The geometric nature of Lie and Noether symmetries

DOI10.1007/s10714-011-1166-xzbMath1216.83049OpenAlexW2062294311WikidataQ125131416 ScholiaQ125131416MaRDI QIDQ542431

Andronikos Paliathanasis, Michael Tsamparlis

Publication date: 10 June 2011

Published in: General Relativity and Gravitation (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s10714-011-1166-x

zbMATH Keywords

dynamical systemsBianchi cosmologiesscalar fieldLie symmetriesNoether symmetriesdark energyprojective collineations

Mathematics Subject Classification ID

Relativistic cosmology (83F05) Geometrodynamics and the holographic principle (83E05) Einstein's equations (general structure, canonical formalism, Cauchy problems) (83C05) Gravitational energy and conservation laws; groups of motions (83C40) Exact solutions to problems in general relativity and gravitational theory (83C15) Lie algebras of linear algebraic groups (17B45) Equations of motion in general relativity and gravitational theory (83C10)

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Cites Work

• Unnamed Item
• Unnamed Item
• Exact solution of scalar field cosmology with exponential potentials and transient acceleration
• Lie symmetries of geodesic equations and projective collineations
• Projective differential geometry and geodesic conservation laws in general relativity. I: Projective actions
• Kepler's problem in constant curvature spaces
• Symmetries of homogeneous cosmologies
• The connection between isometries and symmetries of geodesic equations of the underlying spaces
• Integrable problems of celestial mechanics in spaces of constant curvature
• Projective collineations in Einstein spaces
• Bianchi spacetimes in noncommutative phase space
• Decoupling of the general scalar field mode and the solution space for Bianchi type I and V cosmologies coupled to perfect fluid sources
• Conformal Killing vectors in Robertson-Walker spacetimes
• Affine collineations in Robertson–Walker space-time
• Sl(3,R) and the repulsive oscillator
• On the Lie symmetries of the classical Kepler problem
• Geodesic first integrals with explicit path-parameter dependence in Riemannian space–times
• The Lie group of Newton's and Lagrange's equations for the harmonic oscillator
• A gauge invariant formulation of time-dependent dynamical symmetry mappings and associated constants of motion for Lagrangian particle mechanics. I
• Symmetry groups and conserved quantities for the harmonic oscillator
• Projective transformations and symmetries of differential equation
• Dynamical symmetries and constants of the motion for classical particle systems
• The projective geometric theory of systems of second-order differential equations: straightening and symmetry theorems
• Nöther Symmetries in Bianchi Universes
• Quantum Cosmology. I
• Symmetries and differential equations