Topic
Symmetry (physics)
About: Symmetry (physics) is a research topic. Over the lifetime, 26435 publications have been published within this topic receiving 500189 citations. The topic is also known as: symmetry (physics) & physical symmetry.
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TL;DR: In this article, new identities relating the Euler-Lagrange, Lie-Backlund and Noether operators are obtained, and the symmetry based results deduced from the new identities are used to construct Lagrangians for partial differential equations.
Abstract: New identities relating the Euler–Lagrange, Lie–Backlund and Noether operators are obtained Some important results are shown to be consequences of these fundamental identities Furthermore, we generalise an interesting example presented by Noether in her celebrated paper and prove that any Noether symmetry is equivalent to a strict Noether symmetry, ie a Noether symmetry with zero divergence We then use the symmetry based results deduced from the new identities to construct Lagrangians for partial differential equations In particular, we show how the knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilised to construct a Lagrangian for the equation Several examples are given
137 citations
TL;DR: In this article, the dynamics of the interaction of two modes that are degenerate in a square layer, but non-degenerate in rectangular one was investigated, and it was shown that the symmetry of the fluid cell has dramatic effects on the dynamics.
Abstract: Parametrically excited surface wave modes on a fluid layer driven by vertical forcing can interact with each other when more than one spatial mode is excited. We have investigated the dynamics of the interaction of two modes that are degenerate in a square layer, but non-degenerate in a rectangular one. Novel experimental techniques were developed for this purpose, including the real-time measurement of all relevant slowly varying mode amplitudes, investigation of the phase-space structure by means of transient studies starting from a variety of initial conditions, and automated determination of stability boundaries as a function of driving amplitude and frequency. These methods allowed both stable and unstable fixed points (sinks, sources, and saddles) to be determined, and the nature of the bifurcation sequences to be clearly established. In most of the dynamical regimes, multiple attractors and repellers (up to 16) were found, including both pure and mixed modes. We found that the symmetry of the fluid cell has dramatic effects on the dynamics. The fully degenerate case (square cell) yields no time-dependent patterns, and is qualitatively understood in terms of third-order amplitude equations whose basic structure follows from symmetry arguments. In a slightly rectangular cell, where the two modes are separated in frequency by a small amount (about 1%), mode competition produces both periodic and chaotic states organized around unstable pure and mixed-state fixed points.
137 citations
TL;DR: In this article, the authors extend the classification of symmetries necessary to predict the universality class of spectral fluctuations of quantal systems whose classical motion is chaotic, by explaining that a system with neither time-reversal symmetry (T) nor geometric symmetry may display the spectral statistics of the Gaussian orthogonal ensemble (GOE), rather than those of the GUE, provided it possesses instead some combination of symmetry which includes T. Such combinations constitute invariance under anti-unitary transformations (whose classical analogue are called anticanonical).
Abstract: The authors extend the classification of symmetries necessary to predict the universality class of spectral fluctuations of quantal systems whose classical motion is chaotic, by explaining that a system with neither time-reversal symmetry (T) nor geometric symmetry may display the spectral statistics of the Gaussian orthogonal ensemble (GOE), rather than those of the Gaussian unitary ensemble (GUE), provided it possesses instead some combination of symmetries which includes T. Such combinations constitute invariance under anti-unitary transformations (whose classical analogue are called anticanonical). For a particle in a magnetic field B plus scalar potential V, an example is TSx where Sx is a mirror reflection under which B and V are invariant. The authors illustrate this numerically for a single flux line in a hard-walled enclosure (Aharonov-Bohm quantum billiards), which also provides an example of an anti-unitary symmetry of non-geometrical origin; the spectral fluctuations are, as predicted, GOE rather than GUE.
136 citations
TL;DR: Using a recent classification of local symmetries of the vacuum Einstein equations, it is shown that there can be no observables for the vacuum gravitational field built as spatial integrals of local functions of Cauchy data and their derivatives.
Abstract: Using a recent classification of local symmetries of the vacuum Einstein equations, it is shown that there can be no observables for the vacuum gravitational field (in a closed universe) built as spatial integrals of local functions of Cauchy data and their derivatives.
136 citations
TL;DR: In this paper, it was shown that the rotating black holes in an arbitrary number of dimensions possess the same hidden symmetry as the four-dimensional Kerr metric, besides the spacetime symmetries generated by the Killing vectors they also admit the (antisymmetric) Killing-Yano and symmetric Killing tensors.
Abstract: We demonstrate that the rotating black holes in an arbitrary number of dimensions and without any restrictions on their rotation parameters possess the same hidden symmetry as the four-dimensional Kerr metric. Namely, besides the spacetime symmetries generated by the Killing vectors they also admit the (antisymmetric) Killing-Yano and symmetric Killing tensors.
136 citations