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Normal Modes

Normal Modes. Eigenvalues. The general EL equation becomes a matrix equation. q is a vector of generalized coordinates Equivalent to solving for the determinant The number of solution will match the number of variables. Eigenfrequencies Normal mode vectors. Pendulum Eigenfrequencies.

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Normal Modes

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  1. Normal Modes

  2. Eigenvalues • The general EL equation becomes a matrix equation. • q is a vector of generalized coordinates • Equivalent to solving for the determinant • The number of solution will match the number of variables. • Eigenfrequencies • Normal mode vectors

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

  4. Pendulum Modes • The normal modes come from the vector equation. • Factor out single pendulum frequency w • The normal mode equations correspond to • q1 = q2 • q1 = -q2

  5. Triple Pendulum • Couple three plane pendulums of the same mass and length. • Three couplings • Identical values • Define angles q1, q2, q3 as generalized variables. • Similar restrictions as with two pendulums. l l l q1 q2 q3 m m m

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

  7. Normal Coordinates • Solve the equations for ratios q1/q3, q2/q3. • Use single root • Find one eigenvector • Matches a normal coordinate • Solve for the double root. • All equations are equivalent • Pick q2= 0 • Find 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. • Degenerate states allow choice in coordinates • n-fold degeneracy involves n(n-1)/2 parameter choices • 2-fold for triple pendulum involved one choice next

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