Central force

For central potential $V(r)=\beta r^k$ and nonzero angular momentum $J$, a circular orbit satisfies $\beta k{r}_0^{k+2}=J^2/m$, so attraction requires $\beta k>0$. Twice differentiating the radial effective potential $V+J^2/(2m{r}^2)$ gives $V_{\mathrm{eff}}^{\prime \prime }(r_0)=(k+2)J^2/(m{r}_0^4)$. Its harmonic approximation therefore has radial angular frequency ${\omega }_r=\sqrt{k+2}|J|/(m{r}_0^2)$, whereas $|{\omega }_{\theta }|=|J|/(m{r}_0^2)$. This proves the ratio and shows stability precisely when $k>-2$. At $k=-2$ the tuned effective potential is flat; at $k<-2$ the radial disturbance grows exponentially. Rationality of this local frequency ratio is only a necessary condition for all nearby orbits to close; Bertrand's theorem makes a stronger finite-amplitude assertion.

For potential energy $\Phi (r)=-mk/r^2$, the inward central force has magnitude $2mk/r^3$. With conserved specific angular momentum $h=r^2\dot{\theta }$, the radial equation is $\ddot{r}=(h^2-2k)/r^3$. At the critical value $h^2=2k$, this reduces to $\ddot{r}=0$. An inward initial radial velocity $-v$, with $v>0$ and $r(0)=a$, therefore gives $r=a-vt$ until collision at $t_{\ast }=a/v$.

For $h>0$, integration of $\dot{\theta }=h/(a-vt{)}^2$ gives $\theta -\theta (0)=(h/v)(1/r-1/a)$. Thus the orbit is a hyperbolic spiral and the particle completes infinitely many turns before the finite collision time. The tangential speed $h/r$ diverges. The potential energy and kinetic energy individually diverge with opposite signs, while their sum remains the finite conserved mechanical energy $m{v}^2/2$.

A planar orbit whose radius oscillates as its polar angle increases, producing successive lobes around the force center. For nonconstant $r$ with fundamental angular radial period $2\pi /\nu$ and nonzero specific angular momentum, the full position and velocity repeat precisely when $\nu$ is a rational number, provided the radius is bounded above and away from zero. If $\nu =p/q$ in lowest terms, $p$ radial periods advance the angle by $2\pi q$; $dt=r^{2}d\theta /h$ then gives a finite time period. This proves closure rather than merely a spatial self-intersection.

Joseph Bertrand's theorem states that, among smooth attractive central potentials admitting stable circular orbits, requiring every bound orbit to be closed singles out the inverse-distance potential energy and the isotropic quadratic potential energy, up to additive constants. Rational radial-to-angular frequency ratios near a circular orbit provide only a necessary local closure test; finite-amplitude closure is the additional content. In $U=B{r}^n$, the radial-to-angular frequency ratio near a stable circle is $\sqrt{n+2}$, and stability requires $nB>0$ and $n>-2$. The all-bound-orbit cases are $n=-1$ and $n=2$.

For a taut elastic string with potential $k(r-L{)}^2/2$, a circular orbit has ${\omega }_0^2=k(r_0-L)/(m{r}_0)$. A radial impulse preserves angular momentum, so the curvature of the fixed-angular-momentum effective potential gives ${\Omega }^2=k/m+3{\omega }_0^2$. The ratio is three at $r_0=6L/5$ and approaches two at $r_0\gg L$. Rational linearized frequency ratios describe leading small-amplitude closed sketches without proving exact closure at finite amplitude.

For $\mathbf{F}=-A\mathbf{r}$, the turning radii obey $L^2=mA{r}_{\min }^2{r}_{\max }^2$. A radial impulse preserves angular momentum. Its circular orbit therefore has $r_c^4=L^2/(mA)$, or $r_c=\sqrt{r_{\min }{r}_{\max }}$. Cancel the radial velocity where the original orbit crosses $r_c$.

A radial force proportional to $r^{-3}$. For potential $V=B/r^2$, its outward radial component is $2B/r^3$: positive $B$ is repulsive and negative $B$ is attractive.

For nonzero angular momentum $L$, put $u=1/r$ and $\lambda =1+2mB/L^2$. The Binet equation is $u_{\theta \theta }+\lambda u=0$. Positive $\lambda$ gives cosine scattering branches, zero gives linear reciprocal radius, and negative gives exponential combinations, with energy distinguishing sinh, exponential and cosh branches. Only intervals with $u>0$ are physical.

At $B=-L^2/(2m)$ the centrifugal and force potentials cancel. With zero radial speed, any radius is a circular solution, but there is no radial restoring force; radial-velocity perturbations are not confined.

Let $p$ be the perpendicular distance from the force centre to the tangent of a regular planar orbit. Conserved physical angular momentum has magnitude $|h|=mpv$, while conserved energy is $E=m{v}^2/2+V(r)$. Eliminating speed gives the displayed equation when $h\ne 0$. For a differentiable polar curve, $p=r^2/\sqrt{r^2+(dr/d\theta {)}^2}$, connecting the geometric tangent distance to the radial energy equation.