Hamilton's principle - Revision history

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	[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]].		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]].

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	[[Hamilton's principle/Applications/The brachistochrone\|the brachistochrone]] to the path integral at
	[[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]], each worked		[[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]], each worked
	rather than described.		rather than described.

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Of all the ways a ball could fly from your hand to a window, it takes one. Of all
the paths light could follow from a lamp to your eye, it takes one. For three
centuries physics has had a single answer to why: along the true path, a quantity
called the '''action''' is '''stationary'''.

This volume is about that statement — what the quantity is, what ''stationary''
means and why it is not the same as ''least'', how far the idea reaches, and why
it turned out to be more fundamental than the laws it was invented to reformulate.

== What is stationary ==

[[File:OPL-HP-F-01.png|thumb|right|400px|A true path and several comparison paths between fixed endpoints.]]

Take a system with a starting configuration and an ending configuration, and a
fixed time to get from one to the other. For '''every''' path between them — not
just the physical one, but every geometrically imaginable curve — compute

:<math>S = \int_{t_0}^{t_1} \bigl( T - V \bigr)\, dt</math>

the accumulated difference between kinetic and potential energy along that path.
Each path gets a number. The true path is the one where that number stops
changing to first order: nudge the path slightly and <math>S</math> does not
change, to the accuracy of the nudge.

That is '''Hamilton's principle'''. It is equivalent to Newton's laws for
ordinary mechanical systems, and it is stated in one line rather than one line
per particle.

The one-line version conceals a real strangeness. To evaluate <math>S</math> for
a path you need the whole path — where the ball goes at the end is part of the
input. Newton's laws are local in time: given position and velocity now, they say
what happens next. Hamilton's principle is a statement about a completed journey.
The two give the same answer, and they do not sound like the same kind of claim.

== Why "stationary" and not "least" ==

[[File:OPL-HP-F-21.png|thumb|right|400px|Chronology of the action principle.]]

The principle was born as the principle of '''least''' action, and the name has
stuck for nearly three centuries. It is wrong, and this volume takes some care
over how wrong.

Stationary means the first variation vanishes — the same condition that a
function's derivative is zero at a maximum, a minimum, or a saddle. It does not
say the action is smallest. Sometimes it is. Sometimes it is a maximum, and
sometimes it is neither.

The clearest case needs no mathematics. On a sphere, the shortest route between
two cities is the short way round the great circle through them. But the '''long'''
way round the same great circle is also a valid stationary path: it satisfies the
same equation, at every point, and no local test distinguishes it from the short
one. It is stationary and it is emphatically not shortest.

Optics gives a case where a real physical path is a '''maximum'''. Light
reflecting from a suitably curved mirror takes longer than nearby paths would,
and it goes that way regardless. Fermat's principle, stated as least time, is
simply false there.

Hamilton knew this and said so, in 1834, in a passing correction he made no fuss
about — 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.</ref> The mathematics behind his correction took the rest of the
century, and it is the subject of this volume's second mathematics track.

'''Where the distinction bites.''' It is not pedantry. The point past which a
path stops being shortest is a '''conjugate point''', and conjugate points are
physical: they are the foci of a lens, the multiple images produced by a
gravitational lens, and the places where the classical approximation to quantum
mechanics breaks down. A theory that says ''least'' cannot describe any of them.

== Why the idea matters ==

[[File:OPL-HP-F-58.png|thumb|right|400px|The two long gaps in the subject's history.]]

Four reasons, in increasing order of consequence.

'''It is economical.''' One scalar function replaces a vector equation per
particle, and coordinates can be chosen freely — angles, distances, whatever suits
the problem — because the recipe does not care. For a constrained system the
constraint forces, which dominate a Newtonian treatment, never appear.

'''It explains conservation laws.''' Momentum, energy and angular momentum are
conserved because the action does not change under translations, time shifts and
rotations. That is Noether's theorem of 1918, and it converts conservation laws
from experimental facts into consequences of symmetry. It also works in reverse:
find a symmetry and you have found a conserved quantity.

'''It builds theories.''' Write the most general action consistent with the
symmetries you require, and the equations of motion follow. Electromagnetism comes
out of demanding that a phase can be chosen independently at each point. General
relativity comes out of demanding invariance under arbitrary coordinate changes.
In both cases the symmetry chose the theory and almost nothing else had to be
assumed.

'''It is why classical mechanics is true.''' In quantum mechanics '''every''' path
contributes, each carrying a phase equal to its action in units of
<math>\hbar</math>. For anything larger than an atom those phases oscillate so
violently that neighbouring paths cancel — except near a path where the action is
stationary, where neighbours share a phase and reinforce. The classical path is
not selected. It is the one that survives interference.

That last point turns the principle inside out. The question is not why nature
extremises the action. Nature does not extremise anything: it sums over
everything, and stationary phase is where a sum of oscillating terms fails to
cancel. Hamilton's principle is a theorem about interference — and it explains, as
a bonus, why the correct word is ''stationary''. Constructive interference needs
only stationary phase. A maximum reinforces exactly as a minimum does.

== How to read this volume ==

Every page carrying hard mathematics has a plain-language counterpart, named at
the top of the page and linking back. You can read either level throughout, or
switch between them.

'''The history.''' From Fermat's least time in the 1650s to Feynman's sum over
paths in 1948, by way of the 1744 priority dispute, Lagrange's reformulation of
mechanics, Hamilton's two essays, Jacobi, and Noether.
[[Hamilton's principle/History/Hamilton 1834 and 1835]] is the pivot;
[[Hamilton's principle/History/Noether 1918]] is where the subject stops being a
technique and becomes an explanation.

