**Transcribed Image Text: **1. (9 pts) In a model of the benzene ring, the hydrogen atoms are ignored and the six carbon atoms are treated as individual particles with angular positions, \( \theta_{j}(t), j=1,2, \ldots, 6 \) and mass \( m \). Each atom is connected to its neighbour by a bond, modelled as a spring with constant \( k \), and the absorption of incoming photons applies a sinusoidal external force with frequency \( \Omega>0 \). Newton's law for the atoms states that \[ \begin{array}{l} m \bar{\theta}_{1}=k\left(\theta_{2}-\theta_{1}\right)-k\left(\theta_{1}-\theta_{6}-2 \pi\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{2}=k\left(\theta_{3}-\theta_{2}\right)-k\left(\theta_{2}-\theta_{1}\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{3}=k\left(\theta_{4}-\theta_{3}\right)-k\left(\theta_{3}-\theta_{2}\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{4}=k\left(\theta_{5}-\theta_{4}\right)-k\left(\theta_{4}-\theta_{3}\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{5}=k\left(\theta_{6}-\theta_{5}\right)-k\left(\theta_{5}-\theta_{4}\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{6}=k\left(2 \pi+\theta_{1}-\theta_{6}\right)-k\left(\theta_{6}-\theta_{5}\right)+\alpha \sin \Omega t \end{array} \] where \( \alpha \) is the amplitude of the photon forcing. Write these ODEs as the system, \[ \ddot{\mathbf{x}}=A \mathbf{x}+\mathbf{c}+\mathbf{a} \sin \Omega t, \] identifying the coefficient matrix \( A \) and the two constant vectors \( \mathbf{c} \) and a. Verify (by substitution and performing a matrix multiplication) that there is a time-independent particular solution of \( \theta_{j}=\frac{1}{3}(j-1) \pi \) for \( \alpha=0 \) (corresponding to the atoms sitting evenly spaced at equilibrium). Confirm (again by substitution) that A has the eigenvectors, \[ \mathbf{e}_{1}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}\right), \mathbf{e}_{2}=\left(\begin{array}{c} 1 \\ -1 \\ 1 \\ -1 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{3}=\left(\begin{array}{c} 1 \\ -1 \\ 0 \\ 1 \\ -1 \\ 0 \end{array}\right), \mathbf{e}_{4}=\left(\begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{5}=\left(\begin{array}{c} 1 \\ 1 \\ 0 \\ -1 \\ -1 \\ 0 \end{array}\right), \mathrm{e}_{6}=\left(\begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \\ -1 \\ -1 \end{array}\right), \] and find the corresponding eigenvalues, \( \lambda \), in terms of \( \omega=\sqrt{k / m} \). Briefly interpret to what motions of the atoms these eigensolutions correspond. At what forcing frequencies does the benzene ring resonate under the photon irradience? Using various matrices, but without performing any detailed algebra or computing any inverses, find the general solution of the problem when resonance does not occur. Highlight the problem with this formal solution when the forcing frequency is resonant.

**More** **Transcribed Image Text: **1. (9 pts) In a model of the benzene ring, the hydrogen atoms are ignored and the six carbon atoms are treated as individual particles with angular positions, \( \theta_{j}(t), j=1,2, \ldots, 6 \) and mass \( m \). Each atom is connected to its neighbour by a bond, modelled as a spring with constant \( k \), and the absorption of incoming photons applies a sinusoidal external force with frequency \( \Omega>0 \). Newton's law for the atoms states that \[ \begin{array}{l} m \bar{\theta}_{1}=k\left(\theta_{2}-\theta_{1}\right)-k\left(\theta_{1}-\theta_{6}-2 \pi\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{2}=k\left(\theta_{3}-\theta_{2}\right)-k\left(\theta_{2}-\theta_{1}\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{3}=k\left(\theta_{4}-\theta_{3}\right)-k\left(\theta_{3}-\theta_{2}\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{4}=k\left(\theta_{5}-\theta_{4}\right)-k\left(\theta_{4}-\theta_{3}\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{5}=k\left(\theta_{6}-\theta_{5}\right)-k\left(\theta_{5}-\theta_{4}\right)+\alpha \sin \Omega t \\ m \bar{\theta}_{6}=k\left(2 \pi+\theta_{1}-\theta_{6}\right)-k\left(\theta_{6}-\theta_{5}\right)+\alpha \sin \Omega t \end{array} \] where \( \alpha \) is the amplitude of the photon forcing. Write these ODEs as the system, \[ \ddot{\mathbf{x}}=A \mathbf{x}+\mathbf{c}+\mathbf{a} \sin \Omega t, \] identifying the coefficient matrix \( A \) and the two constant vectors \( \mathbf{c} \) and a. Verify (by substitution and performing a matrix multiplication) that there is a time-independent particular solution of \( \theta_{j}=\frac{1}{3}(j-1) \pi \) for \( \alpha=0 \) (corresponding to the atoms sitting evenly spaced at equilibrium). Confirm (again by substitution) that A has the eigenvectors, \[ \mathbf{e}_{1}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array}\right), \mathbf{e}_{2}=\left(\begin{array}{c} 1 \\ -1 \\ 1 \\ -1 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{3}=\left(\begin{array}{c} 1 \\ -1 \\ 0 \\ 1 \\ -1 \\ 0 \end{array}\right), \mathbf{e}_{4}=\left(\begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 1 \\ -1 \end{array}\right), \mathbf{e}_{5}=\left(\begin{array}{c} 1 \\ 1 \\ 0 \\ -1 \\ -1 \\ 0 \end{array}\right), \mathrm{e}_{6}=\left(\begin{array}{c} 0 \\ 1 \\ 1 \\ 0 \\ -1 \\ -1 \end{array}\right), \] and find the corresponding eigenvalues, \( \lambda \), in terms of \( \omega=\sqrt{k / m} \). Briefly interpret to what motions of the atoms these eigensolutions correspond. At what forcing frequencies does the benzene ring resonate under the photon irradience? Using various matrices, but without performing any detailed algebra or computing any inverses, find the general solution of the problem when resonance does not occur. Highlight the problem with this formal solution when the forcing frequency is resonant.