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	{{Plebeian\|Hamilton's principle/Explainers/Comparing paths}}		{{Plebeian\|Hamilton's principle/Explainers/Comparing paths}}

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{{Plebeian|Hamilton's principle/Explainers/Comparing paths}}

The '''first variation''' of a functional is the derivative of that functional
along a one-parameter family of paths, evaluated where the family passes through
the path of interest. It is the exact analogue of the first derivative in
ordinary calculus, and it plays the same role: it vanishes at an extremum, and
its vanishing is necessary but not sufficient.

== The variation defined ==

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

Take <math>J</math>, <math>\mathcal{A}</math> and <math>\mathcal{V}</math> as on
[[Hamilton's principle/Mathematics/Functionals and the space of paths]]. Fix
<math>y \in \mathcal{A}</math> and <math>\eta \in \mathcal{V}</math>, and define
the real function

:<math>\Phi(\varepsilon) = J[y + \varepsilon\eta]
= \int_a^b L\bigl(x,\, y + \varepsilon\eta,\, y' + \varepsilon\eta'\bigr)\, dx
\qquad\qquad\mathrm{(1)}</math>

<math>\Phi</math> is an ordinary function of one real variable. The '''first
variation of <math>J</math> at <math>y</math> in the direction
<math>\eta</math>''' is

:<math>\delta J[y;\eta] = \Phi'(0) =
\left. \frac{d}{d\varepsilon} J[y + \varepsilon\eta] \right|_{\varepsilon = 0}
\qquad\qquad\mathrm{(2)}</math>

<math>y</math> is '''stationary''' for <math>J</math> if
<math>\delta J[y;\eta] = 0</math> for every <math>\eta \in \mathcal{V}</math>.

The definition is a reduction, and that is its whole value. A question about an
infinite-dimensional set of competing paths becomes a question about the
derivative of one real function at one point.

== Computing it ==

[[File:OPL-HP-F-02.png|thumb|right|400px|A one-parameter family of paths through the true path.]]

Differentiate (1) under the integral sign. The hypotheses on <math>L</math> from
the previous page license this: <math>L</math> and its first partials are
continuous on a compact set, so the difference quotients converge uniformly.

:<math>\Phi'(\varepsilon) = \int_a^b
\left[ \frac{\partial L}{\partial y}\,\eta
+ \frac{\partial L}{\partial y'}\,\eta' \right] dx
\qquad\qquad\mathrm{(3)}</math>

where both partials are evaluated at
<math>(x,\, y + \varepsilon\eta,\, y' + \varepsilon\eta')</math>. Setting
<math>\varepsilon = 0</math>,

:<math>\delta J[y;\eta] = \int_a^b
\left[ L_y\,\eta + L_{y'}\,\eta' \right] dx
\qquad\qquad\mathrm{(4)}</math>

with <math>L_y</math> and <math>L_{y'}</math> now evaluated along
<math>y</math> itself. Equation (4) is the first variation in its raw form. Both
<math>\eta</math> and <math>\eta'</math> appear, which is the obstacle: they are
not independent, so no conclusion about the integrand follows yet.

== Removing the derivative of the variation ==

[[File:OPL-HP-F-40.png|thumb|right|400px|The boundary term left by integration by parts.]]

Integrate the second term by parts:

:<math>\int_a^b L_{y'}\,\eta'\, dx
= \Bigl[ L_{y'}\,\eta \Bigr]_a^b
- \int_a^b \frac{d}{dx}\bigl( L_{y'} \bigr)\,\eta\, dx
\qquad\qquad\mathrm{(5)}</math>

The boundary term vanishes, because <math>\eta \in \mathcal{V}</math> and
therefore <math>\eta(a) = \eta(b) = 0</math>. This is the only place the fixed
endpoints are used, and it is where they earn their keep. Substituting into (4),

:<math>\delta J[y;\eta] = \int_a^b
\left[ L_y - \frac{d}{dx} L_{y'} \right] \eta \, dx
\qquad\qquad\mathrm{(6)}</math>

Only <math>\eta</math> now appears, and it appears multiplied by an expression
that does not involve <math>\eta</math> at all. Write

:<math>E(x) = L_y - \frac{d}{dx} L_{y'} \qquad\qquad\mathrm{(7)}</math>

so that <math>\delta J[y;\eta] = \int_a^b E(x)\,\eta(x)\, dx</math>. Noether
calls <math>E</math> the Lagrange expression and it recurs under that name on
[[Hamilton's principle/Mathematics/Noether's first theorem]].

== What has and has not been shown ==

Stationarity says (6) vanishes for ''every'' admissible <math>\eta</math>. It
does not yet say <math>E \equiv 0</math>. An integral can vanish because its
integrand vanishes, or because positive and negative contributions cancel. Ruling
out the second possibility is a separate theorem, and it is the subject of
[[Hamilton's principle/Mathematics/The fundamental lemma]].

