Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 7 / a

The four spring extensions are $x_1$, $x_2-x_1$, $x_3-x_2$ and $-x_3$. Hooke's law therefore gives potential energy $k[x_1^2+(x_2-x_1{)}^2+(x_3-x_2{)}^2+x_3^2]/2$, while the kinetic energy is $m{\sum }_i{\dot{x}}_i^2/2$. Thuswith the matrices in the requested quadratic formThe equilibrium offsets disappear because the expansion is in displacements from static equilibrium.

Write $d=2k-m{\omega }^2$. The characteristic determinant of the tridiagonal matrix isThe three values of $d$ are $\sqrt{2}k$, $0$ and $-\sqrt{2}k$. Therefore the normal mode frequencies and convenient Euclidean-normalized generalized eigenvectors areEach pair satisfies $V\mathbf{u}={\omega }^{2}T\mathbf{u}$. The first mode moves all masses in phase, the middle mode leaves the central mass fixed, and the last has the central mass moving opposite to the outer masses. All squared frequencies are positive, consistent with the positive spring potential and fixed endpoints.

Using the three real eigenvectors and positive frequencies in part (ii), the general motion isThere are six independent real constants, $A_1,A_2,A_3,B_1,B_2,B_3$, corresponding to the three initial displacements and three initial velocities. With the displayed orthonormal eigenvectors, $A_j={\mathbf{u}}_j^T\mathbf{x}(0)$ and $B_j={\mathbf{u}}_j^T\dot{\mathbf{x}}(0)/{\omega }_j$. This spans all solutions because the normal-mode eigenvectors form a basis.