Hamilton's principle/Explainers/Strong and weak - Revision history

PMA Publisher: Published from the Principia Machina source

2026-10-15T19:30:21Z

Published from the Principia Machina source

← Older revision		Revision as of 19:30, 15 October 2026
Line 1:		Line 1:
	{{DISPLAYTITLE:Strong and weak}}		{{DISPLAYTITLE:Strong and weak}}
	{{Technical\|Hamilton's principle/Mathematics/The Weierstrass excess function}}		{{Technical\|Hamilton's principle/Mathematics/The Weierstrass excess function}}
			{{Hamilton's principle map}}

	This page explains a second way a stationary path can fail to be the best. It		This page explains a second way a stationary path can fail to be the best. It

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

2026-09-16T04:58:48Z

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

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

2026-09-16T04:46:57Z

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

← Older revision		Revision as of 04:46, 16 September 2026
Line 1:		Line 1:
			{{DISPLAYTITLE:Strong and weak}}
	{{Technical\|Hamilton's principle/Mathematics/The Weierstrass excess function}}		{{Technical\|Hamilton's principle/Mathematics/The Weierstrass excess function}}

Line 148:		Line 149:
	The volume now turns to reorganising what it has, starting at		The volume now turns to reorganising what it has, starting at
	[[Hamilton's principle/Explainers/Reading a curve two ways]].		[[Hamilton's principle/Explainers/Reading a curve two ways]].

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-10

2026-09-08T12:00:00Z

Initial import of B-10

New page

{{Technical|Hamilton's principle/Mathematics/The Weierstrass excess function}}

This page explains a second way a stationary path can fail to be the best. It
uses no mathematics. The technical page it goes with is named above.

== A different kind of failure ==

[[File:OPL-HP-F-13.png|thumb|right|400px|A sawtooth competitor close in position and far in slope.]]

[[Hamilton's principle/Explainers/Where the minimum fails]]
described one way things go wrong: run a path far enough and nearby paths refocus,
and minimality is lost.

There is a second way, and it is not a variation on the first. It fails for a
different reason and needs a different test.

The clue is that there are two different ways for paths to be close, and they are
not the same.

== Two kinds of nearby ==

[[File:OPL-HP-F-34.png|thumb|right|400px|The Weierstrass excess function as a gap above the tangent.]]

Two paths are close in '''position''' if they never get far apart.

Two paths are close in '''direction''' if they also point the same way all along.

A zigzag that criss-crosses a straight road every few centimetres is close in
position and far in direction. It never leaves the road's vicinity. It is
constantly heading somewhere else.

That gap is where this page lives.

== The blind spot ==

[[File:OPL-HP-F-49.png|thumb|right|400px|The four conditions of the sufficiency theory.]]

Everything on
[[Hamilton's principle/Explainers/Is it really a minimum]] and
[[Hamilton's principle/Explainers/Where the minimum fails]] tested against
competitors that are close in '''both''' senses. Gentle nudges. Paths that stay
near and keep roughly the same heading.

So those tests have a blind spot. They say nothing about zigzag competitors.

A path can therefore beat everything gently nearby and still lose badly to
something that stays close in position while wobbling furiously in direction.

Beating only the gentle competitors is called a '''weak''' minimum. Beating the
zigzags too is called a '''strong''' minimum.

The names are unfortunate. A weak minimum is the weaker claim about a stronger
restriction on competitors. Reading the two words the wrong way round is a common
slip.

== A case where it happens ==

There is a setup where a path passes every earlier test and still loses.

Arrange a cost rule that charges a little for gentle movement but '''pays out''' for
violent movement. Nothing forbids such a rule; it is a perfectly ordinary formula.

Now consider staying still. Staying still costs nothing.

Check it against gentle competitors and it wins every time — small movements cost
a little, so doing nothing is best.

Now try a fine sawtooth: move up and down rapidly, with a very small amplitude,
hundreds of times over. In position this path barely leaves the original. In
direction it is nothing like it.

Because it moves violently, the rule pays out. The total comes out '''negative''' —
better than staying still.

And you can make the teeth as small as you like. There are competitors as close in
position as you please, all beating the original.

