Hamilton's principle/Applications/Geodesics - Revision history

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	\qquad\qquad\mathrm{(8)}</math>		\qquad\qquad\mathrm{(8)}</math>

	<math>L</math> carries no explicit <math>\lambda</math>, so ~~the Beltrami identity~~		<math>L</math> carries no explicit <math>\lambda</math>, so [[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] gives
	of [[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] gives

	:<math>\dot\theta^2 + \sin^2\theta\,\dot\varphi^2 = 1		:<math>\dot\theta^2 + \sin^2\theta\,\dot\varphi^2 = 1
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	== Where this goes ==		== Where this goes ==

	~~The Einstein-Hilbert action on~~		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|The Einstein-Hilbert action]] makes free fall
	[[Hamilton's principle/Applications/The Einstein-Hilbert action\|~~the~~ Einstein-Hilbert action]] makes free fall
	a geodesic of spacetime, so (6) becomes the equation of motion of a body under		a geodesic of spacetime, so (6) becomes the equation of motion of a body under
	gravity, and (7) is derived from the same variational principle. Conjugate points		gravity, and (7) is derived from the same variational principle. Conjugate points

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			{{DISPLAYTITLE:Geodesics}}
	A '''geodesic''' is a stationary curve of arc length on a surface or in a curved		A '''geodesic''' is a stationary curve of arc length on a surface or in a curved
	space. Geodesics are the straight lines of curved geometry, and they are the		space. Geodesics are the straight lines of curved geometry, and they are the
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	<references />		<references />

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A '''geodesic''' is a stationary curve of arc length on a surface or in a curved
space. Geodesics are the straight lines of curved geometry, and they are the
setting in which this volume's central distinction is easiest to see: every
great-circle arc on a sphere is a geodesic, and only the short ones are shortest.

== The functional ==

[[File:OPL-HP-F-05.png|thumb|right|400px|Geodesics from a pole refocusing at the antipode.]]

On a surface with metric

:<math>ds^2 = \sum_{i,k} g_{ik}(q)\, dq_i\, dq_k
\qquad\qquad\mathrm{(1)}</math>

the length of a curve <math>q(\lambda)</math> is

:<math>J[q] = \int_{\lambda_0}^{\lambda_1}
\sqrt{ \sum_{i,k} g_{ik}\, \dot q_i \dot q_k }\; d\lambda
\qquad\qquad\mathrm{(2)}</math>

with the dot denoting <math>d/d\lambda</math>.

'''The integrand is homogeneous of degree one in the velocities.''' That has a
consequence worth naming before any calculation: (2) is unchanged under
reparametrisation <math>\lambda \to p(\lambda)</math>. By Noether's second
theorem on
[[Hamilton's principle/Mathematics/Noether's second theorem]], invariance under a
group of arbitrary functions forces an identity among the Lagrange expressions —
here exactly her Weierstrass parametric example,
<math>\sum\psi_i \dot q_i = 0</math>. The Euler-Lagrange equations of (2) are
therefore '''not independent''', and the Hessian in the velocities is singular.

The standard remedy is to use the '''energy functional''' instead:

:<math>E[q] = \tfrac{1}{2}\int \sum_{i,k} g_{ik}\,\dot q_i \dot q_k \; d\lambda
\qquad\qquad\mathrm{(3)}</math>

whose integrand is quadratic and non-degenerate. Extremals of (3) are extremals
of (2) traversed at constant speed, so nothing is lost, and the parametrisation
degeneracy is fixed rather than carried. This is the practical reason (3) is used
throughout differential geometry.

== The geodesic equation ==

[[File:OPL-HP-F-22.png|thumb|right|400px|Geodesic behaviour on flat, spherical and saddle surfaces.]]

Apply the Euler-Lagrange equation of
[[Hamilton's principle/Mathematics/Several dependent variables]] to (3) with
<math>L = \tfrac{1}{2}g_{ik}\dot q_i\dot q_k</math>. Then

:<math>\frac{\partial L}{\partial \dot q_j} = g_{jk}\dot q_k,
\qquad
\frac{\partial L}{\partial q_j}
= \tfrac{1}{2}\,\partial_j g_{ik}\, \dot q_i \dot q_k
\qquad\qquad\mathrm{(4)}</math>

summation implied. The equation is

:<math>\frac{d}{d\lambda}\bigl( g_{jk}\dot q_k \bigr)
- \tfrac{1}{2}\,\partial_j g_{ik}\,\dot q_i\dot q_k = 0
\qquad\qquad\mathrm{(5)}</math>

