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			{{DISPLAYTITLE:The catenary}}
	The '''catenary''' is the shape of a uniform flexible chain hanging under its own		The '''catenary''' is the shape of a uniform flexible chain hanging under its own
	weight between two fixed points. It is the standard worked example of an		weight between two fixed points. It is the standard worked example of an
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	<references />		<references />

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The '''catenary''' is the shape of a uniform flexible chain hanging under its own
weight between two fixed points. It is the standard worked example of an
isoperimetric problem: the chain's length is prescribed, and the shape minimises
potential energy subject to that constraint.

[[Hamilton's principle/Mathematics/The Beltrami identity]] derived the curve with
the constraint suppressed, and
[[Hamilton's principle/Mathematics/Isoperimetric problems]] restored it. This
page completes the problem: the constants, the tension, and the physical
readings.

== The functional and the constraint ==

[[File:OPL-HP-F-23.png|thumb|right|400px|A catenary and the directrix from which its tension is measured.]]

Let the chain have uniform linear density <math>\rho</math> and hang between
<math>(x_0, y_0)</math> and <math>(x_1, y_1)</math>, with <math>y</math>
measured upward. Its potential energy is

:<math>J[y] = \rho g_0 \int_{x_0}^{x_1} y\,\sqrt{1 + y'^2}\; dx
\qquad\qquad\mathrm{(1)}</math>

since the mass of an arc element is <math>\rho\,ds</math> at height
<math>y</math>. The chain is inextensible, so its length is prescribed:

:<math>K[y] = \int_{x_0}^{x_1} \sqrt{1 + y'^2}\; dx = \ell
\qquad\qquad\mathrm{(2)}</math>

This is one condition on the whole curve, not one at each point, so its
multiplier is a '''constant''' — the distinction drawn on
[[Hamilton's principle/Mathematics/Lagrange multipliers]]. Dropping the constant
factor <math>\rho g_0</math>, the augmented integrand is

:<math>L^* = (y + \mu)\sqrt{1 + y'^2}
\qquad\qquad\mathrm{(3)}</math>

== The shape ==

<math>L^*</math> carries no explicit <math>x</math>, so Beltrami applies. The
same collapse as in (6) of
[[Hamilton's principle/Applications/The brachistochrone]] gives

:<math>\frac{y + \mu}{\sqrt{1 + y'^2}} = C
\qquad\qquad\mathrm{(4)}</math>

Solve for the slope and separate:

:<math>\frac{dy}{\sqrt{(y+\mu)^2/C^2 - 1}} = dx
\qquad\qquad\mathrm{(5)}</math>

Substituting <math>y + \mu = C\cosh w</math> makes the radicand
<math>\sinh^2 w</math> and reduces the left side to <math>C\,dw</math>, so
<math>w = (x - x_0')/C</math> and

:<math>y = -\mu + C \cosh\!\left( \frac{x - x_0'}{C} \right)
\qquad\qquad\mathrm{(6)}</math>

The curve is a hyperbolic cosine. It is '''not''' a parabola, a point Galileo got
wrong and Huygens, Leibniz and Johann Bernoulli corrected in 1691. The two agree
to second order near the vertex, which is why the error survived so long.

== Fixing the three constants ==

<math>C</math>, <math>x_0'</math> and <math>\mu</math> are fixed by the two
endpoint conditions and by the length (2). The length of (6) integrates cleanly,
because <math>\sqrt{1+y'^2} = \cosh((x-x_0')/C)</math>:

:<math>\ell = C\left[ \sinh\!\left(\frac{x_1 - x_0'}{C}\right)
- \sinh\!\left(\frac{x_0 - x_0'}{C}\right) \right]
\qquad\qquad\mathrm{(7)}</math>

For the symmetric case, endpoints at <math>x = \pm d</math> and equal height,
symmetry gives <math>x_0' = 0</math> and (7) becomes

:<math>\ell = 2C\sinh\!\left( \frac{d}{C} \right)
\qquad\qquad\mathrm{(8)}</math>

Equation (8) determines <math>C</math> and is transcendental. Writing
<math>u = d/C</math>, it reads
<math>\sinh u / u = \ell/2d</math>. The left side
increases monotonically from 1, so a solution exists precisely when
<math>\ell > 2d</math> — the chain must be longer than the straight-line
distance — and it is then unique.

