Hamilton's principle/Explainers/Two different actions - Revision history

PMA Publisher: Published from the Principia Machina source

2026-10-15T19:30:23Z

Published from the Principia Machina source

← Older revision		Revision as of 19:30, 15 October 2026
Line 1:		Line 1:
	{{DISPLAYTITLE:Two different actions}}		{{DISPLAYTITLE:Two different actions}}
	{{Technical\|Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton}}		{{Technical\|Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton}}
			{{Hamilton's principle map}}

	This page separates two things that are both called the action. It uses no		This page separates two things that are both called the action. It uses no

BFP Rauch: Pipe internal links so cross-references read as titles, not paths

2026-09-16T04:59:00Z

Pipe internal links so cross-references read as titles, not paths

← Older revision		Revision as of 04:59, 16 September 2026
Line 68:		Line 68:
	That is a genuine reformulation, not a metaphor. And it is the same statement as		That is a genuine reformulation, not a metaphor. And it is the same statement as
	the optical one on		the optical one on
	[[Hamilton's principle/Explainers/Light and particles]]: light bends in glass		[[Hamilton's principle/Explainers/Light and particles\|Light and particles]]: light bends in glass
	because glass warps the effective geometry, and a particle bends near a mass for		because glass warps the effective geometry, and a particle bends near a mass for
	the same kind of reason.		the same kind of reason.
Line 91:		Line 91:
	equation in it. This volume flags the point at each place a reader could trip over		equation in it. This volume flags the point at each place a reader could trip over
	it, and the symbols are tabulated at		it, and the symbols are tabulated at
	[[Hamilton's principle/Notation]].		[[Hamilton's principle/Notation\|Notation]].

	== Neither is "least" ==		== Neither is "least" ==
Line 106:		Line 106:
	The mathematical justification for his correction took the rest of the century,		The mathematical justification for his correction took the rest of the century,
	and this volume works through it in the pages from		and this volume works through it in the pages from
	[[Hamilton's principle/Explainers/Is it really a minimum]] onwards.		[[Hamilton's principle/Explainers/Is it really a minimum\|Is it really a minimum]] onwards.

	The quantum picture at		The quantum picture at
	[[Hamilton's principle/Explainers/Sum over histories]] then supplies an		[[Hamilton's principle/Explainers/Sum over histories\|Sum over histories]] then supplies an
	independent reason: what matters is that the quantity barely changes, and a		independent reason: what matters is that the quantity barely changes, and a
	maximum does that just as well as a minimum.		maximum does that just as well as a minimum.
Line 129:		Line 129:

	The optical version of the same correspondence, and what came of it, is at		The optical version of the same correspondence, and what came of it, is at
	[[Hamilton's principle/Explainers/Light and particles]].		[[Hamilton's principle/Explainers/Light and particles\|Light and particles]].

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

2026-09-16T04:47:01Z

Add DISPLAYTITLE: title the page by its section rather than its path

← Older revision		Revision as of 04:47, 16 September 2026
Line 1:		Line 1:
			{{DISPLAYTITLE:Two different actions}}
	{{Technical\|Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton}}		{{Technical\|Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton}}

Line 129:		Line 130:
	The optical version of the same correspondence, and what came of it, is at		The optical version of the same correspondence, and what came of it, is at
	[[Hamilton's principle/Explainers/Light and particles]].		[[Hamilton's principle/Explainers/Light and particles]].

BFP Rauch: 1 revision imported

2026-09-09T05:05:17Z

1 revision imported

← Older revision	Revision as of 05:05, 9 September 2026
(No difference)

hamiltons-principle>HPVBuild: Initial import of B-17

2026-09-08T12:00:00Z

Initial import of B-17

New page

{{Technical|Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton}}

This page separates two things that are both called the action. It uses no
mathematics. The technical page it goes with is named above.

== A confusion worth clearing up ==

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

There are two different quantities in this subject.

Both are called '''the action'''. Both have a principle built on them. Both
principles are called '''the principle of least action'''.