Two further cautions.

'''Stationary is not extremal.''' <math>\Phi'(0) = 0</math> constrains
<math>\Phi</math> only to first order. Whether <math>y</math> minimises,
maximises, or does neither depends on <math>\Phi''(0)</math> and beyond, treated
on [[Hamilton's principle/Mathematics/The second variation]].

'''The family is a probe, not a physical path.''' The paths
<math>y + \varepsilon\eta</math> for <math>\varepsilon \neq 0</math> generally
violate the equations of motion. They are comparison curves, and Hamilton
described them in exactly those terms: geometrically imaginable, but dynamically
impossible motion.<ref>Hamilton, ''On a General Method in Dynamics'' (1834), §3.
Verbatim.</ref>

== The <math>\delta</math> notation ==

The classical literature writes <math>\delta y</math> for
<math>\varepsilon\eta</math> and <math>\delta J</math> for the first variation,
and manipulates <math>\delta</math> as an operator commuting with
<math>d/dx</math> and with <math>\int</math>. Hamilton's papers are written this
way throughout; his fundamental formula reads

:<math>\Sigma\, . \, m(x''\,\delta x + y''\,\delta y + z''\,\delta z) = \delta U
\qquad\qquad\mathrm{(8)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (1.).
Verbatim, with Hamilton's own notation preserved: <math>\Sigma</math> runs over
the points of the system, primes are derivatives with respect to time, and
<math>U</math> is the force-function.</ref>

The operator notation is compact and correct, and this volume uses it in the
history track where the sources use it. The <math>\varepsilon</math>-family
notation is used in the mathematics track because it makes the hypotheses
visible: it is clear what is differentiated, with respect to what, and where the
result is evaluated.

== Variations at free endpoints ==

If one or both endpoint values are not prescribed, <math>\mathcal{V}</math> is
enlarged and the boundary term in (5) no longer vanishes. It then supplies an
extra condition on the extremal, called a natural boundary condition. That case
is worked on
[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]].

== Next ==

[[Hamilton's principle/Mathematics/The fundamental lemma]] proves that a
continuous <math>E</math> whose integral against every admissible
<math>\eta</math> vanishes must itself vanish identically.

== References ==

<references />

← Older revision		Revision as of 04:56, 16 September 2026
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	Take <math>J</math>, <math>\mathcal{A}</math> and <math>\mathcal{V}</math> as on		Take <math>J</math>, <math>\mathcal{A}</math> and <math>\mathcal{V}</math> as on
	[[Hamilton's principle/Mathematics/Functionals and the space of paths]]. Fix		[[Hamilton's principle/Mathematics/Functionals and the space of paths\|Functionals and the space of paths]]. Fix
	<math>y \in \mathcal{A}</math> and <math>\eta \in \mathcal{V}</math>, and define		<math>y \in \mathcal{A}</math> and <math>\eta \in \mathcal{V}</math>, and define
	the real function		the real function
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	so that <math>\delta J[y;\eta] = \int_a^b E(x)\,\eta(x)\, dx</math>. Noether		so that <math>\delta J[y;\eta] = \int_a^b E(x)\,\eta(x)\, dx</math>. Noether
	calls <math>E</math> the Lagrange expression and it recurs under that name on		calls <math>E</math> the Lagrange expression and it recurs under that name on
	[[Hamilton's principle/Mathematics/Noether's first theorem]].		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]].

	== What has and has not been shown ==		== What has and has not been shown ==
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	integrand vanishes, or because positive and negative contributions cancel. Ruling		integrand vanishes, or because positive and negative contributions cancel. Ruling
	out the second possibility is a separate theorem, and it is the subject of		out the second possibility is a separate theorem, and it is the subject of
	[[Hamilton's principle/Mathematics/The fundamental lemma]].		[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]].

	Two further cautions.		Two further cautions.
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	<math>\Phi</math> only to first order. Whether <math>y</math> minimises,		<math>\Phi</math> only to first order. Whether <math>y</math> minimises,
	maximises, or does neither depends on <math>\Phi''(0)</math> and beyond, treated		maximises, or does neither depends on <math>\Phi''(0)</math> and beyond, treated
	on [[Hamilton's principle/Mathematics/The second variation]].		on [[Hamilton's principle/Mathematics/The second variation\|the second variation]].

	'''The family is a probe, not a physical path.''' The paths		'''The family is a probe, not a physical path.''' The paths
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	extra condition on the extremal, called a natural boundary condition. That case		extra condition on the extremal, called a natural boundary condition. That case
	is worked on		is worked on
	[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]].		[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions\|Boundary terms and natural boundary conditions]].

	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/The fundamental lemma]] proves that a		[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] proves that a
	continuous <math>E</math> whose integral against every admissible		continuous <math>E</math> whose integral against every admissible
	<math>\eta</math> vanishes must itself vanish identically.		<math>\eta</math> vanishes must itself vanish identically.