An important class of central fields is formed by those in which the potential energy is inversely proportional to \(r\), and the force accordingly inversely proportional to \(r^2\). They include the fields of Newtonian gravitational attraction and of Coulomb electrostatic interaction; the latter may be either attractive or repulsive.
Let us first consider an attractive field, where
\[\label{equations/15.1}\tag*{(15.1)}U=-\alpha/r\]
with a a positive constant. The “effective” potential energy
\[\label{equations/15.2}\tag*{(15.2)}U_\text{eff}=-\frac{\alpha}{r}+\frac{M^2}{2mr^2}\]
is of the form shown in fig10. As \(r\to 0\), \(U_\text{eff}\) tends to \(+\infty\), and as \(r\to \infty\) it tends to zero from
negative values ; for \(r =
M^2/m\alpha\) it has a minimum value
\[\label{equations/15.3}\tag*{(15.3)}U_{\text{eff,min}}=-m\alpha^2/2M^2\]
10It is seen at once from fig10 that the motion is finite
for \(E <0\) and infinite for \(E > 0\).
The shape of the path is obtained from the general formula (14.7) \(\phi=\int\frac{M\mathrm{d}^{}r/r^2}{\sqrt{\frac{2}{m}(E-U(r))-\frac{M^2}{m^2r^2}}}+\text{constant}\) . Substituting there \(U=-\alpha/r\) and effecting the elementary integration, we have
\[ \phi=\cos^{-1}\frac{(M/r)-(m\alpha/M)}{\sqrt{2mE+\frac{m^2\alpha^2}{M^2}}} +\text{constant} \]
Taking the origin of such that the constant is zero, and putting
\[\label{equations/15.4}\tag*{(15.4)}p=M^2/m\alpha,\qquad e=\sqrt{1+(2EM^2/m\alpha^2)}\]
we can write the equation of the path as
\[\label{equations/15.5}\tag*{(15.5)}p/r=1+e\cos\phi\]
This is the equation of a conic section with one focus at the origin; \(2p\) is called the latus rectum of the orbit and \(e\) the eccentricity. Our choice of the origin of is seen from (15.5) \(p/r=1+e\cos\phi\) to be such that the point where \(\phi = 0\) is the point nearest to the origin (called the perihelion).
In the equivalent problem of two particles interacting according to the law (15.1) \(U=-\alpha/r\) , the orbit of each particl\(e\) is a conic section, with one focus at the centre of mass of the two particles.
It is seen from
(15.4)
\(p=M^2/m\alpha,\qquad
e=\sqrt{1+(2EM^2/m\alpha^2)}\)
that, if \(E\lt 0\), then
the eccentricity \(e \lt 1\), i.e. the
orbit is an ellipse fig11 and the motion is finite, in
accordance with what has been said earlier in this section. According to
the formulae of analytical geometry, the major and minor semi-axes of
the ellipse are
\[\label{equations/15.6}\tag*{(15.6)}a=p/(1-e^2)=\alpha/2|E|,\qquad b=p/\sqrt{1-e^2}=M/\sqrt{2m|E|}\]
11The least possible value of the energy is (15.3) \(U_{\text{eff,min}}=-m\alpha^2/2M^2\) , and then \(e = 0\), i.e. the ellipse becomes a circle. It may be noted that the major axis of the ellipse depends only on the energy of the particle, and not on its angular momentum. The least and greatest distances from the centre of the field (the focus of the ellipse) are
\[\label{equations/15.7}\tag*{(15.7)}r_\text{min}=p/(1+e)=a(1-e),\qquad r_\text{max}=p/(1-e)=a(1+e)\]
These expressions, with \(a\) and \(e\) given by (15.6) \(a=p/(1-e^2)=\alpha/2|E|,\qquad b=p/\sqrt{1-e^2}=M/\sqrt{2m|E|}\) and (15.4) \(p=M^2/m\alpha,\qquad e=\sqrt{1+(2EM^2/m\alpha^2)}\) , can, of course, also be obtained directly as the roots of the equation \(U_\text{eff}(r) = E\).