They are not the same, and treating them as one is the most common error in the
whole area. This page keeps them apart.

== What each one holds fixed ==

The difference is not in the formulas. It is in what you are allowed to vary when
you compare paths.

'''Hamilton's version.''' Fix where the journey starts, where it ends, and '''how
long it takes'''. Compare all paths that get from one to the other in that
time. Their energies may differ.

'''Maupertuis' version.''' Fix where the journey starts, where it ends, and the
'''energy'''. Compare all paths with that energy. How long they take may differ.

One holds the clock and lets the energy float. The other holds the energy and lets
the clock float.

These are different competitions over the same set of curves. The winner turns out
to be the same curve, which is why the two get confused, but the questions being
asked are not the same question.

== What each one is good for ==

'''Hamilton's version is the general one.'''

It needs no conservation of energy. It works for systems where the energy changes,
where things are pushed and driven. It extends to fields, and hence to
electromagnetism and gravity. Everything later in this volume uses it.

'''Maupertuis' version is the specialised one.'''

It requires energy to be conserved, so it applies to a narrower class of problems.
In exchange, it removes time from the question entirely.

That trade is sometimes exactly what you want. If you care about the '''shape''' of
a planet's orbit and not about where it will be next Tuesday, the version without
time in it is simpler. You get the ellipse and you never mention the clock.

Recovering the timing afterwards is possible, and is extra work.

== Turning motion into geometry ==

The specialised version has a consequence worth stating on its own.

With time removed, the problem becomes purely about shape. And it turns out to be
exactly the problem of finding the '''shortest path''' — not in ordinary space, but
in a space that has been stretched and distorted by the forces acting.

So a particle moving through a landscape of hills and valleys is travelling as
straight as it can, in a warped space whose warping the landscape supplies.

That is a genuine reformulation, not a metaphor. And it is the same statement as
the optical one on
[[Hamilton's principle/Explainers/Light and particles]]: light bends in glass
because glass warps the effective geometry, and a particle bends near a mass for
the same kind of reason.

The idea that gravity '''is''' geometry, rather than being a force acting within
it, has this as its ancestor.

== Which one Hamilton used ==

Here is the trap for anyone reading the original sources.

Hamilton's famous 1834 essay uses the '''specialised''' quantity — the one usually
attributed to Maupertuis — throughout. His central function is that one.

The quantity that modern textbooks call Hamilton's action appears in his
'''second''' essay, a year later, introduced as a technical convenience for
avoiding awkward steps.

So the thing named after Hamilton is not the thing he built his first paper on.

Anyone who opens the 1834 essay carrying the modern definition will misread every
equation in it. This volume flags the point at each place a reader could trip over
it, and the symbols are tabulated at
[[Hamilton's principle/Notation]].

== Neither is "least" ==

The word ''least'' is wrong for both, and this volume has now shown that several
times over.

Maupertuis in 1744 meant ''least'' seriously. His argument for the principle was
that nature is economical, and economy was the point of it.

Hamilton corrected the name in 1834, calling it the law of least, or as it might
be better called, of stationary action. He did it in passing, without emphasis.

The mathematical justification for his correction took the rest of the century,
and this volume works through it in the pages from
[[Hamilton's principle/Explainers/Is it really a minimum]] onwards.

The quantum picture at
[[Hamilton's principle/Explainers/Sum over histories]] then supplies an
independent reason: what matters is that the quantity barely changes, and a
maximum does that just as well as a minimum.

== How to keep them apart ==

Three habits to hold on to, and they cover almost every case.

'''Ask what is held fixed.''' Time fixed means Hamilton's. Energy fixed means
Maupertuis'.

'''Ask whether time appears in the answer.''' If the result is a shape with no
schedule, it is the specialised version.

'''Distrust the word "action" without qualification.''' Especially in older
sources, and especially in Hamilton's own.

== Where to go next ==

The optical version of the same correspondence, and what came of it, is at
[[Hamilton's principle/Explainers/Light and particles]].