1.5 Covariant Derivation

2011 ◽  
Keyword(s):  
Covariant Derivation

scholarly journals Manifold Calculus in System Theory and Control—Fundamentals and First-Order Systems

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2092
Author(s):  
Simone Fiori
Keyword(s):  
Control Systems ◽  
Linear Systems ◽  
Linear Control Systems ◽  
First Order ◽  
Curved Manifolds ◽  
Non Linear ◽  
Covariant Derivation ◽  
Non Linear Systems ◽  
And Control

The aim of the present tutorial paper is to recall notions from manifold calculus and to illustrate how these tools prove useful in describing system-theoretic properties. Special emphasis is put on embedded manifold calculus (which is coordinate-free and relies on the embedding of a manifold into a larger ambient space). In addition, we also consider the control of non-linear systems whose states belong to curved manifolds. As a case study, synchronization of non-linear systems by feedback control on smooth manifolds (including Lie groups) is surveyed. Special emphasis is also put on numerical methods to simulate non-linear control systems on curved manifolds. The present tutorial is meant to cover a portion of the mentioned topics, such as first-order systems, but it does not cover topics such as covariant derivation and second-order dynamical systems, which will be covered in a subsequent tutorial paper.

scholarly journals The Spectrum of Teleparallel Gravity

Universe ◽  
2019 ◽  
Vol 5 (3) ◽  
pp. 80 ◽  
Author(s):  
Tomi Koivisto ◽  
Georgios Tsimperis
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Quadratic Form ◽  
Degrees Of Freedom ◽  
Arbitrary Dimension ◽  
Scalar Potential ◽  
Ultra Violet ◽  
Teleparallel Gravity ◽  
Frame Field ◽  
Covariant Derivation

The observer’s frame is the more elementary description of the gravitational field than the metric. The most general covariant, even-parity quadratic form for the frame field in arbitrary dimension generalises the New General Relativity by nine functions of the d’Alembertian operator. The degrees of freedom are clarified by a covariant derivation of the propagator. The consistent and viable models can incorporate an ultra-violet completion of the gravity theory, an additional polarisation of the gravitational wave, and the dynamics of a magnetic scalar potential.

scholarly journals On Kilmister's Conditions for the Existence of Linear Integrals of Dynamical Systems

1965 ◽  
Vol 14 (3) ◽  
pp. 243-244 ◽  
Author(s):  
R. H. Boyer
Keyword(s):  
Dynamical Systems ◽  
Singular Integral ◽  
First Integral ◽  
Killing Field ◽  
Summation Convention ◽  
Covariant Derivation ◽  
Image Position ◽  
Generally Covariant ◽  
Linear Integrals

Kilmister (1) has considered dynamical systems specified by coordinates q( = 1, 2, , n) and a Lagrangian(with summation convention). He sought to determine generally covariant conditions for the existence of a first integral, , linear in the velocities. He showed that it is not, as is usually stated, necessary that there must exist an ignorable coordinate (equivalently, that b must be a Killing field:where covariant derivation is with respect to a). On the contrary, a singular integral, in the sense that for all time if satisfied initially, need not be accompanied by an ignorable coordinate.

A gauge-covariant derivation of the strong-field approximation

Laser Physics ◽  
2009 ◽  
Vol 19 (8) ◽  
pp. 1621-1625 ◽  
Author(s):  
W. Becker ◽  
D. B. Milošević
Keyword(s):  
Strong Field ◽  
Field Approximation ◽  
Covariant Derivation

A GENERALIZATION OF GAUGE COUPLING CONSTANTS TO GAUGE COUPLING FIELDS

1990 ◽  
Vol 05 (21) ◽  
pp. 1633-1637 ◽  
Author(s):  
LORA NIKOLOVA ◽  
V.A. RIZOV
Keyword(s):  
Gauge Group ◽  
Gauge Coupling ◽  
Natural Generalization ◽  
Coupling Constants ◽  
Inner Product ◽  
Coupling Constant ◽  
Derivation Operator ◽  
Covariant Derivation ◽  
Gauge Coupling Constant

A natural generalization of the notion of the gauge coupling constant appearing in the covariant derivation operator is obtained by replacing it with a field Γ which takes values in the linear hermitian invertible mappings [Formula: see text] ([Formula: see text] is the Lie algebra of the gauge group G equipped with a G-invariant inner product). In this case the eigenvalues of Γ(x) for each point x from the space-time take the role of the usual single gauge coupling constant.

scholarly journals Quantization of Gravity and Finite Temperature Effects

Particles ◽  
2021 ◽  
Vol 4 (4) ◽  
pp. 468-488
Author(s):  
I. Y. Park
Keyword(s):  
Perturbation Theory ◽  
Finite Temperature ◽  
Quantum Electrodynamics ◽  
Scattering Amplitude ◽  
Asymptotic Freedom ◽  
Loop Analysis ◽  
Optimal Perturbation ◽  
Quantization Of Gravity ◽  
Gauge Fixing ◽  
Covariant Derivation

Gravity is perturbatively renormalizable for the physical states which can be conveniently defined via foliation-based quantization. In recent sequels, one-loop analysis was explicitly carried out for Einstein-scalar and Einstein-Maxwell systems. Various germane issues and all-loop renormalizability have been addressed. In the present work we make further progress by carrying out several additional tasks. Firstly, we present an alternative 4D-covariant derivation of the physical state condition by examining gauge choice-independence of a scattering amplitude. To this end, a careful dichotomy between the ordinary, and large gauge symmetries is required and appropriate gauge-fixing of the ordinary symmetry must be performed. Secondly, vacuum energy is analyzed in a finite-temperature setup. A variant optimal perturbation theory is implemented to two-loop. The renormalized mass determined by the optimal perturbation theory turns out to be on the order of the temperature, allowing one to avoid the cosmological constant problem. The third task that we take up is examination of the possibility of asymptotic freedom in finite-temperature quantum electrodynamics. In spite of the debates in the literature, the idea remains reasonable.

Covariant derivation of the Maxwell equations

Physica ◽  
1965 ◽  
Vol 31 (12) ◽  
pp. 1713-1727 ◽  
Author(s):  
S.R. De Groot ◽  
L.G. Suttorp
Keyword(s):  
Maxwell Equations ◽  
Covariant Derivation
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