scholarly journals A VARIATIONAL FORMULATION OF SYMPLECTIC NONCOMMUTATIVE MECHANICS

2007 ◽  
Vol 04 (05) ◽  
pp. 789-805 ◽  
Author(s):  
IGNACIO CORTESE ◽  
J. ANTONIO GARCÍA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Variational Principle ◽  
Configuration Space ◽  
Variational Formulation ◽  
Noncommutative Space ◽  
First Order ◽  
Dynamical Properties ◽  
Definition Of ◽  
Space Noncommutativity

The standard lore in noncommutative physics is the use of first order variational description of a dynamical system to probe the space noncommutativity and its consequences in the dynamics in phase space. As the ultimate goal is to understand the inherent space noncommutativity, we propose a variational principle for noncommutative dynamical systems in configuration space, based on results of our previous work [18]. We hope that this variational formulation in configuration space can be of help to elucidate the definition of some global and dynamical properties of classical and quantum noncommutative space.

scholarly journals Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space

2015 ◽  
Vol 22 (04) ◽  
pp. 1550021 ◽  
Author(s):  
Fabio Benatti ◽  
Laure Gouba
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Classical Limit ◽  
Coherent States ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Wigner Functions ◽  
Quantum Mechanical Oscillators ◽  
Mechanical Oscillators ◽  
Space Noncommutativity

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

Phenomenology of noncommutative phase space via the anomalous Zeeman effect in hydrogen atom

2014 ◽  
Vol 29 (31) ◽  
pp. 1450177 ◽  
Author(s):  
Willien O. Santos ◽  
Andre M. C. Souza
Keyword(s):  
Phase Space ◽  
Hydrogen Atom ◽  
Zeeman Effect ◽  
Nonrelativistic Limit ◽  
Noncommutative Phase Space ◽  
G Factor ◽  
First Order ◽  
Anomalous Zeeman Effect ◽  
Space Noncommutativity ◽  
First Order Perturbation

The Hamiltonian describing the anomalous Zeeman effect for the hydrogen atom on noncommutative (NC) phase space is studied using the nonrelativistic limit of the Dirac equation. To preserve gauge invariance, space noncommutativity must be dropped. By using first-order perturbation theory, the correction to the energy is calculated for the case of a weak external magnetic field. We also obtained the orbital and spin g-factors on the NC phase space. We show that the experimental value for the spin g-factor puts an upper bound on the magnitude of the momentum NC parameter of the order of [Formula: see text], 34 μ eV /c. On the other hand, the experimental value for the spin g-factor was used to establish a correction introduced by NC phase space to the presently accepted value of Planck's constant with an uncertainty of 2 part in 1035.

scholarly journals Lagrangian coherent structures and plasma transport processes

2015 ◽  
Vol 81 (5) ◽  
Author(s):  
M. V. Falessi ◽  
F. Pegoraro ◽  
T. J. Schep
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Coherent Structures ◽  
Transport Processes ◽  
Lagrangian Coherent Structures ◽  
Plasma Transport ◽  
Periodic Systems ◽  
Transport Barriers ◽  
Definition Of ◽  
General Time

A dynamical system framework is used to describe transport processes in plasmas embedded in a magnetic field. For periodic systems with one degree of freedom, the Poincaré map provides a splitting of the phase space into regions where particles have different kinds of motion: periodic, quasi-periodic or chaotic. The boundaries of these regions are transport barriers, i.e. a trajectory cannot cross such boundaries throughout the evolution of the system. Lagrangian coherent structures generalize this method to systems with the most general time dependence, splitting the phase space into regions with different qualitative behaviours. This leads to the definition of finite-time transport barriers, i.e. trajectories cannot cross the barrier for a finite amount of time. This methodology can be used to identify fast recirculating regions in the dynamical system and to characterize the transport between them.

