Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton - Revision history

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	[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]] in the
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	{{Plebeian\|Hamilton's principle/Explainers/Two different actions}}		{{Plebeian\|Hamilton's principle/Explainers/Two different actions}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/Two different actions}}

Two distinct quantities are called the action, two distinct statements are called
the principle of least action, and the two are routinely conflated. This page
separates them. The distinction is not pedantic: the principles hold different
things fixed, are stationary over different classes of paths, and one of them
cannot even be stated without conservation of energy.

== The two quantities ==

[[File:OPL-HP-F-36.png|thumb|right|400px|What each action principle holds fixed.]]

'''Hamilton's action''' is

:<math>S = \int_{t_0}^{t_1} L\, dt
= \int_{t_0}^{t_1} (T - V)\, dt
\qquad\qquad\mathrm{(1)}</math>

'''The abbreviated action''', also called Maupertuis' action, is

:<math>W = \int \sum_i p_i\, dq_i
= \int 2T\, dt
\qquad\qquad\mathrm{(2)}</math>

the second equality holding when <math>T</math> is a homogeneous quadratic in the
velocities. They are related by the Legendre transform of
[[Hamilton's principle/Mathematics/The Legendre transform]] in the
<math>(t, H)</math> pair:

:<math>W = S + \int H\, dt = S + E\,(t_1 - t_0)
\qquad\qquad\mathrm{(3)}</math>

the last form holding when energy is conserved.

'''Hamilton's own <math>V</math> is <math>W</math>, not <math>S</math>.''' His
characteristic function on
[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] is
defined as the accumulated living force, <math>\int_0^t 2T\,dt</math>, which is
(2). Readers who assume Hamilton's action means <math>\int L\,dt</math>
misread every equation in the 1834 essay. He introduced the function of the other
kind, with time entering explicitly, only in the 1835 second essay, and named it
<math>S</math>.

== The two principles ==

{| class="wikitable"
! !! Hamilton's principle !! Maupertuis' principle
|-
| quantity || <math>S = \int L\,dt</math> || <math>W = \int \sum p\,dq</math>
|-
| held fixed || endpoints and both times || endpoints and the '''energy'''
|-
| free to vary || the path, at fixed duration || the path '''and''' the duration
|-
| energy || need not be conserved || conservation is a hypothesis
|-
| applies to || any Lagrangian system || conservative systems only
|-
| determines || the path and its timing || the path in space only
|}

The row that matters is the third. Hamilton's principle compares paths traversed
in the same time and lets the energy differ. Maupertuis' principle compares paths
of the same energy and lets the transit time differ. These are different
comparison classes over the same set of curves, so they are different variational
problems that happen to share their solutions.

== Deriving Maupertuis from Hamilton ==

Start from (3) and vary, holding the endpoints and <math>E</math> fixed but
allowing <math>t_1</math> to vary:

:<math>\delta W = \delta S + E\,\delta(t_1 - t_0)
\qquad\qquad\mathrm{(4)}</math>

By the law of varying action, quoted on
[[Hamilton's principle/Mathematics/Action as a function of its endpoints]], the
variation of <math>S</math> at fixed endpoints with variable end time is
<math>\delta S = -H\,\delta t_1 = -E\,\delta t_1</math>. Substituting into (4),
the two terms cancel and

:<math>\delta W = 0
\qquad\qquad\mathrm{(5)}</math>

So <math>W</math> is stationary among energy-conserving paths joining the fixed
endpoints. That is '''Jacobi's form''' of the principle of least action.

The cancellation in (4) is the whole relationship between the two principles, and
it is why they select the same curves while making different statements about
them.

== Jacobi's form ==

For <math>T = \tfrac{1}{2}\sum g_{ik}\dot q_i \dot q_k</math> and potential
<math>V</math>, energy conservation gives
<math>T = E - V</math>, and (2) becomes a purely geometric integral:

:<math>W = \int \sqrt{2\bigl(E - V(q)\bigr)}\; ds,
\qquad
ds^2 = \sum_{i,k} g_{ik}\, dq_i\, dq_k
\qquad\qquad\mathrm{(6)}</math>

No time appears. The trajectory is a '''geodesic''' of the conformally rescaled
metric

:<math>\tilde g_{ik} = 2\bigl(E - V(q)\bigr)\, g_{ik}
\qquad\qquad\mathrm{(7)}</math>

This is a genuinely different way of seeing mechanics: a particle in a potential
travels in a straight line, in a curved space whose curvature the potential
supplies. It also makes the optical analogy exact — compare (6) with Fermat's
integral <math>\int n\,ds</math> and the correspondence
<math>n \leftrightarrow \sqrt{2(E - V)}</math> of
[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]] is read off
directly.

The timing is lost and must be recovered afterwards from
<math>dt = ds/\sqrt{2(E-V)}</math>. That loss is the price of the geometric form,
and it is the practical difference between the two principles.

== Least, and the failure of least ==

Neither principle asserts a minimum, and this is where the two histories diverge.

Maupertuis in 1744 argued from a metaphysical premise — that nature acts with
economy — and the word ''least'' was load-bearing in his argument. Euler gave the
mathematical statement in the same year and treated it as a variational condition.
The subsequent priority dispute is treated on
[[Hamilton's principle/History/The Maupertuis-Koenig affair]].

Hamilton, ninety years later, states the correction plainly, describing the older
result as the law of least, or as it might be better called, of stationary
action.<ref>Hamilton, ''On a General Method in Dynamics'' (1834), §3. Verbatim.
ESTABLISHED source.</ref>

The failure is exhibited, not asserted, on
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]]:
past a conjugate point the action is stationary and is not minimal. For
Maupertuis' principle in Jacobi form (6) that statement is geometrically
transparent — the long way round a great circle is a geodesic and is not the
shortest path — and it is the same counterexample, because for a free particle on
a sphere (6) ''is'' arc length.

