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	{{Plebeian\|Hamilton's principle/Explainers/Action from the endpoints}}		{{Plebeian\|Hamilton's principle/Explainers/Action from the endpoints}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/Action from the endpoints}}

Every use of the action so far has treated it as a functional to be varied over
paths, with the endpoints held fixed. Hamilton's own idea was the opposite:
evaluate the action '''on the actual motion''', so that it is no longer a
functional at all but an ordinary function of where the motion starts and ends.
Its derivatives then reproduce the whole of dynamics. He called it the
'''characteristic function''', and he called the result the '''law of varying
action'''.

This page follows his construction. The 1834 essay is a verified source for this
volume, so the equations below are quoted rather than reconstructed.

== The characteristic function ==

[[File:OPL-HP-F-25.png|thumb|right|400px|Action on the true motion as a function of its endpoints.]]

Hamilton defines <math>V</math> as the accumulated living force of the system,
that is, twice the kinetic energy integrated over the motion:

:<math>V = \int \Sigma\, . \, m(x'\,dx + y'\,dy + z'\,dz)
= \int_0^t 2T\, dt
\qquad\qquad\mathrm{(1)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (B.).
Verbatim. Primes are time derivatives; <math>\Sigma</math> runs over the points
of the system. Hamilton calls this quantity the action of the system, from its
initial to its final position. ESTABLISHED source.</ref>

Note what (1) is not. It is not <math>\int L\,dt</math>. Hamilton's
<math>V</math> is the '''abbreviated''' action <math>\int 2T\,dt</math>, and the
distinction between it and <math>\int L\,dt</math> is the subject of
[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]].
Confusing the two is the most common error in reading Hamilton, and the volume
flags it here as well as there.

<math>V</math> is evaluated along the true motion. It therefore depends on the
initial coordinates <math>a, b, c</math>, the final coordinates
<math>x, y, z</math>, and the energy constant <math>H</math> — in total
<math>6n + 1</math> arguments for a system of <math>n</math> points, and not on
any path.

== The law of varying action ==

Hamilton's central result is the expression for how <math>V</math> changes when
the endpoints and the energy are varied:

:<math>\delta V = \Sigma\, . \, m(x'\,\delta x + y'\,\delta y + z'\,\delta z)
- \Sigma\, . \, m(a'\,\delta a + b'\,\delta b + c'\,\delta c)
+ t\,\delta H
\qquad\qquad\mathrm{(2)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (A.).
Verbatim. Hamilton names this the equation of the characteristic function, or the
law of varying action. ESTABLISHED source.</ref>

Reading (2) coefficient by coefficient gives three groups of relations. First,
the final momenta:

:<math>\frac{\delta V}{\delta x_1} = m_1 x_1',
\qquad
\frac{\delta V}{\delta y_1} = m_1 y_1',
\qquad \ldots
\qquad\qquad\mathrm{(3)}</math>

second, the initial momenta, with the opposite sign:

:<math>\frac{\delta V}{\delta a_1} = -m_1 a_1',
\qquad
\frac{\delta V}{\delta b_1} = -m_1 b_1',
\qquad \ldots
\qquad\qquad\mathrm{(4)}</math>

and third, the time:

:<math>\frac{\delta V}{\delta H} = t
\qquad\qquad\mathrm{(5)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equations (C.), (D.)
and (E.). Verbatim, with Hamilton's <math>\delta</math> denoting partial
differentiation. ESTABLISHED source.</ref>

In modern notation (3) and (4) read
<math>\partial S/\partial q = p</math> and
<math>\partial S/\partial q_0 = -p_0</math>: '''the gradient of the action with
respect to the final position is the final momentum, and with respect to the
initial position is minus the initial momentum.'''

== Why this is the whole solution ==

Equations (3) are <math>3n</math> relations connecting the final coordinates,
their velocities, the initial coordinates and <math>H</math>. Eliminating
<math>H</math> between them and (5) gives the intermediate integrals. Equations
(4) with (5) give the final integrals. Hamilton states the consequence plainly:
the general problem of dynamics is reduced to the search and differentiation of a
single function.

The value of that reduction is not computational. Hamilton is explicit that it
may bring no practical facility, and rests the case elsewhere: on the
intellectual pleasure of reducing all researches respecting the forces and
motions of body to the study of one characteristic function.<ref>Hamilton, ''On a
General Method in Dynamics'' (1834), Introductory Remarks. Verbatim phrases.
ESTABLISHED source.</ref>

== The equations <math>V</math> must satisfy ==

Combining (3) with conservation of energy <math>T = U + H</math> gives a partial
differential equation for <math>V</math> in the final coordinates, and combining
(4) with the initial form gives a second one in the initial coordinates. These
are equations (F.) and (G.), quoted on
[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]]. Hamilton
describes them as of the first order and the second degree, and says they furnish
the principal means of discovering the form of that function.