Expanding the total derivative gives
<math>g_{jk}\ddot q_k + \partial_i g_{jk}\dot q_i \dot q_k</math>. Symmetrising
the middle term over <math>i</math> and <math>k</math>, which is legitimate since
it multiplies the symmetric <math>\dot q_i \dot q_k</math>, and contracting with
the inverse metric <math>g^{mj}</math>,

:<math>\ddot q_m + \Gamma^m_{ik}\, \dot q_i \dot q_k = 0
\qquad\qquad\mathrm{(6)}</math>

with

:<math>\Gamma^m_{ik} = \tfrac{1}{2} g^{mj}
\bigl( \partial_i g_{jk} + \partial_k g_{ji} - \partial_j g_{ik} \bigr)
\qquad\qquad\mathrm{(7)}</math>

Equation (6) is the '''geodesic equation''' and (7) are the '''Christoffel
symbols'''. They are not an extra structure imposed on the metric; they fall out
of the variational problem, and this derivation is the shortest honest route to
them.

== The sphere ==

Take the unit sphere, <math>ds^2 = d\theta^2 + \sin^2\theta\, d\varphi^2</math>,
so <math>L = \tfrac{1}{2}(\dot\theta^2 + \sin^2\theta\,\dot\varphi^2)</math>.

<math>\varphi</math> is cyclic, so by (7) of
[[Hamilton's principle/Mathematics/Several dependent variables]] its conjugate
momentum is conserved:

:<math>p_\varphi = \sin^2\theta\;\dot\varphi = h = \text{const}
\qquad\qquad\mathrm{(8)}</math>

<math>L</math> carries no explicit <math>\lambda</math>, so the Beltrami identity
of [[Hamilton's principle/Mathematics/The Beltrami identity]] gives

:<math>\dot\theta^2 + \sin^2\theta\,\dot\varphi^2 = 1
\qquad\qquad\mathrm{(9)}</math>

taking the constant as 1, which fixes <math>\lambda</math> as arc length.
Eliminating <math>\dot\varphi</math> between (8) and (9),

:<math>\dot\theta^2 = 1 - \frac{h^2}{\sin^2\theta}
\qquad\qquad\mathrm{(10)}</math>

and dividing by (8) to get the orbit equation,

:<math>\frac{d\varphi}{d\theta}
= \frac{h}{\sin^2\theta \sqrt{1 - h^2/\sin^2\theta}}
= \frac{h}{\sin\theta\,\sqrt{\sin^2\theta - h^2}}
\qquad\qquad\mathrm{(11)}</math>

The substitution <math>u = \cot\theta</math> gives
<math>du = -d\theta/\sin^2\theta</math> and turns (11) into

:<math>d\varphi = \frac{-h\,du}{\sqrt{1 - h^2 - h^2u^2}}
\qquad\qquad\mathrm{(12)}</math>

which integrates to an arccosine. Writing
<math>\alpha^2 = (1-h^2)/h^2</math>,

:<math>\cot\theta = \alpha \cos(\varphi - \varphi_0)
\qquad\qquad\mathrm{(13)}</math>

Multiplying (13) by <math>\sin\theta</math> and expanding the cosine gives a
relation linear in <math>\sin\theta\cos\varphi</math>,
<math>\sin\theta\sin\varphi</math> and <math>\cos\theta</math> — that is, linear
in the Cartesian coordinates. So the curve lies in a plane through the centre,
and the geodesics of the sphere are exactly the '''great circles'''.

A shorter argument exists by symmetry: any point and direction lie in a unique
plane through the centre, and reflection in that plane fixes both while mapping
geodesics to geodesics. The calculation above is given because it produces (13)
in a usable form and shows the machinery working.

== Stationary is not shortest ==

'''Every''' great-circle arc satisfies (6). The long way round from London to
Tokyo is a geodesic. It is not the shortest path.

The Jacobi analysis on
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]]
gives the exact boundary. The accessory equation on the unit sphere is
<math>u'' + u = 0</math>, so the first conjugate point of any starting point is
its antipode, at arc length <math>\pi</math>. Therefore:

* arcs shorter than <math>\pi</math> are the unique shortest paths;
* arcs of exactly <math>\pi</math> are stationary and non-unique — every meridian between the poles has the same length;
* arcs longer than <math>\pi</math> are stationary and not shortest.

