Hamilton's principle/Mathematics/The Hamilton-Jacobi equation - Revision history

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{{Plebeian|Hamilton's principle/Explainers/Waves and rays}}

The '''Hamilton-Jacobi equation''' is a single first-order partial differential
equation whose complete solution integrates the entire mechanical problem. It
completes the sequence <math>n</math> second-order equations, then <math>2n</math>
first-order equations, then one partial differential equation — and it is the
point at which mechanics acquires a wave description, which is why this page is
the ancestor of the Schrödinger equation.

== The idea ==

[[File:OPL-HP-F-12.png|thumb|right|400px|Wavefronts and the rays orthogonal to them.]]

Hamilton's equations are easy in one case: if <math>H \equiv 0</math>, then every
coordinate and momentum is constant and the motion is trivial. So look for a
change of variables, preserving the canonical form (7) of
[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]],
that transforms the system into that case. All the difficulty moves into finding
the transformation.

Such a change is a '''canonical transformation''', and one way to generate one is
by a function <math>S(t, q, P)</math> of the old coordinates and the new momenta.
The generating relations are

:<math>p_i = \frac{\partial S}{\partial q_i},
\qquad
Q_i = \frac{\partial S}{\partial P_i},
\qquad
K = H + \frac{\partial S}{\partial t}
\qquad\qquad\mathrm{(1)}</math>

with <math>K</math> the new Hamiltonian. Demanding <math>K = 0</math> and
substituting the first relation into the third gives the equation.

== Statement ==

[[File:OPL-HP-F-32.png|thumb|right|400px|From n second-order equations to one partial differential equation.]]

:<math>\frac{\partial S}{\partial t}
+ H\!\left( t,\, q_1, \ldots, q_n,\,
\frac{\partial S}{\partial q_1}, \ldots, \frac{\partial S}{\partial q_n} \right)
= 0
\qquad\qquad\mathrm{(2)}</math>

This is the '''Hamilton-Jacobi equation''' for the function
<math>S(t, q_1, \ldots, q_n)</math>. It is first order in all
<math>n+1</math> variables and generally nonlinear, since <math>H</math> is
typically quadratic in the momenta.

A '''complete integral''' is a solution containing <math>n</math> non-additive
constants <math>\alpha_1, \ldots, \alpha_n</math>. Given one, the motion follows
by differentiation alone: setting the new momenta to the <math>\alpha_i</math>,
the second relation of (1) gives

:<math>\frac{\partial S}{\partial \alpha_i} = \beta_i = \text{const}
\qquad\qquad\mathrm{(3)}</math>

which are <math>n</math> algebraic relations determining the <math>q_i</math> as
functions of <math>t</math> and the <math>2n</math> constants
<math>\alpha, \beta</math>. No integration is required at this stage. That is
Jacobi's theorem, and it is the payoff: '''the entire integration of the
mechanical problem is traded for one partial differential equation.'''

== Separation of variables ==

The trade is worth making only when (2) can be solved, and the practical route is
separation. If <math>H</math> carries no explicit <math>t</math>, put

:<math>S(t,q) = W(q) - Et
\qquad\qquad\mathrm{(4)}</math>

and (2) reduces to the '''time-independent''' form

:<math>H\!\left( q, \frac{\partial W}{\partial q} \right) = E
\qquad\qquad\mathrm{(5)}</math>

<math>W</math> is '''Hamilton's characteristic function''', and it is the
abbreviated action treated on
[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]].
Each cyclic coordinate separates further, contributing a term
<math>\alpha_k q_k</math> to <math>W</math>. A system that separates completely
is '''integrable''', and the Kepler problem on
[[Hamilton's principle/Applications/Central force and the Kepler problem]] is the
classical instance.

Whether a given system separates depends on the coordinates chosen, and no
general procedure finds them. Most systems separate in no coordinates at all.

== The optical analogy ==

Equation (5) with <math>H = p^2/2m + V</math> reads

:<math>\vert \nabla W \vert^2 = 2m\bigl( E - V(q) \bigr)
\qquad\qquad\mathrm{(6)}</math>

Compare the eikonal equation of geometrical optics,
<math>\vert\nabla\Psi\vert^2 = n^2(q)</math>, with <math>\Psi</math> the optical
path and <math>n</math> the refractive index. The two are the same equation with

:<math>n(q) \;\longleftrightarrow\; \sqrt{2m\bigl(E - V(q)\bigr)}
\qquad\qquad\mathrm{(7)}</math>

So surfaces of constant <math>W</math> are wavefronts and trajectories are the
rays orthogonal to them. Mechanics ''is'' geometrical optics for a medium of
refractive index (7).

Hamilton saw this before he wrote the mechanics. He notes that the dynamical
principle is only another form of the idea already applied to optics in his
theory of systems of rays.<ref>Hamilton, ''On a General Method in Dynamics''
(1834), Introductory Remarks. Verbatim in substance; ESTABLISHED source.</ref>
The optical papers came first and the mechanics is their transcription. This is
developed on
[[Hamilton's principle/Applications/The optical-mechanical analogy]].

'''The consequence that matters.''' Geometrical optics is the short-wavelength
limit of wave optics. If mechanics is geometrical optics, then classical
mechanics should be the short-wavelength limit of some wave mechanics, and
equation (2) should be the limit of a wave equation. Substituting
<math>\psi = A\exp(iS/\hbar)</math> into the Schrödinger equation and keeping the
leading order in <math>\hbar</math> returns exactly (2). Schrödinger constructed
the wave equation by running that argument backwards.

