Decomposition of Vector Fields Beyond Problems of First Order and Their Applications

2017 ◽  
pp. 205-219
Author(s):  
Wieland Reich ◽  
Mario Hlawitschka ◽  
Gerik Scheuermann
Keyword(s):  
Vector Fields ◽  
First Order

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.

scholarly journals CONSISTENT INTERACTIONS BETWEEN GAUGE FIELDS AND LOCAL BRST COHOMOLOGY: THE EXAMPLE OF YANG-MILLS MODELS

1994 ◽  
Vol 03 (01) ◽  
pp. 139-144 ◽  
Author(s):  
G. BARNICH ◽  
M. HENNEAUX ◽  
R. TATAR
Keyword(s):  
Vector Fields ◽  
Jacobi Identity ◽  
Second Order ◽  
Gauge Fields ◽  
Coupling Constant ◽  
Yang Mills ◽  
First Order ◽  
Gauge Symmetries ◽  
Brst Cohomology ◽  
Structure Constants

Recent results on the cohomological reformulation of the problem of consistent interactions between gauge fields are illustrated in the case of the Yang-Mills models. By evaluating the local BRST cohomology through descent equation techniques, it is shown (i) that there is a unique local, Poincaré invariant cubic vertex for free gauge vector fields which preserves the number of gauge symmetries to first order in the coupling constant; and (ii) that consistency to second order in the coupling constant requires the structure constants appearing in the cubic vertex to fulfill the Jacobi identity. The known uniqueness of the Yang-Mills coupling is therefore rederived through cohomological arguments.

scholarly journals Variational formulas for submanifolds of fixed degree

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Giovanna Citti ◽  
Gianmarco Giovannardi ◽  
Manuel Ritoré
Keyword(s):  
Vector Fields ◽  
Riemannian Metric ◽  
Curvature Operator ◽  
Lagrange Equations ◽  
Fixed Degree ◽  
Third Order ◽  
First Order ◽  
Area Functional ◽  
Mean Curvature Operator ◽  
Graded Manifold

AbstractWe consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. As in the case of singular curves in sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the Euler–Lagrange equations. The resulting mean curvature operator can be of third order.

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.

Bifurcation of Limit Cycles from the Center of a Family of Cubic Polynomial Vector Fields

2018 ◽  
Vol 28 (05) ◽  
pp. 1850063 ◽  
Author(s):  
Shiyou Sui ◽  
Liqin Zhao
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Abelian Integral ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Cubic Polynomial

In this paper, we consider the number of zeros of Abelian integral for the system [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are arbitrary polynomials of degree [Formula: see text]. We obtain that [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the maximum number of limit cycles bifurcating from the period annulus up to the first order in [Formula: see text]. So, the bounds for [Formula: see text] or [Formula: see text], [Formula: see text], [Formula: see text] are exact.

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