1. Normal Modes

2. Eigenvalues  The general EL equation becomes a 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  The number of solution will match the number of variables. EigenfrequenciesEigenfrequencies Normal mode vectorsNormal mode vectors

3. Pendulum Eigenfrequencies  The double pendulum problem has two real solutions. Fold mass and length into generalized variables Approximation for small 

4. Pendulum Modes  The normal modes come from the vector equation. Factor out single pendulum frequency Factor out single pendulum frequency   The normal mode equations correspond to                

5. Triple Pendulum  Couple three plane pendulums of the same mass and length. Three couplings Identical values  Define angles  1,  2,  3 as generalized variables. Similar restrictions as with two pendulums.   mm ll  m l

6. Degenerate Solutions Two frequencies are equal Solve two of the equations Frequencies normalized to single pendulum value

7. 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

8. 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