1. Double Pendulum

2.  The double pendulum is a conservative system. Two degrees of freedom  The exact Lagrangian can be written without approximation.   l l m m

3. Dimensionless Form  Make substitutions: Divide by mglDivide by mgl t  t(g/l) 1/2t  t(g/l) 1/2  Find conjugate momenta as angular momenta.

4. Hamilton’s Equations  Make substitutions: Divide by mglDivide by mgl t  t(g/l) 1/2t  t(g/l) 1/2  Find conjugate momenta as angular momenta.

5. Small Angle Approximation  For small angles the Lagrangian simplifies. The energy is E = -3.  The mode frequencies can be found from the matrix form. The winding number  is irrational.

6. Phase Space  The cotangent manifold T* Q is 4-dimensional. Q is a torus T 2. Energy conservation constrains T* Q to an n- torus  Take a Poincare section. Hyperplane   Select d  /dt > 0   JJ 12

7. Boundaries  The greatest motion in  - space occurs when there is no energy in the  -dimension  Points must lie within a boundary curve.  JJ  

8. Fixed Points  For small angle deflections there should be two fixed points. Correspond to normal modesCorrespond to normal modes  JJ  

9. Invariant Tori  An orbit on the Poincare section corresponds to a torus. The motion does not leave the torus. Motion is “invariant”  Orbits correspond to different energies. Mixture of normal modes next  JJ