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	<math>\theta</math> is cyclic, giving <math>mr^2\dot\theta = \ell</math>, which is		<math>\theta</math> is cyclic, giving <math>mr^2\dot\theta = \ell</math>, which is
	(2). <math>L</math> carries no explicit time, so ~~the Beltrami identity of~~		(2). <math>L</math> carries no explicit time, so [[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] gives energy
	[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] gives energy
	conservation. Eliminating <math>\dot\theta</math>,		conservation. Eliminating <math>\dot\theta</math>,

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			{{DISPLAYTITLE:Central force and the Kepler problem}}
	The '''Kepler problem''' — two bodies attracting by an inverse-square law — is the		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		case Hamilton worked in detail in the 1834 essay, to show his method producing
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	<references />		<references />

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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 ==

[[File:OPL-HP-F-30.png|thumb|right|400px|Lambert's theorem: chord and sum of radii.]]

For two points <math>m_1</math>, <math>m_2</math> attracting according to a
function of their distance, Hamilton takes the force-function

:<math>U = m_1 m_2\, f(r)
\qquad\qquad\mathrm{(1)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (71.).
Verbatim. ESTABLISHED source.</ref>

and separates the motion of the centre of gravity from the relative motion, so
that the characteristic function splits as <math>V = V_0 + V_{00}</math>, with
<math>V_0</math> 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
<math>\xi = r\cos\theta</math>, <math>\eta = r\sin\theta</math> in the plane of
the motion, Hamilton obtains

:<math>r^2\theta' = r_0^2\theta_0' = h
\qquad\qquad\mathrm{(2)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (97.).
Verbatim. ESTABLISHED source.</ref>

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

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 ==

[[File:OPL-HP-F-47.png|thumb|right|400px|The effective potential of a central-force problem.]]

Specialise to Newtonian attraction by setting

:<math>f(r) = \frac{1}{r}
\qquad\qquad\mathrm{(3)}</math>

and abbreviate

:<math>m_1 + m_2 = \mu,
\qquad
\frac{h^2}{\mu} = p,
\qquad
\frac{-m_1m_2}{2H_0} = a
\qquad\qquad\mathrm{(4)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equations (101.) and
(102.). Verbatim. ESTABLISHED source.</ref>

The characteristic function of the relative motion is then

:<math>V_0 = \frac{m_1m_2}{\sqrt{\mu}}
\left( \vartheta\sqrt{p}
+ \int_{r_0}^{r} \pm \sqrt{ \frac{2}{r} - \frac{1}{a} - \frac{p}{r^2} }\; dr
\right)
\qquad\qquad\mathrm{(5)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (A3.).
Verbatim. <math>\vartheta</math> is the angle between the initial and final
radii; the sign follows whether <math>r</math> is increasing. ESTABLISHED
source.</ref>

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

:<math>e = \sqrt{1 - \frac{p}{a}},
\qquad
\upsilon = \cos^{-1}\!\left( \frac{a - r}{ae} \right)
\qquad\qquad\mathrm{(6)}</math>

so that <math>r = a(1 - e\cos\upsilon)</math>, the time integral becomes

:<math>t = \sqrt{\frac{a^3}{\mu}}\;
\bigl( \upsilon - \upsilon_0 - e\sin\upsilon + e\sin\upsilon_0 \bigr)
\qquad\qquad\mathrm{(7)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equations (103.),
(104.), (106.) and (107.). Verbatim. ESTABLISHED source.</ref>

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

The corresponding characteristic function is remarkably compact:

:<math>V_0 = m_1m_2\sqrt{\frac{a}{\mu}}\;
\bigl( \upsilon - \upsilon_0 + e\sin\upsilon - e\sin\upsilon_0 \bigr)
\qquad\qquad\mathrm{(8)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (F3.).
Verbatim. ESTABLISHED source.</ref>

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 <math>V_0</math>
depends on. Introducing the chord <math>\tau</math> joining the two positions and
the sum of the radii <math>\sigma = r + r_0</math>, he shows that
<math>V_0</math> is a function of <math>\sigma</math>, <math>\tau</math> and
<math>a</math> only:

