Consider the bound orbits of a body moving under the influence of a spherically symmetric *n*-dimensional power potential with an arbitrary real exponent. The Hamiltonian for the system is

$H=\frac{{p}^{2}}{2m}+a{r}^{k}=E$

where the variables have their usual definitions as vector squares:

${r}^{2}=(\mathbf{r}\xb7\mathbf{r})=\sum _{i=1}^{n}{r}_{i}^{2}\phantom{\rule{4em}{0ex}}{p}^{2}=(\mathbf{p}\xb7\mathbf{p})=\sum _{i=1}^{n}{p}_{i}^{2}$

The Hamiltonian can be written equivalently in terms of radial and angular momenta:

$H=\frac{{p}_{r}^{2}}{2m}+\frac{{L}^{2}}{2m{r}^{2}}+a{r}^{k}=E$

Since the motion takes place in an invariant plane of two dimensions, the overall dimensionality of the space will not enter the development. The distinctive features of the system are not a result of our physical three-dimensional space but apply to any system with at least two spatial dimensions.

For a given energy, there are two types of limiting orbits with respect to angular momentum. For *L* = 0 orbits are degenerate, passing through the origin, and quantities will be labeled with that word. For *L* > 0 the lower radial extremum becomes nonzero, and as *L* increases the two radial extrema approach each other at the minimum of the effective potential. This maximum allowed angular momentum for bound states occurs for circular orbits and quantities will be labeled accordingly.

Circular orbit parameters are determined by setting the derivative of the effective potential equal to zero, giving the angular momentum in terms of an initial radial value as

$-\frac{{L}^{2}}{m{r}^{3}}+ka{r}^{k-1}=0\phantom{\rule{1em}{0ex}}\to \phantom{\rule{1em}{0ex}}{L}_{\mathrm{circular}}=\sqrt{kma}{r}_{0}^{(k+2)/2}$

with the corresponding angular frequency and period

$\begin{array}{c}{\omega}_{\mathrm{circular}}=\frac{{L}_{\mathrm{circular}}}{m{r}_{0}^{2}}=\sqrt{\frac{ka}{m}}{r}_{0}^{(k-2)/2}\\ {T}_{\mathrm{circular}}=\frac{2\pi}{{\omega}_{\mathrm{circular}}}=2\pi \sqrt{\frac{m}{ka}}{r}_{0}^{-(k-2)/2}\end{array}$

For circular orbits *p _{r}* ≡ 0 , so that the energy is

${E}_{\mathrm{circular}}=\frac{k+2}{2}a{r}_{0}^{k}$

For positive values of the exponent *k*, the coupling constant *a* is positive for bound states. For negative values of the exponent the coupling constant must also become negative to provide a potential well. This means that the product *ka* is always positive for bound states and the formulae given for circular orbits hold for *k* ≥ −2 , *i.e.* for the entire range of physically realizable exponents. The change in the coupling constant for negative exponents correctly indicates that energy is negative for these orbits.

As an aside, an evaluation of the second derivative of the energy for circular orbits

$\frac{3{L}^{2}}{m{r}^{4}}+k(k-1)a{r}^{k-2}\phantom{\rule{1em}{0ex}}\to \phantom{\rule{1em}{0ex}}ka(k+2){r}_{0}^{k-2}$

implies that for *k* < −2 circular orbits are possible but are unstable. This regime will not be considered further.

Now consider the degenerate orbits for *L* = 0 that pass through the origin in this limit. At an extremum of the orbit *p _{r}* = 0 as well, so that the energy is

${E}_{\mathrm{origin}}=a{r}_{0}^{k}$

a result that holds for both positive and negative exponents. The evaluation of frequencies and periods, however, is different for the two cases. This is due to a discontinuous shift of the orbit center, which is located at the center of an ellipse for positive exponents but at the focus of an ellipse for negative exponents.

