1. Vibrations

2. Near Equilibrium  Select generalized coordinates Kinetic energy is a homogeneous quadratic function of generalized velocityKinetic energy is a homogeneous quadratic function of generalized velocity Potential is time- independentPotential is time- independent Coordinates reflect equilibriumCoordinates reflect equilibrium

3. Quadratic Potential  Restrict to a small region of configuration space.  Expand the potential to second order. First term vanishes by choiceFirst term vanishes by choice Second term vanishes from equilibriumSecond term vanishes from equilibrium  Tensors G, V are symmetric and constant at equilibrium

4. Coupled Equations  EL equations Constant G and V imply form of equations of motionConstant G and V imply form of equations of motion Tensor G -1 V is not generally diagonalTensor G -1 V is not generally diagonal  Seek solutions of the matrix equation q is a vector of generalized coordinatesq is a vector of generalized coordinates Equivalent to solving for the determinantEquivalent to solving for the determinant

5. Double Pendulum  Two plane pendulums of the same mass and length. Coupled potentials The displacement of one influences the other  Define two angles  1,  2 as generalized variables.  Restrict the problem to small oscillations.   mm ll lower indices to avoid confusion

6. Eigenvalues  The symmetric matrix has two real solutions.  For small , there are two approximate solutions.  The generalized variables had mass and length folded into them.

7. Normal Modes  The normal modes come from the vector equation.  Normal mode equations correspond to                

8. Triple Pendulum  Three plane pendulums of the same mass and length.  Again define angles  1,  2,  3 as generalized variables. Similar restrictions as with two pendulums.   mm ll  m l

9. Degenerate Solutions Two frequencies are equal Solve two of the equations

10. Normal Coordinates  Solve the equations for ratios  1  3  2  3. Use single rootUse single root Find one eigenvectorFind one eigenvector Matches a normal coordinateMatches a normal coordinate  Solve for the double root All equations are equivalentAll equations are equivalent Pick  2 Pick  2  Find third orthogonal vectorFind third orthogonal vector Any combination of these two is an eigenvector

11. Diagonal Lagrangian  The normal coordinates can be used to construct the Lagrangian No coupling in the potential.No coupling in the potential.  Degenerate states allow choice in coordinates n -fold degeneracy involves n(n-1)/2 parameter choicesn -fold degeneracy involves n(n-1)/2 parameter choices 2-fold for triple pendulum involved one choice2-fold for triple pendulum involved one choice next