So staying still is a weak minimum and is not a strong one. Every earlier test
passed. The failure was invisible to all of them.

== The test that catches it ==

The missing test comes from Weierstrass, later in the nineteenth century, and it
is simple to state.

'''Ask what the rule charges for travelling in a completely different direction,
not just a slightly different one.'''

Earlier tests only ever asked about small changes of heading. Weierstrass's test
asks about '''any''' heading — including a wild one.

If setting off in some wild direction is ever cheaper than the rule's own
prediction, a sawtooth exploiting that direction will beat the path.

That is the whole idea: check all directions, not just nearby ones.

== Why this is not academic ==

Rules that reward violent movement sound artificial. They are not rare.

Anything with a cost that grows and then falls off can behave this way. Some
materials get easier to deform once past a threshold; some engineering problems
have costs that drop when you go fast enough.

When a problem is like that, the smooth answer you compute may be a weak minimum
that a rapidly oscillating solution beats. Physically, the material does not sit
smoothly. It forms fine layers, alternating between two states, at a scale limited
only by whatever the model left out.

This is real. Certain alloys form fine twinned structures for exactly this reason,
and the fineness is set by physics the simple model ignores. The mathematics
predicting sawtooths is describing something you can photograph under a
microscope.

== What the whole sufficiency story amounts to ==

Four conditions have now appeared, and it is worth seeing them together.

'''The main equation''' finds stationary paths. Necessary, never sufficient.

'''The local test''' checks that wobbling never pays anywhere. Necessary, not
sufficient.

'''The refocusing test''' checks the journey is not long enough for nearby paths
to meet. Together with the first two, enough for a weak minimum.

'''The all-directions test''' checks that no wild heading is cheap. Needed for a
strong minimum.

Four tests, and only the last two combinations conclude anything positive. That is
why this volume says '''stationary''' and not '''least''': getting from one to the
other took a century and four conditions, and even then only locally.

== One thing still missing ==

All of these assume a best path '''exists'''. They compare a candidate against its
neighbours. None of them shows there is anything to find.

Sometimes there is not. Stretch a soap film between two rings and pull the rings
apart far enough, and the film snaps into two flat discs. No smooth surface
connects them any more. The answer did not become worse; it stopped existing.

Proving that a best path exists is a separate subject, and this volume does not
treat it.

== Where to go next ==

The volume now turns to reorganising what it has, starting at
[[Hamilton's principle/Explainers/Reading a curve two ways]].

← Older revision		Revision as of 04:58, 16 September 2026
Line 9:		Line 9:
	[[File:OPL-HP-F-13.png\|thumb\|right\|400px\|A sawtooth competitor close in position and far in slope.]]		[[File:OPL-HP-F-13.png\|thumb\|right\|400px\|A sawtooth competitor close in position and far in slope.]]

	[[Hamilton's principle/Explainers/Where the minimum fails]]		[[Hamilton's principle/Explainers/Where the minimum fails\|Where the minimum fails]]
	described one way things go wrong: run a path far enough and nearby paths refocus,		described one way things go wrong: run a path far enough and nearby paths refocus,
	and minimality is lost.		and minimality is lost.
Line 38:		Line 38:

	Everything on		Everything on
	[[Hamilton's principle/Explainers/Is it really a minimum]] and		[[Hamilton's principle/Explainers/Is it really a minimum\|Is it really a minimum]] and
	[[Hamilton's principle/Explainers/Where the minimum fails]] tested against		[[Hamilton's principle/Explainers/Where the minimum fails\|Where the minimum fails]] tested against
	competitors that are close in '''both''' senses. Gentle nudges. Paths that stay		competitors that are close in '''both''' senses. Gentle nudges. Paths that stay
	near and keep roughly the same heading.		near and keep roughly the same heading.
Line 148:		Line 148:

	The volume now turns to reorganising what it has, starting at		The volume now turns to reorganising what it has, starting at
	[[Hamilton's principle/Explainers/Reading a curve two ways]].		[[Hamilton's principle/Explainers/Reading a curve two ways\|Reading a curve two ways]].