The case <math>\ell = 2d</math> is the '''abnormal''' case excluded on
[[Hamilton's principle/Mathematics/Isoperimetric problems]]: the taut straight
segment is the only admissible curve, it is an extremal of the length functional
alone, and the multiplier rule does not apply to it. The exclusion in the theorem
and the physical degenerate case are the same case.

<math>\mu</math> then follows from the endpoint height, and the vertex sits at
<math>y = C - \mu</math>.

== Reading the multiplier ==

The multiplier is not bookkeeping. Restoring the factor <math>\rho g_0</math>,
the horizontal component of the chain's tension is

:<math>T_x = \rho g_0 C
\qquad\qquad\mathrm{(9)}</math>

constant along the chain, as it must be since no horizontal force acts on any
element. The total tension at height <math>y</math> is

:<math>T = \rho g_0 (y + \mu)
\qquad\qquad\mathrm{(10)}</math>

which is (4) multiplied through. So <math>-\mu</math> is the height of the
'''directrix''', the level from which tension is measured, and the tension at any
point is the weight of a length of chain equal to the height above it.

That is a genuinely useful result and it is invisible in the unconstrained
treatment. It is the practical argument for multipliers made on
[[Hamilton's principle/Mathematics/Lagrange multipliers]], in a case where the
answer is a number an engineer wants.

Tension is least at the vertex, where it equals <math>T_x</math>, and greatest at
the supports. A chain hung slack has small <math>C</math> and low tension; pulling
it taut drives <math>C \to \infty</math> and the tension with it, which is why a
horizontal cable cannot be pulled truly straight.

== Is it a minimum? ==

<math>L^*_{y'y'} = (y+\mu)(1+y'^2)^{-3/2}</math>, positive wherever
<math>y + \mu > 0</math>, that is, above the directrix — which is everywhere on a
physical chain, since <math>y + \mu = T/\rho g_0 > 0</math>. So the strengthened
Legendre condition holds.

The Jacobi condition requires care here, because the second variation must be
tested only against variations preserving the length to first order, as noted at
the end of
[[Hamilton's principle/Mathematics/The second variation]]. Testing against
unrestricted variations gives the wrong answer. With the restriction imposed, the
catenary is a minimum.

== The related surface ==

Rotating (6) about the horizontal axis gives the '''catenoid''', the minimal
surface spanning two coaxial rings. That problem minimises area,

:<math>A = 2\pi\int y\sqrt{1+y'^2}\; dx
\qquad\qquad\mathrm{(11)}</math>

which is (1) without the constraint — so the unconstrained catenary of
[[Hamilton's principle/Mathematics/The Beltrami identity]] is the soap-film
problem, and the constrained one is the chain. The same integrand, two different
problems, distinguished only by whether a constraint is present.

The catenoid problem also exhibits the failure of existence. Pull the rings
sufficiently far apart and no catenoid connects them: the extremals cease to
exist and the physical film snaps to two discs. That is the '''Goldschmidt
solution''', and it is a discontinuous competitor no variational condition
detects, since it is not near any extremal in any norm. It is the standard
warning that stationarity theory presupposes existence, as noted on
[[Hamilton's principle/Mathematics/The Weierstrass excess function]].

<blockquote>
Provenance: the attributions to Galileo, and to the 1691 solutions of Huygens,
Leibniz and Johann Bernoulli, are editorial. No primary source for any of them
has been verified for this volume. Every equation here is derived on this page.
See [[Hamilton's principle/Sources]].
</blockquote>

== Next ==

[[Hamilton's principle/Applications/Geodesics]] treats the shortest-path problem
on a curved surface.

== References ==

<references />

← Older revision		Revision as of 04:58, 16 September 2026
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	potential energy subject to that constraint.		potential energy subject to that constraint.

