Parallel transport and geodesics

2021 ◽  
pp. 160-177
Author(s):  
Andrew M. Steane
Keyword(s):  
Parallel Transport ◽  
Equation Of Motion ◽  
Geodesic Equation ◽  
Gravitational Redshift ◽  
Conserved Quantities ◽  
Lagrange Method ◽  
Killing Vectors ◽  
Christoffel Symbols

The mathematics of parallel transport and of affine and metric geodesics is presented. The geodesic equation is obtained in several different ways, bringing out its role both as a geometric statement and as an equation of motion. The Euler-Lagrange method to find metric geodesics, and hence Christoffel symbols, is explained. The role of conserved quantities is discussed. Killing’s equation and Killing vectors are introduced. Fermi-Walker transport (the non-rotating freely falling cabin) is defined and discussed. Gravitational redshift is calculated, first in general and then in specific cases.

Metric Spaces and Geodesic Motion

2019 ◽  
pp. 355-384
Author(s):  
David D. Nolte
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Parallel Transport ◽  
Variational Calculus ◽  
Geodesic Equation ◽  
Geodesic Curve ◽  
Geodesic Motion ◽  
Metric Tensor ◽  
Christoffel Symbols ◽  
Derivatives Of

The metric tensor uniquely defines the geometric properties of a metric space, while differential geometry is concerned with the derivatives of vectors and tensors within the metric space. Derivatives of vector quantities include the derivatives of basis vectors, which lead to Christoffel symbols that are directly connected to the metric tensor. This connection between tensor derivatives and the metric tensor provides the tools necessary to define geodesic curves. The geodesic equation is derived from variational calculus and from parallel transport. Geodesic motion is the trajectory of a particle through a metric space defined by an action metric that includes a potential function. In this way, dynamics converts to geometry as a trajectory in a potential converts to a geodesic curve through an appropriately defined metric space.

Conservation laws

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Angular Momentum ◽  
Conservation Laws ◽  
Equations Of Motion ◽  
Geodesic Equation ◽  
First Integrals ◽  
Spacetime Symmetries ◽  
Conserved Quantities ◽  
Killing Vectors ◽  
Invariance Properties ◽  
Matter Energy

This chapter studies how the ‘spacetime symmetries’ can generate first integrals of the equations of motion which simplify their solution and also make it possible to define conserved quantities, or ‘charges’, characterizing the system. As already mentioned in the introduction to matter energy–momentum tensors in Chapter 3, the concepts of energy, momentum, and angular momentum are related to the invariance properties of the solutions of the equations of motion under spacetime translations or rotations. The chapter explores these in greater detail. It first turns to isometries and Killing vectors. The chapter then examines the first integrals of the geodesic equation, and Noether charges.

scholarly journals A Monolithic Approach of Fluid–Structure Interaction by Discrete Mechanics

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 95
Author(s):  
Stéphane Vincent ◽  
Jean-Paul Caltagirone
Keyword(s):  
Solid Mechanics ◽  
Mechanical Energy ◽  
Fluid Structure Interaction ◽  
Equation Of Motion ◽  
Discrete Equation ◽  
Discrete Mechanics ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Vector Potentials

The unification of the laws of fluid and solid mechanics is achieved on the basis of the concepts of discrete mechanics and the principles of equivalence and relativity, but also the Helmholtz–Hodge decomposition where a vector is written as the sum of divergence-free and curl-free components. The derived equation of motion translates the conservation of acceleration over a segment, that of the intrinsic acceleration of the material medium and the sum of the accelerations applied to it. The scalar and vector potentials of the acceleration, which are the compression and shear energies, give the discrete equation of motion the role of conservation law for total mechanical energy. Velocity and displacement are obtained using an incremental time process from acceleration. After a description of the main stages of the derivation of the equation of motion, unique for the fluid and the solid, the cases of couplings in simple shear and uniaxial compression of two media, fluid and solid, make it possible to show the role of discrete operators and to find the theoretical results. The application of the formulation is then extended to a classical validation case in fluid–structure interaction.

Visible-light Photoresponse of Nitrogen-doped TiO2: Excited State Studies Using Time-dependent Density Functional Theory and Equation-of-Motion Coupled Cluster Methods

2010 ◽  
Vol 1263 ◽  
Author(s):  
Niranjan Govind ◽  
Roger Rousseau ◽  
Amity Andersen ◽  
Karol Kowalski
Keyword(s):  
Excited State ◽  
Density Functional ◽  
Equation Of Motion ◽  
Coupled Cluster ◽  
Functional Theory ◽  
Experimental Findings ◽  
High Level ◽  
Cluster Methods ◽  
Compare And Contrast

AbstractTo shed light on the nature of the electronic states at play in N-doped TiO2 nanoparticles, we have performed detailed ground and excited state calculations on pure and N-doped TiO2 rutile using an embedding model. We have validated our model by comparing ground-state embedded results with those obtained from periodic DFT calculations. Our results are consistent with periodic calculations. Using this embedding model we have performed B3LYP based TDDFT calculations of the excited state spectrum. We have also studied the lowest excitations using high-level equation-of-motion coupled cluster (EOMCC) approaches involving all single and inter-band double excitations. We compare and contrast the nature of the excitations in detail for the pure and doped systems using these calculations. Our calculations indicate a lowering of the bandgap and confirm the role of the N3- states on the UV/Vis spectrum of N-doped TiO2 rutile supported by experimental findings.

