1. Double Pendulum

2. Coupled Motion  Two plane pendulums of the same mass and length. Coupled potentials The displacement of one influences the other Coupling is small  Define two angles  1,  2 as generalized variables.   mm ll 

3. Coupled Equations  The Lagrangian has two variables. Two EL equationsTwo EL equations  The equations are coupled in the generalized coordinates.

4. Uncoupled Variables  Add and subtract the two equations to get a different pair of equations.  Define two new generalized variables.  1,  2  There are two characteristic frequencies. One from each equation

5. Configuration Space  A simple pendulum can move in a circle. 1-dimensional configuration space1-dimensional configuration space Represented by a circle S 1Represented by a circle S 1  The double pendulum moves in two circles. 2-dimensional space Circles are independent Represented by a torus S 1  S 1

6. Local Configuration  Motion near equilibrium takes place in a small region of configuration space. Eg. 2-D patch of the torus  Synchronized oscillations would be a line or ellipse. Lissajous figures  Torus: S 1  S 1   

7. Quadratic Potential  An arbitrary potential may involve many variables. Assume time-independentAssume time-independent Generalized coordinatesGeneralized coordinates  Small oscillations occur near equilibrium. Define as the originDefine as the origin Zero potentialZero potential  Near equilibrium the potential can be expanded to second order.

8. Small Oscillation Lagrangian  The potential and kinetic energies can be expressed with matrix terms. Symmetric matrices  Matrices G and V imply the form of equations of motion. Matrix G -1 V not generally diagonal next