:<math>V_0 = \frac{m_1m_2\, w}{m_1 + m_2},
\qquad
w = w(\sigma, \tau, a)
\qquad\qquad\mathrm{(9)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (H3.), with
<math>\sigma</math> and <math>\tau</math> defined at his (113.) and (109.).
Verbatim. ESTABLISHED source.</ref>

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 <math>w</math> by differentiation, Hamilton obtains

:<math>t = \frac{2a^2}{\mu}\,\frac{\delta w}{\delta a}
\qquad\qquad\mathrm{(10)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (S3.).
Verbatim. He describes the result as a new proof of the well-known theorem that
the elliptic time also depends on the same chord and sum of radii. ESTABLISHED
source.</ref>

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 <math>a \to \infty</math> he gives the closed form

:<math>w = 2\sqrt{\mu}\,\bigl( \sqrt{\sigma + \tau} \mp \sqrt{\sigma - \tau} \bigr)
\qquad\qquad\mathrm{(11)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (U3.).
Verbatim. ESTABLISHED source.</ref>

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

:<math>L = \tfrac{1}{2}m\bigl( \dot r^2 + r^2\dot\theta^2 \bigr) + \frac{k}{r}
\qquad\qquad\mathrm{(12)}</math>

<math>\theta</math> is cyclic, giving <math>mr^2\dot\theta = \ell</math>, which is
(2). <math>L</math> carries no explicit time, so the Beltrami identity of
[[Hamilton's principle/Mathematics/The Beltrami identity]] gives energy
conservation. Eliminating <math>\dot\theta</math>,

:<math>E = \tfrac{1}{2}m\dot r^2 + \frac{\ell^2}{2mr^2} - \frac{k}{r}
\qquad\qquad\mathrm{(13)}</math>

a one-dimensional problem in the effective potential. The substitution
<math>u = 1/r</math> converts the orbit equation to
<math>u'' + u = mk/\ell^2</math>, 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
[[Hamilton's principle/Mathematics/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
[[Hamilton's principle/Sources]] and is not derived here.

== Next ==

[[Hamilton's principle/Applications/The rigid body]] treats a system whose
configuration space is not flat.

== References ==

<references />

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	law, and an instance of the rotational entry in the symmetry-to-conservation		law, and an instance of the rotational entry in the symmetry-to-conservation
	table on		table on
	[[Hamilton's principle/Mathematics/Noether's first theorem]]. The energy relation		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]]. The energy relation
	follows as his equation (100.), fixing the constant <math>H_0</math>.		follows as his equation (100.), fixing the constant <math>H_0</math>.

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	<math>\theta</math> is cyclic, giving <math>mr^2\dot\theta = \ell</math>, which is		<math>\theta</math> is cyclic, giving <math>mr^2\dot\theta = \ell</math>, which is
	(2). <math>L</math> carries no explicit time, so the Beltrami identity of		(2). <math>L</math> carries no explicit time, so the Beltrami identity of
	[[Hamilton's principle/Mathematics/The Beltrami identity]] gives energy		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] gives energy
	conservation. Eliminating <math>\dot\theta</math>,		conservation. Eliminating <math>\dot\theta</math>,

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	apsidal line — corresponds to a symmetry not present for a general central force,		apsidal line — corresponds to a symmetry not present for a general central force,
	and by		and by
	[[Hamilton's principle/Mathematics/Noether's first theorem]] there must be a		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]] there must be a
	group behind it. The group is larger than the rotations, and the closure of Kepler		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		orbits is its signature. That extension is named as further reading on
	[[Hamilton's principle/Sources]] and is not derived here.		[[Hamilton's principle/Sources\|Sources]] and is not derived here.

	== Next ==		== Next ==

	[[Hamilton's principle/Applications/The rigid body]] treats a system whose		[[Hamilton's principle/Applications/The rigid body\|the rigid body]] treats a system whose
	configuration space is not flat.		configuration space is not flat.

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