'''The mathematics''', in three tracks.
The '''machinery''' builds the tools: functionals, the first variation, the
fundamental lemma, and the Euler-Lagrange equation, starting at
[[Hamilton's principle/Mathematics/Functionals and the space of paths]].
'''Constraints and sufficiency''' asks when a stationary path is actually a
minimum, and answers with worked counterexamples rather than assertions, starting
at [[Hamilton's principle/Mathematics/Holonomic constraints]] and reaching the
central case at
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].
'''Structure''' reorganises everything — as a transform, a flow in phase space, a
wave equation, and a function of endpoints — starting at
[[Hamilton's principle/Mathematics/The Legendre transform]].

'''The applications''', from the brachistochrone at
[[Hamilton's principle/Applications/The brachistochrone]] to the path integral at
[[Hamilton's principle/Applications/The Feynman path integral]], each worked
rather than described.

'''The apparatus.''' [[Hamilton's principle/Notation]] is worth reading before any
primary source: three symbol collisions in this subject are severe enough to make
Hamilton's own essays unreadable if they are not declared.
[[Hamilton's principle/Sources]] states exactly which text in this volume is
quoted, which is editorial, and what remains unverified.

== A note on sources ==

This volume quotes verbatim only from sources it has opened and checked. Where a
source could not be verified, it says so, attributes no equation to it, and names
what access would close the gap. No equation anywhere in this volume is a
reconstruction.

That discipline is deliberate. A companion volume on Maxwell's ''On Physical Lines
of Force'' discovered, after it was written, that every reachable electronic copy
of that paper had lost its mathematics — the prose survived and read as complete,
and the equations had to be rebuilt from secondary literature. This volume checked
its sources before it was written rather than after.

== Where to start ==

If you want the idea: read on at
[[Hamilton's principle/Explainers/Comparing paths]].

If you want the mathematics:
[[Hamilton's principle/Mathematics/Functionals and the space of paths]].

If you want the one thing this volume most wants to convince you of:
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]],
where a great-circle arc is stationary and is not shortest, and the word ''least''
stops being tenable.

== References ==

<references />

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	paths in 1948, by way of the 1744 priority dispute, Lagrange's reformulation of		paths in 1948, by way of the 1744 priority dispute, Lagrange's reformulation of
	mechanics, Hamilton's two essays, Jacobi, and Noether.		mechanics, Hamilton's two essays, Jacobi, and Noether.
	[[Hamilton's principle/History/Hamilton 1834 and 1835]] is the pivot;		[[Hamilton's principle/History/Hamilton 1834 and 1835\|Hamilton 1834 and 1835]] is the pivot;
	[[Hamilton's principle/History/Noether 1918]] is where the subject stops being a		[[Hamilton's principle/History/Noether 1918\|Noether 1918]] is where the subject stops being a
	technique and becomes an explanation.		technique and becomes an explanation.

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	The '''machinery''' builds the tools: functionals, the first variation, the		The '''machinery''' builds the tools: functionals, the first variation, the
	fundamental lemma, and the Euler-Lagrange equation, starting at		fundamental lemma, and the Euler-Lagrange equation, starting at
	[[Hamilton's principle/Mathematics/Functionals and the space of paths]].		[[Hamilton's principle/Mathematics/Functionals and the space of paths\|Functionals and the space of paths]].
	'''Constraints and sufficiency''' asks when a stationary path is actually a		'''Constraints and sufficiency''' asks when a stationary path is actually a
	minimum, and answers with worked counterexamples rather than assertions, starting		minimum, and answers with worked counterexamples rather than assertions, starting
	at [[Hamilton's principle/Mathematics/Holonomic constraints]] and reaching the		at [[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]] and reaching the
	central case at		central case at
	[[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]].
	'''Structure''' reorganises everything — as a transform, a flow in phase space, a		'''Structure''' reorganises everything — as a transform, a flow in phase space, a
	wave equation, and a function of endpoints — starting at		wave equation, and a function of endpoints — starting at
	[[Hamilton's principle/Mathematics/The Legendre transform]].		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]].

	'''The applications''', from the brachistochrone at		'''The applications''', from the brachistochrone at
	[[Hamilton's principle/Applications/The brachistochrone]] to the path integral at		[[Hamilton's principle/Applications/The brachistochrone\|the brachistochrone]] to the path integral at
	[[Hamilton's principle/Applications/The Feynman path integral]], each worked		[[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]], each worked
	rather than described.		rather than described.

	'''The apparatus.''' [[Hamilton's principle/Notation]] is worth reading before any		'''The apparatus.''' [[Hamilton's principle/Notation\|Notation]] is worth reading before any
	primary source: three symbol collisions in this subject are severe enough to make		primary source: three symbol collisions in this subject are severe enough to make
	Hamilton's own essays unreadable if they are not declared.		Hamilton's own essays unreadable if they are not declared.
	[[Hamilton's principle/Sources]] states exactly which text in this volume is		[[Hamilton's principle/Sources\|Sources]] states exactly which text in this volume is
	quoted, which is editorial, and what remains unverified.		quoted, which is editorial, and what remains unverified.

Line 160:		Line 160:

	If you want the idea: read on at		If you want the idea: read on at
	[[Hamilton's principle/Explainers/Comparing paths]].		[[Hamilton's principle/Explainers/Comparing paths\|Comparing paths]].

	If you want the mathematics:		If you want the mathematics:
	[[Hamilton's principle/Mathematics/Functionals and the space of paths]].		[[Hamilton's principle/Mathematics/Functionals and the space of paths\|Functionals and the space of paths]].

	If you want the one thing this volume most wants to convince you of:		If you want the one thing this volume most wants to convince you of:
	[[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]],
	where a great-circle arc is stationary and is not shortest, and the word ''least''		where a great-circle arc is stationary and is not shortest, and the word ''least''
	stops being tenable.		stops being tenable.