'''Nothing local distinguishes the three cases.''' They satisfy the same
equation, at every point, with the same Legendre coefficient. Only the total arc
length differs. This is the volume's counterexample in its most concrete form,
and it is why the subject is called stationary action rather than least action.

== Other surfaces ==

'''The plane and the cylinder''' have zero curvature. The accessory equation is
<math>u'' = 0</math>, which has no second zero, so there are no conjugate points
and geodesics minimise without limit. On the cylinder they are helices, and the
absence of conjugate points does not prevent several geodesics joining two
points — the helix may wind any number of times. Uniqueness and minimality are
different claims.

'''Negatively curved surfaces''' give <math>u'' - \vert K \vert u = 0</math>,
whose solutions grow, so again no conjugate points. Geodesics diverge and each is
the unique minimiser.

'''Positively curved surfaces''' give the sphere's behaviour: geodesics reconverge
and minimality fails past the refocusing point. Conjugate points are a symptom of
positive curvature, and that is the geometric content of the Jacobi condition.

== Where this goes ==

The Einstein-Hilbert action on
[[Hamilton's principle/Applications/The Einstein-Hilbert action]] makes free fall
a geodesic of spacetime, so (6) becomes the equation of motion of a body under
gravity, and (7) is derived from the same variational principle. Conjugate points
then acquire physical meaning as the refocusing of light by a gravitational lens.

<blockquote>
Provenance: every equation on this page is derived here. No primary source is
cited and none is claimed.
</blockquote>

== Next ==

[[Hamilton's principle/Applications/Fermat's principle]] treats the optical
analogue, where the same machinery gives Snell's law.

== References ==

<references />

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	reparametrisation <math>\lambda \to p(\lambda)</math>. By Noether's second		reparametrisation <math>\lambda \to p(\lambda)</math>. By Noether's second
	theorem on		theorem on
	[[Hamilton's principle/Mathematics/Noether's second theorem]], invariance under a		[[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]], invariance under a
	group of arbitrary functions forces an identity among the Lagrange expressions —		group of arbitrary functions forces an identity among the Lagrange expressions —
	here exactly her Weierstrass parametric example,		here exactly her Weierstrass parametric example,
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	Apply the Euler-Lagrange equation of		Apply the Euler-Lagrange equation of
	[[Hamilton's principle/Mathematics/Several dependent variables]] to (3) with		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]] to (3) with
	<math>L = \tfrac{1}{2}g_{ik}\dot q_i\dot q_k</math>. Then		<math>L = \tfrac{1}{2}g_{ik}\dot q_i\dot q_k</math>. Then

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	<math>\varphi</math> is cyclic, so by (7) of		<math>\varphi</math> is cyclic, so by (7) of
	[[Hamilton's principle/Mathematics/Several dependent variables]] its conjugate		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]] its conjugate
	momentum is conserved:		momentum is conserved:

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	<math>L</math> carries no explicit <math>\lambda</math>, so the Beltrami identity		<math>L</math> carries no explicit <math>\lambda</math>, so the Beltrami identity
	of [[Hamilton's principle/Mathematics/The Beltrami identity]] gives		of [[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] gives

	:<math>\dot\theta^2 + \sin^2\theta\,\dot\varphi^2 = 1		:<math>\dot\theta^2 + \sin^2\theta\,\dot\varphi^2 = 1
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	The Jacobi analysis on		The Jacobi analysis on
	[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]]		[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points\|the Jacobi condition and conjugate points]]
	gives the exact boundary. The accessory equation on the unit sphere is		gives the exact boundary. The accessory equation on the unit sphere is
	<math>u'' + u = 0</math>, so the first conjugate point of any starting point is		<math>u'' + u = 0</math>, so the first conjugate point of any starting point is
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	The Einstein-Hilbert action on		The Einstein-Hilbert action on
	[[Hamilton's principle/Applications/The Einstein-Hilbert action]] makes free fall		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|the Einstein-Hilbert action]] makes free fall
	a geodesic of spacetime, so (6) becomes the equation of motion of a body under		a geodesic of spacetime, so (6) becomes the equation of motion of a body under
	gravity, and (7) is derived from the same variational principle. Conjugate points		gravity, and (7) is derived from the same variational principle. Conjugate points
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	== Next ==		== Next ==

	[[Hamilton's principle/Applications/Fermat's principle]] treats the optical		[[Hamilton's principle/Applications/Fermat's principle\|Fermat's principle]] treats the optical
	analogue, where the same machinery gives Snell's law.		analogue, where the same machinery gives Snell's law.