The period \(T\) of revolution in an elliptical orbit is conveniently found by using the law of conservation of angular momentum in the form of the area integral (14.3) \(M=2m\dot{f}\) . Integrating this equation with respect to time from zero to \(T\), we have \(2mf = TM\), where \(f\) is the area of the orbit. For an ellipse \(f = \pi ab\), and by using the formulae (15.6) \(a=p/(1-e^2)=\alpha/2|E|,\qquad b=p/\sqrt{1-e^2}=M/\sqrt{2m|E|}\) we find
\[\label{equations/15.8}\tag*{(15.8)}\begin{align} T&=2\pi a^{3/2}\sqrt{m/\alpha} \\ &=\pi\alpha\sqrt{m/2|E|^3} \end{align}\]
The proportionality between the square of the period and the cube of the linear dimension of the orbit has already been demonstrated in §10. Mechanical similarity. It may also be noted that the period depends only on the energy of the particle.
For \(E \gt 0\) the motion is
infinite. If \(E > 0\), the
eccentricity \(e > 1\), i.e. the the
path is a hyperbola with the origin as internal focus
fig12. The distance of the perihelion from the focus is
\[\label{equations/15.9}\tag*{(15.9)}r_\text{min}=p/(e+1)=a(e-1)\]
where \(a = p/(e^2-1) = \alpha/2E\) is the “semi-axis” of the hyperbola.
12If \(E = 0\), the eccentricity \(e = 1\), and the particle moves in a parabola with perihelion distance \(r_\text{min}= \textstyle\frac{1}{2}\displaystyle p\). This case occurs if the particle starts from rest at infinity.
The co-ordinates of the particle as functions of time in the orbit may be found by means of the general formula (14.6) \(t=\int\mathrm{d}^{}r\bigg/\sqrt{\frac{2}{m}(E-U(r))-\frac{M^2}{m^2r^2}}+\text{constant}\) . They may be represented in a convenient parametric form as follows.
Let us first consider elliptical orbits. With \(a\) and \(e\) given by (15.6) \(a=p/(1-e^2)=\alpha/2|E|,\qquad b=p/\sqrt{1-e^2}=M/\sqrt{2m|E|}\) and (15.4) \(p=M^2/m\alpha,\qquad e=\sqrt{1+(2EM^2/m\alpha^2)}\) we can write the integral (14.6) \(t=\int\mathrm{d}^{}r\bigg/\sqrt{\frac{2}{m}(E-U(r))-\frac{M^2}{m^2r^2}}+\text{constant}\) for the time as
\[\begin{align} t &=\sqrt{\frac{m}{2|E|}}\int\frac{r\mathrm{d}^{}r}{\sqrt{-r^2+(\alpha/|E|)r-(M^2/2m|E|)}} \\ &=\sqrt{\frac{ma}{\alpha}}\int\frac{r\mathrm{d}^{}r}{\sqrt{a^2e^2-(r-a)^2}} \end{align}\]
The obvious substitution \(r-a =-ae\cos\xi\) converts the integral to
\[ t=\sqrt{\frac{ma^3}{\alpha}}\int(1-e\cos\xi)\mathrm{d}^{}\xi=\sqrt{\frac{ma^3}{\alpha}}(\xi-e\sin\xi)+\text{constant} \]
If time is measured in such a way that the constant is zero, we have the following parametric dependence of \(r\) on \(t\):
\[\label{equations/15.10}\tag*{(15.10)}r=a(1-e\cos\xi),\qquad t=\sqrt{ma^3/\alpha}(\xi-e\sin\xi),\]
the particle being at perihelion at \(t = 0\). The Cartesian co-ordinates \(x = r\cos\phi, y = r \sin\phi\) (the \(x\) and \(y\) axes being respectively parallel to the major and minor axes of the ellipse) can likewise be expressed in terms of the parameter \(\xi\). From (15.5) \(p/r=1+e\cos\phi\) and (15.10) \(r=a(1-e\cos\xi),\qquad t=\sqrt{ma^3/\alpha}(\xi-e\sin\xi),\) we have
\[ ex=p-r=a(1-e^2)-a(1-e\cos\xi)=ae(\cos\xi-e) \]
\(y\) is equal to \(\sqrt{r^2-x^2}\). Thus
\[\label{equations/15.11}\tag*{(15.11)}x=a(\cos\xi-e),\qquad y=a\sqrt{q-e^2}\sin\xi\]
A complete passage round the ellipse corresponds to an increase of \(\xi\) from \(0\) to \(2\pi\).
