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Stanford AA 218 - Lecture Notes

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PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Stanford University Department of Aeronautics and Astronautics Introduction to Symmetry AnalysisBrian CantwellDepartment of Aeronautics and AstronauticsStanford UniversityChapter 5 - Introduction toOne-Parameter Lie GroupsStanford University Department of Aeronautics and Astronautics 5.1 The Symmetry of FunctionsDefinition 5.1. A mathematical relationship between variables is said to possess a symmetry property if one can subject the variables to a group of transformations and the resulting expression reads the same in the new variables as the original expression. The relationship is said to be invariant under the transformation group.Stanford University Department of Aeronautics and Astronautics Translation along horizontal linesStanford University Department of Aeronautics and Astronautics A reflection and a translationStanford University Department of Aeronautics and Astronautics One parameter Lie groupsStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and Astronautics 5.4 Invariant functionsExample 5.1 Invariance of a parabola under dilationTransformΨ x y,[ ] y x2∕=usingTd i l : x esx~ ; y en sy~= ={ }The result isΨ x y,[ ] y x2∕ es n 2–( )y~x~2∕( )= =if we setn 2=Ψ x y,[ ] y x2∕ y~x~2∕ Ψ x~y~,[ ]= = =Stanford University Department of Aeronautics and Astronautics Definition 5.3A functionΨ x[ ]is said to be invariant under the Lie groupTs: xjFjx~s,[ ] ; j 1 … n, ,= ={ }if and only ifΨ x[ ] Ψ F x~s,[ ][ ] Ψ x~[ ]= =Stanford University Department of Aeronautics and Astronautics 5.5 Infinitesimal form of a Lie groupExpand the groupx~jFjx s,[ ]=In a Taylor series about the identity elements00=x~jxjs∂ Fj∂ s---------s 0=⎝ ⎠⎛ ⎞O s2( ) … ; j+ + + 1 … n, ,= =The infinitesimals of the group areξjx[ ]∂∂ s-----Fjx s,[ ]s 0= ; j 1 … n, ,= =The vector,ξjis also called the vector field of the group.Stanford University Department of Aeronautics and Astronautics Lie series, the group operator and the infinitesimalinvariance condition for functions5.6Substitutex~jFjx s,[ ]=intoΨ x~[ ]Now expand in a Taylor series abouts 0=Use the chain ruleThe expansion becomes the Lie series representation of the functionStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and Astronautics The operatoris called the group operator.The Lie series can be written concisely using the group operator.Any analytic function can be expanded asThe Lie series can be formally written as the exponential map,Stanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and Astronautics 5.9 Multi-parameter groupsThe projective group inndimensionsLetNow assumesis infinitesimally small.Expand and retain only the lowest order term.Stanford University Department of Aeronautics and Astronautics The infinitesimals of the n-dimensional projective group areThe corresponding group operators are:Stanford University Department of Aeronautics and Astronautics The projective group in two-dimensions isStanford University Department of Aeronautics and Astronautics The commutatorStanford University Department of Aeronautics and Astronautics 5.14 ExercisesStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and


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