scholarly journals Higher-Stage Noether Identities and Second Noether Theorems

2015 ◽  
Vol 2015 ◽  
pp. 1-19
Author(s):  
G. Sardanashvily
Keyword(s):  
Regularity Condition ◽  
Chain Complex ◽  
Higher Order ◽  
Boundary Operator ◽  
Lagrangian System ◽  
Brst Operator ◽  
Gauge Symmetries ◽  
Degenerate Lagrangian

The direct and inverse second Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. Such Lagrangian theory is characterized by a hierarchy of nontrivial higher-stage Noether identities which is described in the homology terms. If a certain homology regularity condition holds, one can associate with a reducible degenerate Lagrangian the exact Koszul–Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete nontrivial Noether and higher-stage Noether identities. The second Noether theorems associate with the above-mentioned Koszul–Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. If gauge symmetries are algebraically closed, this operator is extended to the nilpotent BRST operator which brings the above-mentioned cochain sequence into the BRST complex and provides a BRST extension of an original Lagrangian.

NOETHER IDENTITIES OF A DIFFERENTIAL OPERATOR: THE KOSZUL–TATE COMPLEX

2005 ◽  
Vol 02 (05) ◽  
pp. 873-886 ◽  
Author(s):  
G. SARDANASHVILY
Keyword(s):  
Differential Operator ◽  
Fiber Bundle ◽  
Differential Operators ◽  
Chain Complex ◽  
Boundary Operator ◽  
Lagrangian System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Given a generic Lagrangian system, its Euler–Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condition holds, one can associate to a differential operator the exact chain complex with the boundary operator whose nilpotency restarts all the Noether identities characterizing the degeneracy of an original differential operator.

The Impulsive Action Integral for Rigid-Body Mechanical Systems With Impacts

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Kerim Yunt
Keyword(s):  
Rigid Body ◽  
Regularity Condition ◽  
Mechanical Systems ◽  
Momentum Balance ◽  
Lagrangian System ◽  
Analytical Mechanics ◽  
Impulsive Action ◽  
Action Integral ◽  
Finite Dimensional ◽  
Stationarity Conditions

There is a missing link in analytical mechanics which shows that general impactive processes are obtained by extremizing some sort of action integral for which momentum and energy are not necessarily conserved. In this work, the conditions under which general nonconserving impacts become a part of an extremizing solution for mechanical systems, which are scleronomic (not explicitly time depending) and holonomic, are investigated. The stationarity conditions of an impulsive action integral are investigated and the main theorem is proven. The general momentum balance and the total energy change over a collisional impact for a mechanical scleronomic holonomic finite-dimensional Lagrangian system are obtained in the form of stationarity conditions of a modified action integral under a regularity condition on the impactive transition sets.

On BVPS for Strongly Elliptic Systems with Higher Order Boundary Conditions

2007 ◽  
Vol 14 (1) ◽  
pp. 145-167
Author(s):  
Flavia Lanzara
Keyword(s):  
Boundary Conditions ◽  
Explicit Formula ◽  
Singular Integral ◽  
Elliptic Systems ◽  
Higher Order ◽  
Boundary Operator ◽  
Compatibility Conditions ◽  
Layer Potential ◽  
Integral System ◽  
Regular Type

Abstract We consider BVPs for strongly elliptic systems of order 2𝑙 with the boundary conditions of order 𝑙 + 𝑛, 𝑛 ⩾ 0. By representing the solution by means of a simple layer potential of order 𝑛 and by imposing the boundary conditions, we get a singular integral system which is of regular type if and only if the boundary operator satisfies the Lopatinskiĭ condition and which can be solved if suitable compatibility conditions are satisfied. An explicit formula for computing the index of the BVP is given.

scholarly journals Higher Order Coercive Inequalities

2021 ◽  
Author(s):  
Yifu Wang ◽  
Boguslaw Zegarlinski
Keyword(s):  
Regularity Condition ◽  
Higher Order ◽  
Probability Measures ◽  
Class Probability

AbstractWe study the higher order q- Poincaré and other coercive inequalities for a class probability measures satisfying Adam’s regularity condition.

GRADED LAGRANGIAN FORMALISM

2013 ◽  
Vol 10 (05) ◽  
pp. 1350016 ◽  
Author(s):  
G. SARDANASHVILY
Keyword(s):  
Classical Field Theory ◽  
General Setting ◽  
Classical Field ◽  
Lagrangian Formalism ◽  
Lagrangian Systems ◽  
Comprehensive Description ◽  
Variational Bicomplex ◽  
Gauge Symmetries ◽  
Degenerate Lagrangian ◽  
Graded Manifolds

Graded Lagrangian formalism in terms of a Grassmann-graded variational bicomplex on graded manifolds is developed in a very general setting. This formalism provides the comprehensive description of reducible degenerate Lagrangian systems, characterized by hierarchies of non-trivial higher-order Noether identities and gauge symmetries. This is a general case of classical field theory and Lagrangian non-relativistic mechanics.

Inequivalent Lagrangians for the damped harmonic oscillator

2004 ◽  
Vol 82 (7) ◽  
pp. 561-567 ◽  
Author(s):  
S Ghosh ◽  
J Shamanna ◽  
B Talukdar
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Higher Order ◽  
Space Structure ◽  
Phase Space Structure ◽  
Damped Harmonic Oscillator ◽  
Constants Of Motion ◽  
Order Degenerate ◽  
Degenerate Lagrangian ◽  
Dynamical Origin

A constant of the motion, in addition to what exists in the literature, is presented for the damped harmonic oscillator and its dynamical origin is investigated. These two constants of motion are used to construct expressions for a hierarchy of inequivalent Lagrangians. It is shown that each inequivalent Lagrangian may be related to a higher order degenerate Lagrangian. The hierarchical Lagrangians tend to pose some characteristic problems for discussing the corresponding phase-space structure. PACS Nos.: 47.20.Ky, 42.81.Dp

scholarly journals The KT-BRST Complex of a Degenerate Lagrangian System

2008 ◽  
Vol 83 (3) ◽  
pp. 237-252 ◽  
Author(s):  
D. Bashkirov ◽  
G. Giachetta ◽  
L. Mangiarotti ◽  
G. Sardanashvily
Keyword(s):  
Lagrangian System ◽  
Degenerate Lagrangian

scholarly journals A note on higher-order vertices of higher-spin fields in flat and (A)dS space

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Euihun Joung ◽  
Massimo Taronna
Keyword(s):  
Arbitrary Number ◽  
Flat Space ◽  
Higher Order ◽  
Yang Mills ◽  
Homogeneous Solutions ◽  
Gauge Symmetries ◽  
Higher Spin ◽  
Spin Fields ◽  
Higher Spin Fields

Abstract In this work we classify (homogeneous) solutions to the Noether procedure in (A)dS for an arbitrary number of external legs and in general dimensions, analysing also the corresponding deformations of gauge symmetries. This builds upon the corresponding flat space classification [1], which we review and give its relation with the (A)dS result presented here. The role of dimensional dependent identities is studied in detail, which we find do not lead to new solutions for couplings involving more than three fields. For spins one and two our formalism recovers the Yang-Mills and Gravity examples.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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