PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Stanford University Department of Aeronautics and Astronautics Introduction to Symmetry AnalysisBrian CantwellDepartment of Aeronautics and AstronauticsStanford UniversityChapter 8 - Ordinary Differential EquationsStanford University Department of Aeronautics and Astronautics 8.1 Extension of Lie Groups in the PlaneStanford University Department of Aeronautics and Astronautics The Extended Transformation is a GroupTwo transformations of the extended groupCompose the two transformationsStanford University Department of Aeronautics and Astronautics The last relation is rearranged to readDifferentiating F and G givesComparing the expressions in parentheses we haveThe composed transformation is in exactly the same form as the original transformation!Stanford University Department of Aeronautics and Astronautics Finite transformation of the second derivativeThe twice extended finite transformation isStanford University Department of Aeronautics and Astronautics Finite transformation of higher derivativesStanford University Department of Aeronautics and Astronautics The p-th order extended group isStanford University Department of Aeronautics and Astronautics Infinitesimal transformation of the second derivativeRecall the infinitesimal transformation of coordinateswhereSubstituteExpand and retain only the lowest order termsStanford University Department of Aeronautics and Astronautics The once-extended infinitesimal transformation in the plane iswhere the infinitesimal function fully written out isStanford University Department of Aeronautics and Astronautics The twice-extended infinitesimal transformation in the plane iswhereExpand and retain only the lowest order termsStanford University Department of Aeronautics and Astronautics The infinitesimal function transforming third derivatives isStanford University Department of Aeronautics and Astronautics Expand and retain only the lowest order terms. The p times extended infinitesimal transformation isThe infinitesimal transformation of higher order derivativeswhereStanford University Department of Aeronautics and Astronautics 8.2 Expansion of an ODE in a Lie Series - the Invariance Condition for ODEsStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and Astronautics The characteristic equations associated with extended groups areStanford University Department of Aeronautics and Astronautics The general second-order ordinary differential equationis invariant under the twice-extended group if and only ifStanford University Department of Aeronautics and Astronautics Consider the case of the simplest second-order ODEThe invariance condition isFully written out the invariance condition isFor invariance this equation must be satisfied subject to the condition that y is a solution ofStanford University Department of Aeronautics and Astronautics The determining equations of the group areThese equations can be used to work out the unknown infinitesimals.Stanford University Department of Aeronautics and Astronautics Assume that the infinitesimals can be written as a multivariate power seriesInsert these series into the determining equationsStanford University Department of Aeronautics and Astronautics The coefficients must satisfy the following algebraic systemFinally the infinitesimals areStanford University Department of Aeronautics and Astronautics The two parameter group of the Blasius equationThe invariance conditionStanford University Department of Aeronautics and Astronautics Written out the invariance condition isStanford University Department of Aeronautics and Astronautics Now gather coefficients of like products of derivatives of yStanford University Department of Aeronautics and Astronautics The function y[x] is a solution of the Blasius equation. This is a constraint on the invariance condition that can be used to eliminate the third derivative.Stanford University Department of Aeronautics and Astronautics Further simplifyandStanford University Department of Aeronautics and Astronautics Finally the determining equations areFrom which the two parameter group of the Blasius equation is determined to