromagnetic Maxwell Lagrangian broken by the introduction of an explicit mass term and a topological term can be restored to obtain the dual equivalent and gauge-invariant version of the theory.

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