scholarly journals Actions, topological terms and boundaries in first-order gravity: A review

2016 ◽  
Vol 25 (04) ◽  
pp. 1630011 ◽  
Author(s):  
Alejandro Corichi ◽  
Irais Rubalcava-García ◽  
Tatjana Vukašinac
Keyword(s):  
Boundary Conditions ◽  
Symplectic Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Action Principle ◽  
Internal Boundary ◽  
Four Dimensions ◽  
First Order ◽  
Conserved Charges ◽  
Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

scholarly journals Vibration analysis of multi-stepped and multi-damaged parabolic arches using GDQ

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Erasmo Viola ◽  
Marco Miniaci ◽  
Nicholas Fantuzzi ◽  
Alessandro Marzani
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Internal Boundary ◽  
Radius Of Curvature ◽  
Inertia Effects ◽  
Circular Arch

AbstractThis paper investigates the in-plane free vibrations of multi-stepped and multi-damaged parabolic arches, for various boundary conditions. The axial extension, transverse shear deformation and rotatory inertia effects are taken into account. The constitutive equations relating the stress resultants to the corresponding deformation components refer to an isotropic and linear elastic material. Starting from the kinematic hypothesis for the in-plane displacement of the shear-deformable arch, the equations of motion are deduced by using Hamilton’s principle. Natural frequencies and mode shapes are computed using the Generalized Differential Quadrature (GDQ) method. The variable radius of curvature along the axis of the parabolic arch requires, compared to the circular arch, a more complex formulation and numerical implementation of the motion equations as well as the external and internal boundary conditions. Each damage is modelled as a combination of one rotational and two translational elastic springs. A parametric study is performed to illustrate the influence of the damage parameters on the natural frequencies of parabolic arches for different boundary conditions and cross-sections with localizeddamage.Results for the circular arch, derived from the proposed parabolic model with the derivatives of some parameters set to zero, agree well with those published over the past years.

scholarly journals ON BOUNDARY TERMS AND CONFORMAL TRANSFORMATIONS IN CURVED SPACETIMES

2002 ◽  
Vol 11 (05) ◽  
pp. 703-714 ◽  
Author(s):  
R. CASADIO ◽  
A. GRUPPUSO
Keyword(s):  
Hamiltonian Formalism ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Conformal Transformations ◽  
Five Dimensions ◽  
Four Dimensions ◽  
Boundary Terms ◽  
Pure Gravity ◽  
Theories Of Gravity ◽  
Flat Spacetimes

We intend to clarify the interplay between boundary terms and conformal transformations in scalar-tensor theories of gravity. We first consider the action for pure gravity in five dimensions and show that, on compactifing a la Kaluza–Klein to four dimensions, one obtains the correct boundary terms in the Jordan (or String) Frame form of the Brans–Dicke action. Further, we analyze how the boundary terms change under the conformal transformations which lead to the Pauli (or Einstein) frame and to the nonminimally coupled massless scalar field. In particular, we study the behaviour of the total energy in asymptotically flat spacetimes as it results from surface terms in the Hamiltonian formalism.

scholarly journals DIRAC ANALYSIS AND INTEGRABILITY OF GEODESIC EQUATIONS FOR CYLINDRICALLY SYMMETRIC SPACETIMES

2003 ◽  
Vol 12 (08) ◽  
pp. 1431-1444 ◽  
Author(s):  
UGUR CAMCI
Keyword(s):  
Symplectic Structure ◽  
Equations Of Motion ◽  
First Order ◽  
Geodesic Equations ◽  
Cylindrically Symmetric ◽  
Symmetric Spacetimes ◽  
Constraint Analysis ◽  
Stationary Spacetime ◽  
Cylindrically Symmetric Spacetimes ◽  
Geodesic Equations Of Motion

Dirac's constraint analysis and the symplectic structure of geodesic equations are obtained for the general cylindrically symmetric stationary spacetime. For this metric, using the obtained first order Lagrangian, the geodesic equations of motion are integrated, and found some solutions for Lewis, Levi-Civita, and Van Stockum spacetimes.

scholarly journals On quantum-mechanical equations of motion in representation dependent of external sources

2017 ◽  
Vol 32 (32) ◽  
pp. 1750189
Author(s):  
Igor A. Batalin ◽  
Peter M. Lavrov
Keyword(s):  
Normal Form ◽  
Boundary Conditions ◽  
Path Integral ◽  
Equations Of Motion ◽  
Quantum Mechanical ◽  
Motion Equations ◽  
Closed Solution ◽  
Boundary Terms ◽  
External Sources

In the present paper, we consider in detail the aspects of the Heisenberg’s equations of motion, related to their transformation to the representation dependent of external sources. We provide with a closed solution as to the variation-derivative motion equations in the general case of a normal form (symbol) chosen. We show that the action in the path integral does depend actually on a particular choice of a normal symbol. We have determined both the aspects of the latter dependence: the specific boundary conditions for virtual trajectories, and the specific boundary terms in the action.

