Exercise:
A double pulum consists of two mathematical pula connected to each other see figure. center includegraphicswidth.mm#image_path:mathematical-doublpulum-# center For small angles uptheta_ and uptheta_ the differential s can be shown to be approximately ddot uptheta_ -+muomega_^uptheta_ + mulambdaomega_^uptheta_ ddot uptheta_ +mu/lambdaomega_^uptheta_ -+muomega_^uptheta_ with omega_^ fracgell_ omega_^ fracgell_ mu fracm_m_ lambda fracell_ell_ Derive the angular frequencies and eigenvectors for the following two cases: abcliste abc Equal masses and lengths: m_m_m and ell_ell_ell. abc Different masses and lengths: m_mub m_ mub m and ell_labell_labell. abcliste

Solution:
abcliste abc In this case we have omega^ omega_^omega_^fracgell mu lambda The simplified coefficient matrix is thus bf A pmatrix -omega^ & omega^ omega^ & -omega^ pmatrix -omega^ pmatrix & - - & pmatrix The trace and determinant are tau -tauaPomega^ Delta -detaPomega^ so the eigenvalues are lambda_sf -omega^fractauaP pm sqrttauaP^-timesdetaP -omega^solaPpmsqrtrootaP Longrightarrow omega_s sqrtfracgellsolaP-sqrtrootaP quad textrmslow mode omega_f sqrtfracgellsolaP+sqrtrootaP quad textrmfast mode The eigenvectors are hat x_s pmatrix sqrt pmatrix hat x_f pmatrix -sqrt pmatrix abc With mumubO lambdalabO omega_^omega^ and omega_^omega^/labO the coefficient matrix is bf A pmatrix -omega^ & omega^ omega^ & -omega^ pmatrix -omega^ pmatrix & - - & pmatrix The eigenvalues and angular frequencies of the normal modes are found to be lambda_sf -omega^solbPpmsqrtrootbP Longrightarrow omega_s sqrtfracgellsolbP-sqrtrootbP quad textrmslow mode omega_f sqrtfracgellsolbP+sqrtrootbP quad textrmfast mode The corresponding eigenvectors are hat x_s pmatrix sqrt- pmatrix hat x_f pmatrix sqrt+ - pmatrix abcliste