Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 321 / 3

At $x_n=0$ and $y_n=hn$, every radial pair displacement vanishes, so $f_x^n=0$. The particles with indices $n+l$ and $n-l$ give opposite azimuthal accelerations,which cancel pairwise. The acceleration sum has absolute convergence because ${\sum }_{l\ge 1}{l}^{-2}<\infty$. With zero velocities and $x_n=0$, both the Coriolis force and radial tidal force vanish. The equally spaced row is therefore an equilibrium of the infinite gravitating particle lattice. Its absolute gravitational potential would require an additional reference or regularization; the acceleration used here is well defined.

Let ${\mathbf{d}}_{jn}=h(j-n){\mathbf{e}}_y$ and $\Delta \mathbf{\xi }={\mathbf{\xi }}_j-{\mathbf{\xi }}_n$, with ${\mathbf{\xi }}_n=(x_n^{\prime },y_n^{\prime })$. Linearizing the Newton's law of universal gravitation usesFor an azimuthal separation, the matrix is $\mathrm{diag}(1,-2)$. After canceling the equilibrium forces, the linearized shearing sheet equations areThe transverse gravitational response tends to restore a radial displacement, while the longitudinal response reinforces an azimuthal density concentration. Both sums converge for the bounded Fourier modes considered next.

Set $\xi =hk$. For the discrete Fourier mode, the displacement difference in the $l$th neighbor pair carries $e^{il\xi }-1$, andThus the infinite system reduces toThe gravitating lattice Fourier kernel $F$ is nonnegative, even and $2\pi$-periodic, with $F(0)=0$. The uniform convergence and absolute convergence of its differentiated series givesTo prove the maximum, it is not sufficient to maximize each cosine term separately: the even terms do not reach their individual maxima at $\pi$. Instead use the convergent integral representation $l^{-2}={\int }_0^{\infty }t{e}^{-lt}dt$. Summing the geometric series for sines givesFor $0<\xi <\pi$, the denominator is positive and $\sin \xi >0$, so $F^{\prime }>0$. For $\pi <\xi <2\pi$, $F^{\prime }<0$. The interchange is justified by ${\sum }_l{\int }_0^{\infty }t{e}^{-lt}|\sin (l\xi )|dt\le {\sum }_l{l}^{-2}<\infty$. HenceTheir value iswhere $\zeta$ is the Riemann zeta function. This maximizing Fourier mode alternates the signs of neighboring particle displacements.

Put $A=(Gm/h^3)F(hk)$. The two amplitude equations have coefficient matrixIts determinant gives the dispersion relationWriting $q=A/{\Omega }^2\ge 0$, its squared growth rates areand the four growth rates are both square roots of each displayed value. DefineFor $0<q<q_{-}$, the discriminant is positive, the sum of the squared rates is $q-1<0$, and their product is $2q(3-q)>0$; both squared rates are negative. These modes have purely imaginary $s$. For $q_{-}<q<q_{+}$, the squared rates form a nonreal complex conjugate pair; their square roots include roots with positive real part. This is overstability, meaning growing oscillations. For $q\ge q_{+}$, at least the plus squared rate is positive because $q>1$, so a real growing root exists. Consequently every $q>q_{-}$ is exponentially unstable.

Take the maximizing mode $hk=\pi$, so $q=(Gm/(h^3{\Omega }^2))F(\pi )$. It is unstable wheneverThis is also the threshold for the onset of exponential growth among all lattice modes, since $0\le F\le F(\pi )$. At equality the maximizing mode has repeated imaginary roots and can have algebraic growth; it is a marginal collision of oscillatory modes, not a strict-decay assertion. The $k=0$ mode has $F=0$, an epicyclic motion sector and a zero-frequency epicyclic guiding center sector with possible secular shear drift. These neutral or algebraic behaviors are distinct from the strict exponential instability established by the requested inequality.