Effects of Varying Boundary Conditions on the Buried Pipelines

2006 ◽  
Vol 110 ◽  
pp. 183-192 ◽  
Author(s):  
Ouk Sub Lee ◽  
Dong Hyeok Kim
Keyword(s):  
Boundary Conditions ◽  
Probability Of Failure ◽  
Internal Boundary ◽  
Buried Pipeline ◽  
Buried Pipelines ◽  
Safety Level ◽  
First Order ◽  
Reliability Method ◽  
Margin Of Safety ◽  
Corrosion Defects

In this paper, the FORM (first order reliability method) has been employed to estimate the probability of failure for the buried pipeline degraded by corrosion defects. The estimated results are used to assess the reliability of buried pipeline exposed to varying external and internal boundary conditions corresponding to a required target safety level. Furthermore, the effects of distribution types of random variables affecting the strength of buried pipelines on the probability of failure and the reliability of the buried pipeline are systematically investigated. The plant-engineers should be informed about the margin of safety level of existing pipelines corresponding to the target safety level to assess the integrity of the corroded pipeline under operation.

ON THE FIRST ORDER COVARIANT HAMILTONIAN FORMALISM ON GROUP MANIFOLD

1990 ◽  
Vol 05 (04) ◽  
pp. 725-746 ◽  
Author(s):  
A. FOUSSATS ◽  
O. ZANDRON
Keyword(s):  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Covariant Formalism ◽  
Field Equations ◽  
Bianchi Identities ◽  
First Order ◽  
Group Manifold

By considering two explicit examples D=6 and D=11 supergravities we show the applicability, and check the validity of the first order canonical covariant formalism on group manifold. In each case we find the primary constraints, the total Hamiltonian and the field equations of motion. Moreover, we show how the Bianchi identities appear in the covariant Hamiltonian formalism.

scholarly journals Covariant phase space with boundaries

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Daniel Harlow ◽  
Jie-qiang Wu
Keyword(s):  
Phase Space ◽  
Boundary Conditions ◽  
Hamiltonian Dynamics ◽  
Equations Of Motion ◽  
Field Theories ◽  
Boundary Terms ◽  
Space Formalism ◽  
Spatial Boundary ◽  
Original Construction ◽  
Space Method

Abstract The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically treat total derivatives and boundary terms, which has led to some confusion about how exactly to apply the formalism in the presence of boundaries. In particular the original construction of the canonical Hamiltonian relies on the assumed existence of a certain boundary quantity “B”, whose physical interpretation has not been clear. We here give an algorithmic procedure for applying the covariant phase space formalism to field theories with spatial boundaries, from which the term in the Hamiltonian involving B emerges naturally. Our procedure also produces an additional boundary term, which was not present in the original literature and which so far has only appeared implicitly in specific examples, and which is already nonvanishing even in general relativity with sufficiently permissive boundary conditions. The only requirement we impose is that at solutions of the equations of motion the action is stationary modulo future/past boundary terms under arbitrary variations obeying the spatial boundary conditions; from this the symplectic structure and the Hamiltonian for any diffeomorphism that preserves the theory are unambiguously constructed. We show in examples that the Hamiltonian so constructed agrees with previous results. We also show that the Poisson bracket on covariant phase space directly coincides with the Peierls bracket, without any need for non-covariant intermediate steps, and we discuss possible implications for the entropy of dynamical black hole horizons.

Model of vortex flow of charge in a vessel with turbine impeller

1987 ◽  
Vol 52 (8) ◽  
pp. 1888-1904
Author(s):  
Miloslav Hošťálek ◽  
Ivan Fořt
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Theoretical Data ◽  
Eddy Diffusion ◽  
Quantitative Agreement ◽  
Creeping Flow ◽  
Model Parameters ◽  
Mean Flow ◽  
Two Dimensional Flow ◽  
Vorticity Transport

A theoretical model is described of the mean two-dimensional flow of homogeneous charge in a flat-bottomed cylindrical tank with radial baffles and six-blade turbine disc impeller. The model starts from the concept of vorticity transport in the bulk of vortex liquid flow through the mechanism of eddy diffusion characterized by a constant value of turbulent (eddy) viscosity. The result of solution of the equation which is analogous to the Stokes simplification of equations of motion for creeping flow is the description of field of the stream function and of the axial and radial velocity components of mean flow in the whole charge. The results of modelling are compared with the experimental and theoretical data published by different authors, a good qualitative and quantitative agreement being stated. Advantage of the model proposed is a very simple schematization of the system volume necessary to introduce the boundary conditions (only the parts above the impeller plane of symmetry and below it are distinguished), the explicit character of the model with respect to the model parameters (model lucidity, low demands on the capacity of computer), and, in the end, the possibility to modify the given model by changing boundary conditions even for another agitating set-up with radially-axial character of flow.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

scholarly journals Non-Riemannian gravity actions from double field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
A. D. Gallegos ◽  
U. Gürsoy ◽  
S. Verma ◽  
N. Zinnato
Keyword(s):  
Field Theory ◽  
Quantum Gravity ◽  
Condensed Matter ◽  
Equations Of Motion ◽  
Technical Note ◽  
Action Principle ◽  
Relativistic Symmetry ◽  
Gravitational Theories ◽  
Double Field Theory

Abstract Non-Riemannian gravitational theories suggest alternative avenues to understand properties of quantum gravity and provide a concrete setting to study condensed matter systems with non-relativistic symmetry. Derivation of an action principle for these theories generally proved challenging for various reasons. In this technical note, we employ the formulation of double field theory to construct actions for a variety of such theories. This formulation helps removing ambiguities in the corresponding equations of motion. In particular, we embed Torsional Newton-Cartan gravity, Carrollian gravity and String Newton-Cartan gravity in double field theory, derive their actions and compare with the previously obtained results in literature.

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