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

Introduce the fixed endpoint values $x_0=x_{N+1}=0$. The Lagrangian isThus $T=m{I}_N$, $V_{ij}=k(2{\delta }_{ij}-{\delta }_{i,j+1}-{\delta }_{i,j-1})$, andThis is a symmetric tridiagonal matrix, reflecting that only nearest-neighbour masses are coupled.

Expand the tridiagonal determinant along its first row. The diagonal entry contributes $d{D}_{N-1}$. In the off-diagonal cofactor, expansion along its first column supplies the other factor $-k$, and the cofactor sign makes the net contribution $-k^2{D}_{N-2}$. ThereforeThe initial values are $D_1=d$ and $D_2=d^2-k^2$. Writing the same formula at $N=2$ forces $\boxed{D_0=1}$, the empty determinant convention. This is the tridiagonal determinant recurrence.

Substitute $D_N={\beta }^N$ into the recurrence. Since $k>0$, its roots are nonzero and satisfyFor distinct roots, the general recurrence solution is $D_N=c_1{\beta }_1^N+c_2{\beta }_2^N$. The initial values require $c_1+c_2=1$ and $c_1{\beta }_1+c_2{\beta }_2=d={\beta }_1+{\beta }_2$, givingAt a repeated root, $d=\pm 2k$ and ${\beta }_1={\beta }_2=\beta =\pm k$, this quotient must be replaced by its limit:It also follows by solving the repeated-root recurrence as $(c_1+c_{2}N){\beta }^N$. This exceptional case is necessary to avoid introducing a false root in the next part; it is included in the fixed-end spring-chain determinant and its repeated-root limit.

For distinct nonzero recurrence roots, the determinant formula givesThe ratio one is excluded: at that repeated root, part (iii) gives $(N+1)(\pm k{)}^N\ne 0$. Thus the naive extra root corresponding to equal recurrence roots is spurious.

To count the physical solutions without ambiguity from exchanging ${\beta }_1$ and ${\beta }_2$, write ${\beta }_{1,2}=k{e}^{\pm iq}$ and $d=2k\cos q$ with $0<q<\pi$. ThenIts $N$ distinct zeros occur at $q=j\pi /(N+1)$, $j=1,\dots ,N$. They give exactly $N$ distinct positive normal frequencies, or equivalently $N$ distinct roots in ${\omega }^2$:If signed $\omega$ is counted, the characteristic equation has $2N$ roots $\pm {\omega }_j$. The sine quotient is the Chebyshev polynomial of the second kind representation of the fixed-end spring-chain determinant and its repeated-root limit.