	[[Hamilton's principle/Mathematics/The Beltrami identity]] derived the curve with		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] derived the curve with
	the constraint suppressed, and		the constraint suppressed, and
	[[Hamilton's principle/Mathematics/Isoperimetric problems]] restored it. This		[[Hamilton's principle/Mathematics/Isoperimetric problems\|Isoperimetric problems]] restored it. This
	page completes the problem: the constants, the tension, and the physical		page completes the problem: the constants, the tension, and the physical
	readings.		readings.
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	This is one condition on the whole curve, not one at each point, so its		This is one condition on the whole curve, not one at each point, so its
	multiplier is a '''constant''' — the distinction drawn on		multiplier is a '''constant''' — the distinction drawn on
	[[Hamilton's principle/Mathematics/Lagrange multipliers]]. Dropping the constant		[[Hamilton's principle/Mathematics/Lagrange multipliers\|Lagrange multipliers]]. Dropping the constant
	factor <math>\rho g_0</math>, the augmented integrand is		factor <math>\rho g_0</math>, the augmented integrand is

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	<math>L^*</math> carries no explicit <math>x</math>, so Beltrami applies. The		<math>L^*</math> carries no explicit <math>x</math>, so Beltrami applies. The
	same collapse as in (6) of		same collapse as in (6) of
	[[Hamilton's principle/Applications/The brachistochrone]] gives		[[Hamilton's principle/Applications/The brachistochrone\|the brachistochrone]] gives

	:<math>\frac{y + \mu}{\sqrt{1 + y'^2}} = C		:<math>\frac{y + \mu}{\sqrt{1 + y'^2}} = C
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	The case <math>\ell = 2d</math> is the '''abnormal''' case excluded on		The case <math>\ell = 2d</math> is the '''abnormal''' case excluded on
	[[Hamilton's principle/Mathematics/Isoperimetric problems]]: the taut straight		[[Hamilton's principle/Mathematics/Isoperimetric problems\|Isoperimetric problems]]: the taut straight
	segment is the only admissible curve, it is an extremal of the length functional		segment is the only admissible curve, it is an extremal of the length functional
	alone, and the multiplier rule does not apply to it. The exclusion in the theorem		alone, and the multiplier rule does not apply to it. The exclusion in the theorem
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	That is a genuinely useful result and it is invisible in the unconstrained		That is a genuinely useful result and it is invisible in the unconstrained
	treatment. It is the practical argument for multipliers made on		treatment. It is the practical argument for multipliers made on
	[[Hamilton's principle/Mathematics/Lagrange multipliers]], in a case where the		[[Hamilton's principle/Mathematics/Lagrange multipliers\|Lagrange multipliers]], in a case where the
	answer is a number an engineer wants.		answer is a number an engineer wants.

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	tested only against variations preserving the length to first order, as noted at		tested only against variations preserving the length to first order, as noted at
	the end of		the end of
	[[Hamilton's principle/Mathematics/The second variation]]. Testing against		[[Hamilton's principle/Mathematics/The second variation\|the second variation]]. Testing against
	unrestricted variations gives the wrong answer. With the restriction imposed, the		unrestricted variations gives the wrong answer. With the restriction imposed, the
	catenary is a minimum.		catenary is a minimum.
Line 144:		Line 144:

	which is (1) without the constraint — so the unconstrained catenary of		which is (1) without the constraint — so the unconstrained catenary of
	[[Hamilton's principle/Mathematics/The Beltrami identity]] is the soap-film		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] is the soap-film
	problem, and the constrained one is the chain. The same integrand, two different		problem, and the constrained one is the chain. The same integrand, two different
	problems, distinguished only by whether a constraint is present.		problems, distinguished only by whether a constraint is present.
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	detects, since it is not near any extremal in any norm. It is the standard		detects, since it is not near any extremal in any norm. It is the standard
	warning that stationarity theory presupposes existence, as noted on		warning that stationarity theory presupposes existence, as noted on
	[[Hamilton's principle/Mathematics/The Weierstrass excess function]].		[[Hamilton's principle/Mathematics/The Weierstrass excess function\|the Weierstrass excess function]].

	<blockquote>		<blockquote>
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	Leibniz and Johann Bernoulli, are editorial. No primary source for any of them		Leibniz and Johann Bernoulli, are editorial. No primary source for any of them
	has been verified for this volume. Every equation here is derived on this page.		has been verified for this volume. Every equation here is derived on this page.
	See [[Hamilton's principle/Sources]].		See [[Hamilton's principle/Sources\|Sources]].
	</blockquote>		</blockquote>

	== Next ==		== Next ==

	[[Hamilton's principle/Applications/Geodesics]] treats the shortest-path problem		[[Hamilton's principle/Applications/Geodesics\|Geodesics]] treats the shortest-path problem
	on a curved surface.		on a curved surface.