== What each is good for ==

'''Hamilton's principle''' is the general one. It requires no conservation law,
extends to dissipative and time-dependent systems in modified forms, and
generalises to field theory, where the abbreviated action has no natural analogue.
Every application in this volume beyond
[[Hamilton's principle/Applications/Central force and the Kepler problem]] uses
it.

'''Maupertuis' principle''' is the specialised one. Where it applies it removes
time from the problem entirely, turning dynamics into geometry, and it is the
right tool for orbit shapes, for the semiclassical quantisation conditions, and
for anything where the trajectory matters and the schedule does not.

== The confusion, named ==

Three habits produce the conflation, and all three appear in the literature:

* calling both quantities <math>S</math>;
* calling both principles the principle of least action;
* citing Hamilton for a statement about <math>\int L\,dt</math> while quoting an equation from 1834, where <math>V</math> is <math>\int 2T\,dt</math>.

This volume writes <math>S</math> for (1) and <math>W</math> for (2) throughout,
reserves ''Hamilton's principle'' for the first statement and ''Maupertuis'
principle'' for the second, and uses ''stationary'' rather than ''least'' for
both. The concordance on [[Hamilton's principle/Notation]] records the collision
with Hamilton's own usage, in which <math>V</math> means (2).

== O7 closes here ==

The structure track has taken the machinery of O5 and O6 and reorganised it four
times: as a transform, as a flow in phase space, as a wave equation, and as a
function of endpoints. Noether's two theorems explain why the first integrals
found along the way existed. This page separates the two principles that the
whole subject is named after.

== Next ==

[[Hamilton's principle/Applications/The brachistochrone]] opens the applications
track.

== References ==

<references />

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	the second equality holding when <math>T</math> is a homogeneous quadratic in the		the second equality holding when <math>T</math> is a homogeneous quadratic in the
	velocities. They are related by the Legendre transform of		velocities. They are related by the Legendre transform of
	[[Hamilton's principle/Mathematics/The Legendre transform]] in the		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]] in the
	<math>(t, H)</math> pair:		<math>(t, H)</math> pair:

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	'''Hamilton's own <math>V</math> is <math>W</math>, not <math>S</math>.''' His		'''Hamilton's own <math>V</math> is <math>W</math>, not <math>S</math>.''' His
	characteristic function on		characteristic function on
	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] is		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]] is
	defined as the accumulated living force, <math>\int_0^t 2T\,dt</math>, which is		defined as the accumulated living force, <math>\int_0^t 2T\,dt</math>, which is
	(2). Readers who assume Hamilton's action means <math>\int L\,dt</math>		(2). Readers who assume Hamilton's action means <math>\int L\,dt</math>
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	By the law of varying action, quoted on		By the law of varying action, quoted on
	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]], the		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]], the
	variation of <math>S</math> at fixed endpoints with variable end time is		variation of <math>S</math> at fixed endpoints with variable end time is
	<math>\delta S = -H\,\delta t_1 = -E\,\delta t_1</math>. Substituting into (4),		<math>\delta S = -H\,\delta t_1 = -E\,\delta t_1</math>. Substituting into (4),
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	integral <math>\int n\,ds</math> and the correspondence		integral <math>\int n\,ds</math> and the correspondence
	<math>n \leftrightarrow \sqrt{2(E - V)}</math> of		<math>n \leftrightarrow \sqrt{2(E - V)}</math> of
	[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]] is read off		[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation\|the Hamilton-Jacobi equation]] is read off
	directly.		directly.

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	mathematical statement in the same year and treated it as a variational condition.		mathematical statement in the same year and treated it as a variational condition.
	The subsequent priority dispute is treated on		The subsequent priority dispute is treated on
	[[Hamilton's principle/History/The Maupertuis-Koenig affair]].		[[Hamilton's principle/History/The Maupertuis-Koenig affair\|the Maupertuis-Koenig affair]].

	Hamilton, ninety years later, states the correction plainly, describing the older		Hamilton, ninety years later, states the correction plainly, describing the older
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	The failure is exhibited, not asserted, on		The failure is exhibited, not asserted, 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]]:
	past a conjugate point the action is stationary and is not minimal. For		past a conjugate point the action is stationary and is not minimal. For
	Maupertuis' principle in Jacobi form (6) that statement is geometrically		Maupertuis' principle in Jacobi form (6) that statement is geometrically
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	generalises to field theory, where the abbreviated action has no natural analogue.		generalises to field theory, where the abbreviated action has no natural analogue.
	Every application in this volume beyond		Every application in this volume beyond
	[[Hamilton's principle/Applications/Central force and the Kepler problem]] uses		[[Hamilton's principle/Applications/Central force and the Kepler problem\|Central force and the Kepler problem]] uses
	it.		it.

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	reserves ''Hamilton's principle'' for the first statement and ''Maupertuis'		reserves ''Hamilton's principle'' for the first statement and ''Maupertuis'
	principle'' for the second, and uses ''stationary'' rather than ''least'' for		principle'' for the second, and uses ''stationary'' rather than ''least'' for
	both. The concordance on [[Hamilton's principle/Notation]] records the collision		both. The concordance on [[Hamilton's principle/Notation\|Notation]] records the collision
	with Hamilton's own usage, in which <math>V</math> means (2).		with Hamilton's own usage, in which <math>V</math> means (2).

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	== Next ==		== Next ==

	[[Hamilton's principle/Applications/The brachistochrone]] opens the applications		[[Hamilton's principle/Applications/The brachistochrone\|the brachistochrone]] opens the applications
	track.		track.