Double PendulumSlide 2Dimensionless FormHamilton’s EquationsSmall Angle ApproximationPhase SpaceBoundariesFixed PointsInvariant ToriDouble PendulumDouble PendulumDouble PendulumDouble PendulumThe double pendulum is a The double pendulum is a conservative system.conservative system.•Two degrees of freedomTwo degrees of freedomThe exact Lagrangian can The exact Lagrangian can be written without be written without approximation.approximation. 22)sin()cos(2llmT llmm 22))sin()(sin())cos()(cos(2llllmT 222)(cos)(222mlT  )cos(cos2)(cos)(222222 mglmlLDimensionless FormDimensionless FormMake substitutions:Make substitutions:•Divide by mglDivide by mgl•tt  tt((gg//ll))1/21/2 )cos(cos2)(cos)(22222lglglgglLFind conjugate momenta as Find conjugate momenta as angular momenta.angular momenta.)cos(cos2)(21cos)(22L)cos1()cos23()(cos)2(2 LJ )cos1()(cosLJHamilton’s EquationsHamilton’s EquationsMake substitutions:Make substitutions:•Divide by mglDivide by mgl•tt  tt((gg//ll))1/21/2Find conjugate momenta as Find conjugate momenta as angular momenta.angular momenta.)cos(cos22cos3)cos23()cos1(222JJJJHE2cos3)cos1(22JJJH)sin(sin2HJ2cos3)cos23(2)cos1(2JJJH )sin(2cos3)cos23()cos1(22sin22cos3sin22222JJJJJJJHJSmall Angle ApproximationSmall Angle ApproximationFor small angles the For small angles the Lagrangian simplifies.Lagrangian simplifies.•The energy is The energy is EE = -3. = -3.The mode frequencies can The mode frequencies can be found from the matrix be found from the matrix form.form.•The winding number The winding number  is is irrational.irrational.012121532222222122)(21)(T22)(21V222221222212Phase SpacePhase SpaceThe cotangent manifold T*The cotangent manifold T*QQ is 4-dimensional.is 4-dimensional.•QQ is a torus is a torus TT22..•Energy conservation Energy conservation constrains T*constrains T*QQ to an n-torus to an n-torusTake a Poincare section.Take a Poincare section.•Hyperplane Hyperplane  •Select Select dd//dtdt > 0 > 0J1 2BoundariesBoundariesThe greatest motion in The greatest motion in --space occurs when there is space occurs when there is no energy in the no energy in the -dimension-dimensionPoints must lie within a Points must lie within a boundary curve.boundary curve.JJJ )cos1( 02cos3)cos1(22JJcos22),,)cos1(,0(2JEJJHEboundFixed PointsFixed PointsFor small angle deflections For small angle deflections there should be two fixed there should be two fixed points.points.•Correspond to normal Correspond to normal modesmodesJ012121532222iiiiii222122221211112111Invariant ToriInvariant ToriAn orbit on the Poincare An orbit on the Poincare section corresponds to a section corresponds to a torus.torus.•The motion does not leave The motion does not leave the torus.the torus.•Motion is “invariant”Motion is “invariant”Orbits correspond to Orbits correspond to different energies.different energies.•Mixture of normal modesMixture of normal