With the identification ${p}_{r}=m\stackrel{\xb7}{r}$ , the energy equation is

$\sqrt{\frac{m}{2}}\int \frac{dr}{\sqrt{{E}_{\mathrm{origin}}-a{r}^{k}}}=\sqrt{\frac{m}{2}}\int \frac{dr}{\sqrt{a{r}_{0}^{k}-a{r}^{k}}}=t$

For *k* > 0 the period of the motion is four times the value of the left-hand sides evaluated between the origin and the singularity in the denominator of the integrand,

${T}_{\mathrm{origin}}=2\sqrt{\frac{2m}{a}}{r}_{0}^{-(k-2)/2}\underset{0}{\overset{1}{\int}}\frac{du}{\sqrt{1-{u}^{k}}}=2\sqrt{\frac{2m}{a}}\frac{\sqrt{\pi}\Gamma (\frac{1}{k}+1)}{\Gamma (\frac{1}{k}+\frac{1}{2})}{r}_{0}^{-(k-2)/2}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}k>0$

and the corresponding angular frequency is

${\omega}_{\mathrm{origin}}=\frac{2\pi}{{T}_{\mathrm{origin}}}=\sqrt{\frac{\pi a}{2m}}\frac{\Gamma (\frac{1}{k}+\frac{1}{2})}{\Gamma (\frac{1}{k}+1)}{r}_{0}^{(k-2)/2}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}k>0$

The frequencies ω_{circular} and ω_{origin} are equal for *k* = 2 , the simple harmonic oscillator. This potential is unique in this context in that the spatial variables remain in phase for all values of initial conditions.

For *k* < 0 the integral determining the period needs a bit of rearrangement to keep the arguments of resulting gamma functions positive. Additionally the period is only twice the value of the integral due to the discontinuous shift of the orbit center:

${T}_{\mathrm{origin}}=\sqrt{\frac{2m}{\left|a\right|}}{r}_{0}^{\left(\right|k|+2)/2}\underset{0}{\overset{1}{\int}}\frac{{u}^{\left|k\right|/2}du}{\sqrt{1-{u}^{\left|k\right|}}}=\sqrt{\frac{2m}{\left|a\right|}}\frac{\sqrt{\pi}\Gamma (\frac{1}{\left|k\right|}+\frac{1}{2})}{\Gamma \left(\frac{1}{\left|k\right|}\right)}{r}_{0}^{\left(\right|k|+2)/2}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}k<0$

The corresponding angular frequency is

${\omega}_{\mathrm{origin}}=\frac{2\pi}{{T}_{\mathrm{origin}}}=\sqrt{\frac{2\pi \left|a\right|}{m}}\frac{\Gamma \left(\frac{1}{\left|k\right|}\right)}{\Gamma (\frac{1}{\left|k\right|}+\frac{1}{2})}{r}_{0}^{-\left(\right|k|+2)/2}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}k<0$

This expression is not a simple extension to negative values of the frequency for positive exponents. This can be seen using an identity for gamma functions:

$\Gamma (1-x)\Gamma \left(x\right)=\frac{\pi}{sin\pi x}$

Applying this to the ratio of gamma functions in frequency for positive exponents

$\frac{\Gamma (\frac{1}{2}-\frac{1}{\left|k\right|})}{\Gamma (1-\frac{1}{\left|k\right|})}=\frac{\Gamma [1-(\frac{1}{\left|k\right|}+\frac{1}{2}\left)\right]}{\Gamma (1-\frac{1}{\left|k\right|})}=\frac{\Gamma \left(\frac{1}{\left|k\right|}\right)}{\Gamma (\frac{1}{\left|k\right|}+\frac{1}{2})}tan\frac{\pi}{\left|k\right|}$

the factor of the tangent prevents the two expressions for frequency from being analytic continuations of each other. This is one expression of the discontinuous shift of the orbit center that will be explored more thoroughly in a future presentation.

*Uploaded 2017.10.05*
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