scholarly journals Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jacky Cresson ◽  
Fernando Jiménez ◽  
Sina Ober-Blöbaum
Keyword(s):  
Fractional Calculus ◽  
Calculus Of Variations ◽  
Fractional Derivatives ◽  
Dissipative Systems ◽  
Alternative Formulation ◽  
Conserved Quantities ◽  
Lagrange Equations ◽  
Discrete Counterpart ◽  
Fractional Calculus Of Variations

<p style='text-indent:20px;'>We prove a Noether's theorem of the first kind for the so-called <i>restricted fractional Euler-Lagrange equations</i> and their discrete counterpart, introduced in [<xref ref-type="bibr" rid="b26">26</xref>,<xref ref-type="bibr" rid="b27">27</xref>], based in previous results [<xref ref-type="bibr" rid="b11">11</xref>,<xref ref-type="bibr" rid="b35">35</xref>]. Prior, we compare the restricted fractional calculus of variations to the <i>asymmetric fractional calculus of variations</i>, introduced in [<xref ref-type="bibr" rid="b14">14</xref>], and formulate the restricted calculus of variations using the <i>discrete embedding</i> approach [<xref ref-type="bibr" rid="b12">12</xref>,<xref ref-type="bibr" rid="b18">18</xref>]. The two theories are designed to provide a variational formulation of dissipative systems, and are based on modeling irreversbility by means of fractional derivatives. We explicit the role of time-reversed solutions and causality in the restricted fractional calculus of variations and we propose an alternative formulation. Finally, we implement our results for a particular example and provide simulations, actually showing the constant behaviour in time of the discrete conserved quantities outcoming the Noether's theorems.</p>

scholarly journals Symmetry Constrained Decoherence of Conditional Expectation Values

Universe ◽  
2019 ◽  
Vol 5 (2) ◽  
pp. 46 ◽  
Author(s):  
M. Mohammady ◽  
Alessandro Romito
Keyword(s):  
Conditional Expectation ◽  
Sufficient Conditions ◽  
Specific Model ◽  
Quantum Mechanical ◽  
Conserved Quantities ◽  
System State ◽  
Quantum Thermodynamics ◽  
Expectation Values ◽  
Thermal Measurements

Conditional expectation values of quantum mechanical observables reflect unique non-classical correlations, and are generally sensitive to decoherence. We consider the circumstances under which such sensitivity to decoherence is removed, namely, when the measurement process is subjected to conservation laws. Specifically, we address systems with additive conserved quantities and identify sufficient conditions for the system state such that its coherence plays no role in the conditional expectation values of observables that commute with the conserved quantity. We discuss our findings for a specific model where the system-detector coupling is given by the Jaynes-Cummings interaction, which is relevant to experiments tracking trajectories of qubits in cavities. Our results clarify, among others, the role of coherence in thermal measurements in current architectures for quantum thermodynamics experiments.

scholarly journals Classical Integrability of Chiral QCD2 and Classical Curves

1997 ◽  
Vol 12 (32) ◽  
pp. 2445-2453 ◽  
Author(s):  
Robert De Mello Koch ◽  
João P. Rodrigues
Keyword(s):  
Large Class ◽  
Gauge Group ◽  
Infinite Number ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Conserved Quantities ◽  
Large N ◽  
Fermionic Fields

In this letter, classical chiral QCD 2 is studied in the lightcone gauge A-=0. The once integrated equation of motion for the current is shown to be of the Lax form, which demonstrates an infinite number of conserved quantities. Specializing to gauge group SU(2), we show that solutions to the classical equations of motion can be identified with a very large class of curves. We demonstrate this correspondence explicitly for two solutions. The classical fermionic fields associated with these currents are then obtained. Finally, we conclude by showing how 't Hooft's large-N solution is obtained from one of our solutions.

scholarly journals Fractional Killing-Yano Tensors and Killing Vectors Using the Caputo Derivative in Some One- and Two-Dimensional Curved Space

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Ehab Malkawi ◽  
D. Baleanu
Keyword(s):  
Fractional Calculus ◽  
Exact Solutions ◽  
Caputo Derivative ◽  
Dimensional Space ◽  
Two Dimensional ◽  
Curved Space ◽  
Killing Vectors ◽  
Constant Of Motion ◽  
Christoffel Symbols

The classical free Lagrangian admitting a constant of motion, in one- and two-dimensional space, is generalized using the Caputo derivative of fractional calculus. The corresponding metric is obtained and the fractional Christoffel symbols, Killing vectors, and Killing-Yano tensors are derived. Some exact solutions of these quantities are reported.

HIGHER ORDER SYMMETRIES AND THE KOUTRAS ALGORITHM

2002 ◽  
Vol 11 (03) ◽  
pp. 337-351 ◽  
Author(s):  
G. AMERY ◽  
S. D. MAHARAJ
Keyword(s):  
Conformal Factor ◽  
Higher Order ◽  
Conformal Killing ◽  
Killing Vectors ◽  
Killing Tensors ◽  
Conformal Killing Vectors

We investigate the form of Killing tensors, constructed from conformal Killing vectors of a given spacetime (M, g), by utilizing the Koutras algorithm. As an example we find irreducible Killing tensors in Robertson–Walker spacetimes. A number of theorems are given for the existence of Killing tensors in the conformally related spacetime [Formula: see text]. The form of the conformally related Killing tensors are explicitly determined. The conditions on the conformal factor Ω relating the two spacetimes (M, g) and [Formula: see text] are determined for the existence of the tensors. Also we briefly consider the role of recurrent vectors, inheriting conformal vectors and gradient conformal vectors in building Killing tensors.

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