Entirely similar calculations for the hyperbolic orbits give
\[\label{equations/15.12}\tag*{(15.12)}\begin{align} r &= a(e\mathop{\mathrm{ch}}\xi-1), & t&=\sqrt{ma^3/\alpha}(e\mathop{\mathrm{sh}}\xi-\xi), \\ x &= a(e-\mathop{\mathrm{ch}}\xi)), & t&=a\sqrt{e^2-1}\mathop{\mathrm{sh}}\xi, \end{align}\]
where the parameter \(\xi\) varies from \(-\infty\) to \(+\infty\). Let us now consider motion in a repulsive field, where
\[\label{equations/15.13}\tag*{(15.13)}U=\alpha/r,\quad(\alpha\gt0)\]
Here the effective potential energy is
\[ U_\text{eff}=\frac{\alpha}{r}+\frac{M^2}{2mr^2} \]
and decreases monotonically from \(+\infty\) to zero as \(r\) varies from zero to infinity. The energy of the particle must be positive, and the motion is always infinite. The calculations are exactly similar to those for the attractive field. The path is a hyperbola (or, if \(E = 0\), a parabola):
\[\label{equations/15.14}\tag*{(15.14)}p/r=-1+e\cos\phi\]
where \(p\) and \(e\) are again given by
(15.4)
\(p=M^2/m\alpha,\qquad
e=\sqrt{1+(2EM^2/m\alpha^2)}\)
. The path passes the centre of the field in the manner
shown in fig13 The perihelion distance is
\[\label{equations/15.15}\tag*{(15.15)}r_\text{min}=p/(e-1)=a(e+1)\]
The time dependence is given by the parametric equations
\[\label{equations/15.16}\tag*{(15.16)}\begin{align} r &= a(e\mathop{\mathrm{ch}}\xi+1), & t&=\sqrt{ma^3/\alpha}(e\mathop{\mathrm{sh}}\xi+\xi), \\ x &= a(\mathop{\mathrm{ch}}\xi+e)), & t&=a\sqrt{e^2-1}\mathop{\mathrm{sh}}\xi, \end{align}\]
To conclude this section, we shall show that there is an integral of the motion which exists only in fields \(U = \alpha/r\) (with either sign of \(\alpha\)). It is easy to verify by direct calculation that the quantity
\[\label{equations/15.17}\tag*{(15.17)}\boldsymbol{v}\times\boldsymbol{M}+\alpha\boldsymbol{r}/r\]
is constant. For its total time derivative is \(\dot{\boldsymbol{v}}\times\boldsymbol{M}+\alpha\boldsymbol{v}/r-\alpha\boldsymbol{r}(\boldsymbol{v}\cdot\boldsymbol{r})/r^3\) or,
\[ m\boldsymbol{r}(\boldsymbol{v}\cdot\dot{\boldsymbol{v}})-m\boldsymbol{v}(\boldsymbol{r}\cdot\dot{\boldsymbol{v}})+\alpha\boldsymbol{v}/r-\alpha\boldsymbol{r}(\boldsymbol{r}\cdot\boldsymbol{r})/r^3 \]
Putting \(m\dot{\boldsymbol{v}} = \alpha\boldsymbol{r}/r^3\) from the equation of motion, we find that this expression vanishes.
13The direction of the conserved vector (15.17) \(\boldsymbol{v}\times\boldsymbol{M}+\alpha\boldsymbol{r}/r\) is along the major axis from the focus to the perihelion, and its magnitude is \(\alpha e\). This is most simply seen by considering its value at perihelion.
It should be emphasised that the integral (15.17) \(\boldsymbol{v}\times\boldsymbol{M}+\alpha\boldsymbol{r}/r\) of the motion, like \(M\) and \(E\), is a one-valued function of the state (position and velocity) of the particle. We shall see in §50-general-properties-of-motion-in-s-dimensions that the existence of such a further one-valued integral is due to the degeneracy of the motion.