VibrationsNear EquilibriumQuadratic PotentialCoupled EquationsDouble PendulumEigenvaluesNormal ModesTriple PendulumDegenerate SolutionsNormal CoordinatesDiagonal LagrangianVibrationsVibrationsNear EquilibriumNear EquilibriumSelect generalized Select generalized coordinatescoordinates•Kinetic energy is a Kinetic energy is a homogeneous quadratic homogeneous quadratic function of generalized function of generalized velocityvelocity•Potential is time-Potential is time-independentindependent•Coordinates reflect Coordinates reflect equilibriumequilibriumjiijqqGT21),,(1 fqqVV 00jqVQuadratic PotentialQuadratic PotentialRestrict to a small region of Restrict to a small region of configuration space.configuration space.Expand the potential to Expand the potential to second order.second order.•First term vanishes by First term vanishes by choicechoice•Second term vanishes from Second term vanishes from equilibriumequilibriumTensors Tensors GG, , VV are symmetric are symmetric and constant at equilibriumand constant at equilibriumjijijjqqqqVqqVVV00021jiijjijiqqVqqqqVV21021jiijVV jiijGG Coupled EquationsCoupled EquationsEL equationsEL equations•Constant Constant GG and and VV imply imply form of equations of motionform of equations of motion•Tensor Tensor GG-1-1VV is not generally is not generally diagonaldiagonalSeek solutions of the matrix Seek solutions of the matrix equationequation•qq is a vector of generalized is a vector of generalized coordinatescoordinates•Equivalent to solving for the Equivalent to solving for the determinantdeterminantjiijjiijqqVqqGL21210jijjijqVqG 0det2ijijGV qqVG21 01 jjkkqVGqDouble PendulumDouble PendulumTwo plane pendulums of the Two plane pendulums of the same mass and length.same mass and length.•Coupled potentialsCoupled potentials•The displacement of one The displacement of one influences the otherinfluences the otherDefine two angles Define two angles 11, , 22 as as generalized variables.generalized variables.Restrict the problem to small Restrict the problem to small oscillations.oscillations.m ml l 222121T 212221212Vlower indices to avoid confusionEigenvaluesEigenvaluesThe symmetric matrix has The symmetric matrix has two real solutions.two real solutions.For small For small , there are two , there are two approximate solutions.approximate solutions.The generalized variables The generalized variables had mass and length folded had mass and length folded into them.into them.2111ijV221101001ijG212121lgNormal ModesNormal ModesThe normal modes come The normal modes come from the vector equation.from the vector equation.Normal mode equations Normal mode equations correspond tocorrespond to• •  02jijijqGV021 0100111121021Triple PendulumTriple PendulumThree plane pendulums of Three plane pendulums of the same mass and length.the same mass and length.Again define angles Again define angles 11, , 22, , 33 as generalized variables.as generalized variables.•Similar restrictions as with Similar restrictions as with two pendulums.two pendulums.m ml l 23222121T  133221232221212VmlDegenerate SolutionsDegenerate Solutions100010001ijG   02131322322111222Two frequencies are equalSolve two of the equations111ijV2221110   010101322132213212Normal CoordinatesNormal CoordinatesSolve the equations for Solve the equations for ratios ratios 11332233..•Use single rootUse single root•Find one eigenvectorFind one eigenvector•Matches a normal Matches a normal coordinatecoordinateSolve for the double rootSolve for the double root•All equations are equivalentAll equations are equivalent•Pick Pick 22 •Find third orthogonal vectorFind third orthogonal vector 1,1,1311x1323121202023213210321 12 1,0,1212x 1,2,1613xAny combination of these two is an eigenvector 32131Diagonal LagrangianDiagonal LagrangianThe normal coordinates can be used to construct the The normal coordinates can be used to construct the LagrangianLagrangian•No coupling in the potential.No coupling in the potential.Degenerate states allow choice in coordinatesDegenerate states allow choice in coordinates•nn-fold degeneracy involves -fold degeneracy involves nn((nn-1)/2-1)/2 parameter choices parameter choices•2-fold for triple pendulum involved one choice2-fold for triple pendulum involved one choice       23222121232221211121