The gap between Hamilton in 1834 and Schrödinger in 1926 was ninety-two years,
and the mathematics was in place the whole time. What was missing was any reason
to believe the wavelength was not zero.

== Conjugate points return ==

<math>S</math> is not globally single-valued. Where extremals through a common
starting point refocus, several trajectories reach the same
<math>(t,q)</math>, <math>S</math> becomes multivalued, and the derivatives in
(1) fail. Those are precisely the conjugate points of
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]],
seen from the wave side: they are the caustics of the ray family, where the
wavefront folds.

The same singularities appear in every setting the analogy touches: a focus in
optics, a caustic in the ray picture, a conjugate point in the variational
theory, and the location of the phase correction in the path integral on
[[Hamilton's principle/Applications/The Feynman path integral]]. One phenomenon,
four names.

== Hamilton's pair ==

Hamilton derived two partial differential equations rather than one, a final and
an initial equation, because his characteristic function depends on both
endpoints. In his notation they are

:<math>\tfrac{1}{2}\,\Sigma\, \tfrac{1}{m}
\left\{ \left( \tfrac{\delta V}{\delta x} \right)^2
+ \left( \tfrac{\delta V}{\delta y} \right)^2
+ \left( \tfrac{\delta V}{\delta z} \right)^2 \right\} = U + H
\qquad\qquad\mathrm{(8)}</math>

and the same expression in the initial coordinates equal to
<math>U_0 + H</math>.<ref>Hamilton, ''On a General Method in Dynamics'' (1834),
equations (F.) and (G.). Verbatim, with Hamilton's <math>\delta</math> denoting
partial differentiation and <math>U</math> the force-function. He describes them
as two partial differential equations, initial and final, of the first order and
the second degree, which the characteristic function must identically satisfy.
ESTABLISHED source.</ref>

Equation (8) is (5) with <math>V</math> for <math>W</math> and
<math>U + H</math> for <math>E</math>, allowing for his sign convention on the
force-function. The pairing is the subject of the next page.

== Next ==

[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] takes
Hamilton's own route: define the action on the actual motion, as a function of
where it starts and ends, and read the mechanics off its derivatives.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	coordinate and momentum is constant and the motion is trivial. So look for a		coordinate and momentum is constant and the motion is trivial. So look for a
	change of variables, preserving the canonical form (7) of		change of variables, preserving the canonical form (7) of
	[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]],		[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations\|the Hamiltonian and Hamilton's equations]],
	that transforms the system into that case. All the difficulty moves into finding		that transforms the system into that case. All the difficulty moves into finding
	the transformation.		the transformation.
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	<math>W</math> is '''Hamilton's characteristic function''', and it is the		<math>W</math> is '''Hamilton's characteristic function''', and it is the
	abbreviated action treated on		abbreviated action treated on
	[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton]].		[[Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton\|Abbreviated action: Maupertuis versus Hamilton]].
	Each cyclic coordinate separates further, contributing a term		Each cyclic coordinate separates further, contributing a term
	<math>\alpha_k q_k</math> to <math>W</math>. A system that separates completely		<math>\alpha_k q_k</math> to <math>W</math>. A system that separates completely
	is '''integrable''', and the Kepler problem on		is '''integrable''', and the Kepler problem on
	[[Hamilton's principle/Applications/Central force and the Kepler problem]] is the		[[Hamilton's principle/Applications/Central force and the Kepler problem\|Central force and the Kepler problem]] is the
	classical instance.		classical instance.

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	The optical papers came first and the mechanics is their transcription. This is		The optical papers came first and the mechanics is their transcription. This is
	developed on		developed on
	[[Hamilton's principle/Applications/The optical-mechanical analogy]].		[[Hamilton's principle/Applications/The optical-mechanical analogy\|the optical-mechanical analogy]].

	'''The consequence that matters.''' Geometrical optics is the short-wavelength		'''The consequence that matters.''' Geometrical optics is the short-wavelength
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	<math>(t,q)</math>, <math>S</math> becomes multivalued, and the derivatives in		<math>(t,q)</math>, <math>S</math> becomes multivalued, and the derivatives in
	(1) fail. Those are precisely the conjugate points of		(1) fail. Those are precisely the conjugate points 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]],
	seen from the wave side: they are the caustics of the ray family, where the		seen from the wave side: they are the caustics of the ray family, where the
	wavefront folds.		wavefront folds.
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	optics, a caustic in the ray picture, a conjugate point in the variational		optics, a caustic in the ray picture, a conjugate point in the variational
	theory, and the location of the phase correction in the path integral on		theory, and the location of the phase correction in the path integral on
	[[Hamilton's principle/Applications/The Feynman path integral]]. One phenomenon,		[[Hamilton's principle/Applications/The Feynman path integral\|the Feynman path integral]]. One phenomenon,
	four names.		four names.

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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]] takes		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]] takes
	Hamilton's own route: define the action on the actual motion, as a function of		Hamilton's own route: define the action on the actual motion, as a function of
	where it starts and ends, and read the mechanics off its derivatives.		where it starts and ends, and read the mechanics off its derivatives.