Special bracket versus Jacobi bracket on the classical phase space of general relativistic test particle

2014 ◽  
Vol 11 (07) ◽  
pp. 1460020 ◽  
Author(s):  
Josef Janyška
Keyword(s):  
Phase Space ◽  
Test Particle ◽  
Jet Space ◽  
Hamiltonian Approach ◽  
One Dimensional ◽  
Lie Bracket ◽  
General Relativistic ◽  
Jacobi Bracket ◽  
Classical Phase Space ◽  
Phase Functions

The classical phase space of general relativistic classical test particle (here called, for short, "phase space") is defined as the first jet space of motions regarded as timelike one-dimensional submanifolds of spacetime. By the projectability assumption, we define the subsheaf of special phase functions with a special Lie bracket and we compare the Lie algebra of special phase functions with the structures obtained on the phase space by the standard Hamiltonian approach.

Remarks on infinitesimal symmetries of geometrical structures of the classical phase space of general relativistic test particle

2015 ◽  
Vol 12 (08) ◽  
pp. 1560020 ◽  
Author(s):  
Josef Janyška
Keyword(s):  
Electromagnetic Field ◽  
Phase Space ◽  
Test Particle ◽  
Contact Structure ◽  
Jet Space ◽  
Geometrical Structures ◽  
Lorentzian Metric ◽  
Dimensional Phase Space ◽  
General Relativistic ◽  
Classical Phase Space

The phase space of general relativistic test particle is defined as the 1-jet space of motions. A Lorentzian metric and an electromagnetic field define the joined almost-cosymplectic-contact structure on the odd-dimensional phase space. In this paper, we study infinitesimal symmetries (ISs) of this phase structure. We prove that there are no hidden ISs.

scholarly journals GEOMETRIC STRUCTURES OF THE CLASSICAL GENERAL RELATIVISTIC PHASE SPACE

2008 ◽  
Vol 05 (05) ◽  
pp. 699-754 ◽  
Author(s):  
JOSEF JANYŠKA ◽  
MARCO MODUGNO
Keyword(s):  
Electromagnetic Field ◽  
Phase Space ◽  
Relativistic Particle ◽  
Jet Space ◽  
Geometric Structures ◽  
Geometric Properties ◽  
Geometric Conditions ◽  
General Relativistic ◽  
Relativistic Phase

This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1-dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialize these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.

QUANTUM DENSITY OF PROBABILITY AT THE CLASSICAL PECULIAR POINT

1996 ◽  
Vol 10 (11) ◽  
pp. 1285-1291
Author(s):  
L. BUONANNO ◽  
M. RENNA ◽  
I.P. PAVLOTSKY
Keyword(s):  
Phase Space ◽  
Particle Dynamics ◽  
Classical Particle ◽  
Dimensional System ◽  
Rectilinear Motion ◽  
Initial Condition ◽  
Speed Of Light ◽  
Singular Surfaces ◽  
One Dimensional ◽  
Classical Phase Space

The so-called no-interaction theorem of D.G. Currie, T.F. Jordan, E.C. Sudarshan, H. Leutwyler, G. Marmo and N. Mukunda makes it possible to construct relativistic quasi-classical particle dynamics in the post-Galilean approximation only.1−4 In this approximation the Lagrangians are singular on some surfaces of the phase space. The dynamical properties are essentially peculiar on the singular surfaces.5−8 In the particular case of the rectilinear motion of two electrons the peculiar point appears when the distance between the particles r=r0, where r0=e2/mc2 (the so-called “radius of an electron”). Here m and e are respectively the mass and the charge of the electron, c is the speed of light. In this paper it is shown that in the simple case of a one-dimensional system of two electrons with the symmetrical initial condition v1=−v2 (v1 and v2 are the velocities of the particles), the density of probability tends to zero when the distance between electrons tends to r0. In other words, the point of the classical phase-space, which cannot be crossed by the trajectory of the system, reflects at the point where the corresponding quantum system has the vanishing probability.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Beyond the Schwarzschild Metric

2018 ◽  
Author(s):  
David M. Wittman
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Gravitational Waves ◽  
Test Particle ◽  
Direct Detection ◽  
Relativistic Effects ◽  
Big Bang ◽  
Clusters Of Galaxies ◽  
General Relativistic ◽  
The Universe

General relativity explains much more than the spacetime around static spherical masses.We briefly assess general relativity in the larger context of physical theories, then explore various general relativistic effects that have no Newtonian analog. First, source massmotion gives rise to gravitomagnetic effects on test particles.These effects also depend on the velocity of the test particle, which has substantial implications for orbits around black holes to be further explored in Chapter 20. Second, any changes in the sourcemass ripple outward as gravitational waves, and we tell the century‐long story from the prediction of gravitational waves to their first direct detection in 2015. Third, the deflection of light by galaxies and clusters of galaxies allows us to map the amount and distribution of mass in the universe in astonishing detail. Finally, general relativity enables modeling the universe as a whole, and we explore the resulting Big Bang cosmology.

scholarly journals Hamiltonian Approach to Magnetic Fields with Toroidal Surfaces

1985 ◽  
Vol 40 (10) ◽  
pp. 959-967
Author(s):  
A. Salat
Keyword(s):  
Magnetic Field ◽  
Hamiltonian System ◽  
Magnetic Fields ◽  
Constant Pressure ◽  
Time Dependent ◽  
Hamiltonian Approach ◽  
One Dimensional ◽  
Mhd Equilibrium ◽  
Magnetic Surfaces ◽  
Rotational Transform

The equivalence of magnetic field line equations to a one-dimensional time-dependent Hamiltonian system is used to construct magnetic fields with arbitrary toroidal magnetic surfaces I = const. For this purpose Hamiltonians H which together with their invariants satisfy periodicity constraints have to be known. The choice of H fixes the rotational transform η(I). Arbitrary axisymmetric fields, and nonaxisymmetric fields with constant η(I) are considered in detail.Configurations with coinciding magnetic and current density surfaces are obtained. The approach used is not well suited, however, to satisfying the additional MHD equilibrium condition of constant pressure on magnetic surfaces.

Classical nonlinearity and quantum decay: The effect of classical phase-space structures

2001 ◽  
Vol 64 (5) ◽  
Author(s):  
Yosef Ashkenazy ◽  
Luca Bonci ◽  
Jacob Levitan ◽  
Roberto Roncaglia
Keyword(s):  
Phase Space ◽  
Space Structures ◽  
Classical Phase Space

One-dimensional coherent-state representation on a circle in phase space

1994 ◽  
Vol 50 (5) ◽  
pp. 4293-4297 ◽  
Author(s):  
P. Domokos ◽  
P. Adam ◽  
J. Janszky
Keyword(s):  
Phase Space ◽  
Coherent State ◽  
State Representation ◽  
One Dimensional
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