Stanford University Department of Aeronautics and AstronauticsIntroduction to SymmetryAnalysisBrian CantwellDepartment of Aeronautics and AstronauticsStanford UniversityVariational Symmetriesfor the Equation Governing Flexural WavesStanford University Department of Aeronautics and AstronauticsNoether’s TheoremStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and AstronauticsDispersion RelationStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and AstronauticsStanford University Department of Aeronautics and AstronauticsSolutionsIn dimensioned form the equation isTake the origin of coordinates at the center of the rod.Even solutions are of the formand odd solutions areDStanford University Department of Aeronautics and AstronauticsThe wave speed isThe frequency and wave number cannot be selectedindependently. They are related bythusand the wave speed isThe waves are highly dispersive with short wavelengthstraveling much faster than long waves.c =!k=EI"A#$%&'(1/2kStanford University Department of Aeronautics and AstronauticsNondimensionalize using the characteristic wave number and frequencyIn dimensionless variables the equation becomeswherek0=1I!"#$%&1/4!0=E"A#$%&'(1/2Stanford University Department of Aeronautics and AstronauticsWe wish to solveCombine invariance under translation in time and invarianceunder dilation of the dependent variableTo form the group operatorSolution by Separation of variables. With characteristic equationsWith invariantsStanford University Department of Aeronautics and AstronauticsThe solution is of the formWe are seeking time-periodic solutions of the equation. LetThe fourth order ODEhas the general solutionG(x) = Ae!1/2x+ Be"!1/2x+ CSin!1/2x( )+ DCos!1/2x( )Stanford University Department of Aeronautics and AstronauticsEven and odd solutions arewhereDSuperposition of solutions for various frequencies and associatedwave numbers can be used to match the boundary conditions for agiven problem.Stanford University Department of Aeronautics and AstronauticsLagrangianThe equation governing flexural waves in a beam isSubstitute the Lagrangian into the Euler-Lagrange equationsSolutions of this equation minimize the action integralie, solutions minimize the volume integral of the difference betweenkinetic and potential energy. The equation can be generated from theLagrangianThe result isStanford University Department of Aeronautics and AstronauticsSymmetriesThe equation is invariant under a four parameter group of translations and dilationsplus the infinite-dimensional group corresponding to linear superposition of solutions.Stanford University Department of Aeronautics and AstronauticsConservation laws generated from symmetriesThe relations used to generate components of the conserved vectorscorresponding to variational symmetries of this equation are:where the characteristic function isStanford University Department of Aeronautics and AstronauticsInvariance under translation in timeInfinitesimalsCharacteristic functionConserved vectorCheck to see if this vector is indeed conserved!Stanford University Department of Aeronautics and AstronauticsThe quantityis the energy per unit length of the rod: kinetic energy + strain energyNote thatIf the bending moment and shear force vanish at the ends of thebeam, ie the second and third spatial derivatives are zero, then thetotal energy is conserved.Stanford University Department of Aeronautics and AstronauticsInvariance under translation in spaceInfinitesimalsCharacteristic functionConserved vectorCheck to see if this vector is indeed conserved!Stanford University Department of Aeronautics and AstronauticsThe quantitycan be regarded as the effective “momentum per unit length” of the rod.Note thatIf the acceleration, bending moment and shear force vanish at theends of the beam, ie the time, second and third spatial derivativesare zero, then the total momentum is conserved.Stanford University Department of Aeronautics and AstronauticsInvariance under translation in y[x,t]InfinitesimalsCharacteristic functionConserved vectorCheck to see if this vector is indeed conserved!Stanford University Department of Aeronautics and AstronauticsThe quantitycan be regarded as the effective “mass per unit length” of the rod.Note thatIf the shear force vanishes at the ends of the beam, ie the thirdspatial derivative is zero, then the total mass is conserved.Stanford University Department of Aeronautics and AstronauticsInvariance under dilationInfinitesimalsCharacteristic functionConserved vectorCheck to see if this vector is indeed conserved!Stanford University Department of Aeronautics and AstronauticsIntegrate along the beamIf the acceleration, bending moment and shear force vanish at theends of the beam, ie the time, second and third spatial derivativesare zero, then the time integral above is conserved.Note that the integral involves the mass momentum and energy perunit length of the