Central force and the Kepler problem

The Kepler problem — two bodies attracting by an inverse-square law — is the case Hamilton worked in detail in the 1834 essay, to show his method producing known results by a new route. Because that essay is a verified source for this volume, this page can follow his own treatment and quote his equations rather than reconstruct them.

Reduction

Lambert's theorem: chord and sum of radii.

For two points $m_1$, $m_2$ attracting according to a function of their distance, Hamilton takes the force-function

$U=m_1{m}_{2}f(r)\mathrm{(}\mathrm{1}\mathrm{)}$

and separates the motion of the centre of gravity from the relative motion, so that the characteristic function splits as $V=V_0+V_{00}$, with $V_0$ the relative action. The centre of gravity moves uniformly in a straight line, which Hamilton recovers as his equations (83.) and (84.). The problem reduces to the relative motion, described by a characteristic function of the relative coordinates alone.

Two conservation laws come out of that reduction. Setting $\xi =r\cos \theta$, $\eta =r\sin \theta$ in the plane of the motion, Hamilton obtains

$r^2{\theta }^{\prime }=r_0^2{{\theta }_0}^{\prime }=h\mathrm{(}\mathrm{2}\mathrm{)}$

so $\frac{1}{2}h$ is the constant areal velocity — Kepler's second law, and an instance of the rotational entry in the symmetry-to-conservation table on Noether's first theorem. The energy relation follows as his equation (100.), fixing the constant $H_0$.

That the orbit lies in a plane is not assumed. Hamilton derives it: his equations (91.) and (92.) contain, as he puts it, the known law of equable description of areas and the law of a plane relative orbit.

The inverse-square case

The effective potential of a central-force problem.

Specialise to Newtonian attraction by setting

$f(r)=\frac{1}{r}\mathrm{(}\mathrm{3}\mathrm{)}$

and abbreviate

$m_1+m_2=\mu ,\frac{h^2}{\mu }=p,\frac{-m_1{m}_2}{2H_0}=a\mathrm{(}\mathrm{4}\mathrm{)}$

The characteristic function of the relative motion is then

$V_0=\frac{m_1{m}_2}{\sqrt{\mu }}(\vartheta \sqrt{p}+{\int }_{r_0}^r\pm \sqrt{\frac{2}{r}-\frac{1}{a}-\frac{p}{r^2}}dr)\mathrm{(}\mathrm{5}\mathrm{)}$

The orbit equation is the derivative of (5) with respect to $p$, and Hamilton reads off the classification directly: the orbit is an ellipse, hyperbola or parabola according as $a$ is positive, negative or zero, with the origin at a focus and $p$ the semiparameter.

Kepler's first law is therefore a statement about the sign of one constant in the characteristic function.

Kepler's equation

Introducing the eccentricity and the eccentric anomaly,

$e=\sqrt{1-\frac{p}{a}},\upsilon ={\cos }^{-1}\left(\frac{a-r}{ae}\right)\mathrm{(}\mathrm{6}\mathrm{)}$

so that $r=a(1-e\cos \upsilon )$, the time integral becomes

$t=\sqrt{\frac{a^3}{\mu }}(\upsilon -{\upsilon }_0-e\sin \upsilon +e\sin {\upsilon }_0)\mathrm{(}\mathrm{7}\mathrm{)}$

Equation (7) is Kepler's equation, and the factor $\sqrt{a^3/\mu }$ is Kepler's third law: the period depends on the semi-major axis alone, as $a^{3/2}$.

The corresponding characteristic function is remarkably compact:

$V_0=m_1{m}_2\sqrt{\frac{a}{\mu }}(\upsilon -{\upsilon }_0+e\sin \upsilon -e\sin {\upsilon }_0)\mathrm{(}\mathrm{8}\mathrm{)}$

Compare (7) and (8): the same expression with two signs changed. All three of Kepler's laws are consequences of one function.

Lambert's theorem

Hamilton's most striking result in this section concerns what $V_0$ depends on. Introducing the chord $\tau$ joining the two positions and the sum of the radii $\sigma =r+r_0$, he shows that $V_0$ is a function of $\sigma$, $\tau$ and $a$ only:

$V_0=\frac{m_1{m}_{2}w}{m_1+m_2},w=w(\sigma ,\tau ,a)\mathrm{(}\mathrm{9}\mathrm{)}$

The two positions enter only through the chord between them and the sum of their distances from the focus — not through their angular positions, and not through the shape of the arc.

Since the time follows from $w$ by differentiation, Hamilton obtains

$t=\frac{2a^2}{\mu }\frac{\delta w}{\delta a}\mathrm{(}\mathrm{10}\mathrm{)}$

That is Lambert's theorem: the transit time between two points on a Kepler orbit depends only on the semi-major axis, the chord, and the sum of the radii. It is the basis of orbit determination from two observations, and it falls out of the characteristic function as a corollary about which variables that function contains.

Hamilton makes the case for his method in exactly this way: not by computing faster, but by making a structural fact — which variables the action depends on — immediately visible.

For the parabolic limit $a\to \mathrm{\infty }$ he gives the closed form

$w=2\sqrt{\mu }(\sqrt{\sigma +\tau }\mp \sqrt{\sigma -\tau })\mathrm{(}\mathrm{11}\mathrm{)}$

with the corresponding time as his (V3.), which he notes is a known expression.

The modern route, for comparison

The same results follow from the Lagrangian

$L=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\theta }}^2)+\frac{k}{r}\mathrm{(}\mathrm{12}\mathrm{)}$

$\theta$ is cyclic, giving $m{r}^2\dot{\theta }=\ell$, which is (2). $L$ carries no explicit time, so the Beltrami identity gives energy conservation. Eliminating $\dot{\theta }$,

$E=\frac{1}{2}m{\dot{r}}^2+\frac{{\ell }^2}{2m{r}^2}-\frac{k}{r}\mathrm{(}\mathrm{13}\mathrm{)}$

a one-dimensional problem in the effective potential. The substitution $u=1/r$ converts the orbit equation to $u^{″}+u=mk/{\ell }^2$, whose solution is a conic.

Both routes work. The modern one reaches the orbit faster; Hamilton's produces Lambert's theorem as a by-product, and the modern one does not.

Degeneracy

The inverse-square law is special beyond the closure of its orbits. Its extra conserved quantity — the Laplace-Runge-Lenz vector, fixing the orientation of the apsidal line — corresponds to a symmetry not present for a general central force, and by Noether's first theorem there must be a group behind it. The group is larger than the rotations, and the closure of Kepler orbits is its signature. That extension is named as further reading on Sources and is not derived here.

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the rigid body treats a system whose configuration space is not flat.