Double Pendulum • The double pendulum is a conservative system. • Two degrees of freedom • The exact Lagrangian can be written without approximation. l q m l f m
Make substitutions: Divide by mgl tt(g/l)1/2 Find conjugate momenta as angular momenta. Dimensionless Form
Make substitutions: Divide by mgl tt(g/l)1/2 Find conjugate momenta as angular momenta. Hamilton’s Equations
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 W is irrational.
Phase Space • The cotangent manifold T*Q is 4-dimensional. • Q is a torus T2. • Energy conservation constrains T*Q to an n-torus • Take a Poincare section. • Hyperplane q= 0 • Select dq/dt > 0 q f 1 2 Jf
Boundaries • The greatest motion in f-space occurs when there is no energy in the q-dimension • Points must lie within a boundary curve. Jf 2 1 f
Fixed Points • For small angle deflections there should be two fixed points. • Correspond to normal modes Jf 2 1 f
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 Jf f next