So the logical order is: '''the law of varying action comes first, and the
Hamilton-Jacobi equation is derived from it.''' The modern presentation reverses
this, deriving the equation from canonical transformation theory and mentioning
the characteristic function afterwards. Both are correct; only Hamilton's shows
where the idea came from.

== What Hamilton says about least action ==

Hamilton is precise about the relation between his principle and the older one,
and the passage is the source of this volume's terminology. He observes that
Lagrange and others, from the vanishing of the variation of the action integral,
deduced only the well known law of least action — and then corrects the name in
passing, writing of this well known 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>

He goes further, and it is worth stating exactly what he claims. The older
principle, applied to determine the motion, serves only to form the differential
equations of the second order, which can always be otherwise found — and he notes
that Lagrange, Laplace and Poisson had for that reason spoken lightly of its
utility. His own law of varying action, by contrast, expresses by means of a
single function not the mere differential equations of motion, but their
intermediate and their final integrals.<ref>Hamilton, ''On a General Method in
Dynamics'' (1834), §3. Verbatim phrases. ESTABLISHED source.</ref>

That is the distinction the volume's landing page must carry: varying the path
with fixed endpoints gives the equations of motion; varying the endpoints of the
actual motion gives their solution.

== Relation to the modern <math>S</math> ==

The modern on-shell action is
<math>S(q, t; q_0, t_0) = \int_{t_0}^{t} L\, dt</math> evaluated along the true
motion, and its differential is

:<math>dS = \sum_i p_i\, dq_i - \sum_i p_{0i}\, dq_{0i} - H\, dt + H_0\, dt_0
\qquad\qquad\mathrm{(6)}</math>

Equation (6) is (2) with the energy variation traded for a time variation, via
the Legendre transform of
[[Hamilton's principle/Mathematics/The Legendre transform]]. Hamilton's
<math>V</math> holds <math>H</math> as an argument and produces <math>t</math>;
the modern <math>S</math> holds <math>t</math> and produces <math>H</math>. They
are Legendre transforms of one another in the pair
<math>(t, H)</math>, which is why the same information appears in both and why
the sign patterns match.

Hamilton introduced this second function himself in the 1835 second essay,
writing <math>S</math> for the part of the characteristic function into which the
time enters explicitly.

== Singularities ==

<math>V</math> fails to be a well-defined function wherever more than one motion
connects the given endpoints with the given energy. That is again the conjugate
point of
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]],
and the derivatives (3) and (4) fail there. The great-circle example is the
cleanest case: between antipodes, infinitely many motions of the same action
exist, and the characteristic function has no gradient.

== Next ==

[[Hamilton's principle/Mathematics/Noether's first theorem]] returns to the
functional picture and derives conservation laws from symmetry.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	<math>V</math> is the '''abbreviated''' action <math>\int 2T\,dt</math>, and the		<math>V</math> is the '''abbreviated''' action <math>\int 2T\,dt</math>, and the
	distinction between it and <math>\int L\,dt</math> is the subject of		distinction between it and <math>\int L\,dt</math> is the subject of
	[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]].		[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]].
	Confusing the two is the most common error in reading Hamilton, and the volume		Confusing the two is the most common error in reading Hamilton, and the volume
	flags it here as well as there.		flags it here as well as there.
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	(4) with the initial form gives a second one in the initial coordinates. These		(4) with the initial form gives a second one in the initial coordinates. These
	are equations (F.) and (G.), quoted on		are equations (F.) and (G.), quoted on
	[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation]]. Hamilton		[[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation\|the Hamilton-Jacobi equation]]. Hamilton
	describes them as of the first order and the second degree, and says they furnish		describes them as of the first order and the second degree, and says they furnish
	the principal means of discovering the form of that function.		the principal means of discovering the form of that function.
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	Equation (6) is (2) with the energy variation traded for a time variation, via		Equation (6) is (2) with the energy variation traded for a time variation, via
	the Legendre transform of		the Legendre transform of
	[[Hamilton's principle/Mathematics/The Legendre transform]]. Hamilton's		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]]. Hamilton's
	<math>V</math> holds <math>H</math> as an argument and produces <math>t</math>;		<math>V</math> holds <math>H</math> as an argument and produces <math>t</math>;
	the modern <math>S</math> holds <math>t</math> and produces <math>H</math>. They		the modern <math>S</math> holds <math>t</math> and produces <math>H</math>. They
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	connects the given endpoints with the given energy. That is again the conjugate		connects the given endpoints with the given energy. That is again the conjugate
	point of		point 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]],
	and the derivatives (3) and (4) fail there. The great-circle example is the		and the derivatives (3) and (4) fail there. The great-circle example is the
	cleanest case: between antipodes, infinitely many motions of the same action		cleanest case: between antipodes, infinitely many motions of the same action
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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/Noether's first theorem]] returns to the		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]] returns to the
	functional picture and derives conservation laws from symmetry.		functional picture and derives conservation laws from symmetry.

Hamilton's principle/Mathematics/Action as a function of its endpoints - Revision history

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