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.

scholarly journals HAMILTONIAN AND LAGRANGIAN DYNAMICS IN A NONCOMMUTATIVE SPACE

2003 ◽  
Vol 18 (39) ◽  
pp. 2795-2806 ◽  
Author(s):  
R. P. MALIK
Keyword(s):  
Phase Space ◽  
Quantum Groups ◽  
Physical System ◽  
Hamiltonian Formulation ◽  
Noncommutative Space ◽  
Two Dimensional ◽  
Lagrangian Dynamics ◽  
Symplectic Structures ◽  
First Order ◽  
Tangent Manifold

We discuss the dynamics of a particular two-dimensional (2D) physical system in the four-dimensional (4D) (non-)commutative phase space by exploiting the consistent Hamiltonian and Lagrangian formalisms based on the symplectic structures defined on the 4D (non-)commutative cotangent manifolds. The noncommutativity exists equivalently in the coordinate or the momentum planes embedded in the 4D cotangent manifolds. The signature of this noncommutativity is reflected in the derivation of the first-order Lagrangians where we exploit the most general form of the Legendre transformation defined on the (non-)commutative (co-)tangent manifolds. The second-order Lagrangian, defined on the 4D tangent manifold, turns out to be the same irrespective of the noncommutativity present in the 4D cotangent manifolds for the discussion of the Hamiltonian formulation. A connection with the noncommutativity of the dynamics, associated with the quantum groups on the q-deformed 4D cotangent manifolds, is also pointed out.

A variational principle in relativistic magnetofluid dynamics

1979 ◽  
Vol 21 (3) ◽  
pp. 511-517 ◽  
Author(s):  
I. Merches
Keyword(s):  
Charged Particle ◽  
Variational Principle ◽  
Variational Formulation ◽  
Lagrangian Density ◽  
Equation Of Motion ◽  
Covariant Form ◽  
Charged Fluids ◽  
Maxwell’S Equation ◽  
Definition Of ◽  
Maxwell's Equation

The definition of the generalized antipotential four-vector makes it possible to give a relativistically covariant variational formulation in the dynamics of ideal charged fluids. A special relativistically covariant form of Maxwell's equation is given. The antipotential four-vector does not explicitly appear in the Lagrangian density. The derivation of the equation of motion of a single charged particle is given, to illustrate the theory.

Hamiltonian mechanics

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Configuration Space ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Hamiltonian Mechanics ◽  
Lagrange Equations ◽  
First Order ◽  
Point Transformations ◽  
First Order Differential Equations

This chapter gives a brief overview of Hamiltonian mechanics. The complexity of the Newtonian equations of motion for N interacting bodies led to the development in the late 18th and early 19th centuries of a formalism that reduces these equations to first-order differential equations. This formalism is known as Hamiltonian mechanics. This chapter shows how, given a Lagrangian and having constructed the corresponding Hamiltonian, Hamilton’s equations amount to simply a rewriting of the Euler–Lagrange equations. The feature that makes the Hamiltonian formulation superior is that the dimension of the phase space is double that of the configuration space, so that in addition to point transformations, it is possible to perform more general transformations in order to simplify solving the equations of motion.

scholarly journals A short study of a string on a plane: The energy and the effective mass

2016 ◽  
Vol 31 (17) ◽  
pp. 1650103
Author(s):  
S. V. Talalov
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Effective Mass ◽  
Special Form ◽  
Degrees Of Freedom ◽  
Relativistic String ◽  
Strong Correlations ◽  
Classical Level ◽  
Definition Of ◽  
Extended Galilei Group

We investigate the new special class of the finite string on a plane, after the reduction from the relativistic 4D case. The suggested special form of the phase space allows to define the extended Galilei group as a group of the spacetime symmetry for the considered system. The definition of the energy for the studied non-relativistic string through the Cazimir function of this group is suggested. The concept of the effective mass for the investigated dynamical system is introduced. The appearance of strong correlations between the degrees of freedom even on the classical level is discussed.

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

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