Appendix: Some mathematical results

Here we collect some mathematical results concerning the logarithmic potential and mainly related to the limit case $r_{\rm c}\to 0$ for a fixed value of h or, equivalently, $h\to\infty$ for a fixed value of $r_{\rm c}$.

First we consider the collinear problem $\ddot{x}=-x^{-1}$. For other similar problems but with different exponent, for instance, $\ddot{x}=-x^{-\gamma}$, $\gamma\ne 1$, and with Hamiltonian $H=y^2/2+x^{1-\gamma}/(1-\gamma)$, the scalings $x=\alpha z$, $t=\beta s$ lead to ( $'\equiv{\rm d}/ {\rm d}s$) $\alpha\beta^{-2}z''=-\alpha^{-\gamma}z^{-\gamma}$, which has the same form as before if $\beta^2=\alpha^{\gamma+1}.$ Let w=z' and $\widehat{H}=w^2/2+z^{1-\gamma}/(1-\gamma)$. Then the level of energy, H=hand $\widehat{H}=\hat{h}$ are related by $\hat{h}=h\alpha^{\gamma-1}$. Hence, $\hat{h}$ can be scaled to the values $1,\,0$ or -1. Furthermore, a collision solution (i.e. of unbounded acceleration) leaving from (or ending at) x=0 for t=0 is of the form $x(t)=at^b+\cdots$ with $b=2(1+\gamma)^{-1}$, $a=\left((1+\gamma)^2/2(\gamma-1)\right)^{1/(1+\gamma)}$, if $\gamma>1$. For $\gamma=1$ and $H=y^2/2+0.5\ln x^2$, the scalings defined above lead to $\alpha=\beta$ and $h=\hat{h}+\ln\alpha$. So we can always reduce to the level $\hat{h}=0$ setting $\alpha={\rm e}^h$ and then $\vert x\vert\le 1$.

To solve the equations of motion $\dot{x}=y$, $\dot{y}=-x^{-1}$ with $\vert x\vert={\rm e}^{-y^2/2}$, it is convenient to use y as independent variable. Then, for x>0, and assuming that at t=0, $y=+\infty$, $x=\sqrt{2\pi}N(y),\ t=\sqrt{2\pi}F(-y)$, where N, F ( $F\equiv \int_{-\infty}^yN(u){\rm d}u)$ denote, respectively, the density and distribution function of the standard normal law (0 average and standard deviation 1). In particular, a solution leaving from collision ends again at it ($y=-\infty$) after $t=\sqrt{2\pi}$. Furthermore, as

$$% F(y)={{\rm e}^{-y^2/2}\over\sqrt{2\pi}}{1\over \vert y\vert}\sum_{n\ge 0} {(-1)^n(2n-1)!!\over y^{2n}}$$ (37)

is the asymptotic series for $y\to -\infty$ (with error less, in absolute value, than the first neglected term), the relation $x(t)=t\sqrt{\ln t^{-2}} \left(1+{\cal O}\left(\ln(\ln t^{-1})/\ln t^{-1}\right)\right)$for $t\to 0$ follows immediately. Due to the symmetry, the same relation applies when $y\to -\infty$ but $t\to \sqrt{2\pi}-t$. Note that y decreases if x>0 and it increases if x<0. Hence two solutions appear for the limit problem. These solutions are the natural continuation one from the other. This is the suitable "regularization'' to be used for the "collision'' in that problems, as a blow up shows in a simple way. The natural continuation of an orbit (i.e., the one allowing to recover continuity with respect to initial conditions) is just to "cross'' the origin in the same direction that the orbit has. This is in contrast with the classical two-body problem, where the regularization looks like an elastic bounce.

When we consider the problem $\dot{x}=y$, $\dot{y}=-x/(r_{\rm c}^2+x^2)$ on the level h=0 of $H=y^2/2+0.5\ln(r_{\rm c}^2+x^2)$ with $r_{\rm c}$ small, no relevant differences with the previous problem appear unless |x| is small (of the order $Kr_{\rm c}$, K finite). The effect of $r_{\rm c}$ is to bound y(now $\vert y\vert=\sqrt{\ln r_{\rm c}^{-2}}$ at x=0 instead of $\infty$) and "to match'' the two previous solutions in the regions x>0 and x<0. As the time interval to go from x=0 to $\vert x\vert=Kr_{\rm c}$ for any finite K goes to zero with $r_{\rm c}$ in both problems, the limit period is $2\sqrt{2\pi}$ (always on the level h=0; the scaling ${\rm e}^h$ is required for other energies, see Sect. 5).

Now we are interested in the normal variational equations (as presented in (31)) in the limit case $h\to\infty$. In an equivalent way we can reformulate the second of (31), rewritten as $\ddot{\xi}+ \beta(r_{\rm c}^2+x^2)^{-1}\xi=0$ with $r_{\rm c}\to 0$ and x being a solution of the first in (31), rewritten as $\ddot{x}+x/(r_{\rm c}^2+x^2)=0$ on h=0. As we are interested in values of $\beta$ for which the trace of the monodromy matrix is 2 (see Sect. 5.1, Fig. 14 and related discussion), we should have a periodic solution of the normal variational equations. The n-th value of $\beta$, $\beta_n$, should correspond to a periodic solution having n full oscillations in one period, i.e., 2n zeros exactly. Due to symmetry properties we can select one of the zeros at x=0, $y=y_{{\rm max}}$ (and, by symmetry, another at x=0, $y=y_{{\rm min}}$). Hence, we are looking for a solution that vanishes at the extreme values of y and having exactly n-1 zeros between them when x>0. The remaining n-1 zeros occur for x<0.

In the limit, when $r_{\rm c}\to 0$, we have the linear equation $\ddot{\xi}+\beta x^{-2}\xi=0$ where t, x can be expressed as a function of $y=\dot{x}$ as above. Thus, using y as independent variable, we have ( $'\equiv{\rm d}/{\rm d}y$) $\xi''+y\xi'+\beta\xi=0$. Let $\xi={\rm e}^{-y^2/2}g(y)$. Then the equation for g reads $g''-yg'+(\beta-1)g=0$. The solutions of this equation grow slower than ${\rm e}^{y^2/2}$ for $\vert y\vert\to\infty$ only if $\beta-1=m$is an integer. Then we obtain g(y)=He_m(y), the Hermite polynomials (see Abramowitz & Stegun 1972). They can be computed from He₀(y)=0, He₁(y)=y, He_m+1(y)=yHe_m(y)-m He_m-1(y)for $m\ge 1$. As we want n-1 real zeros of $\xi$, one finally has $\beta=n$.