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

A '''functional''' is a rule that assigns one number to a whole function. The
calculus of variations is the study of such rules, and Hamilton's principle is a
statement about one of them. This page fixes the objects the rest of the
mathematics track works with: the admissible set, the functional itself, and what
it means for two paths to be close.

== The admissible set ==

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

Fix two points <math>a < b</math> on the real line and two values
<math>y_a</math>, <math>y_b</math>. Consider the set

:<math>\mathcal{A} = \{\, y \in C^2[a,b] \;:\; y(a) = y_a,\ y(b) = y_b \,\}
\qquad\qquad\mathrm{(1)}</math>

of twice continuously differentiable functions joining the two endpoints. Call
<math>\mathcal{A}</math> the '''admissible set''' and each of its members an
'''admissible path'''.

Two features of <math>\mathcal{A}</math> matter and are easy to miss.

First, <math>\mathcal{A}</math> is '''not a vector space'''. The sum of two
admissible paths takes the value <math>2y_a</math> at <math>a</math>, so it is
not admissible. This is why the variation defined on the next page is built from
a separate space of ''differences'' rather than from <math>\mathcal{A}</math>
itself.

Second, the smoothness class is a choice, not a fact. Taking
<math>C^2</math> makes the Euler-Lagrange equation a genuine second-order
differential equation. Weaker classes admit solutions with corners, and the
brachistochrone with a broken extremal is the standard reason to care. The
Weierstrass theory on
[[Hamilton's principle/Mathematics/The Weierstrass excess function]] works in the
wider class.

== The space of variations ==

Set

:<math>\mathcal{V} = \{\, \eta \in C^2[a,b] \;:\; \eta(a) = \eta(b) = 0 \,\}
\qquad\qquad\mathrm{(2)}</math>

This ''is'' a vector space. Its members vanish at both endpoints, so for any
admissible <math>y</math> and any real <math>\varepsilon</math> the path

:<math>y_\varepsilon = y + \varepsilon\eta \qquad\qquad\mathrm{(3)}</math>

is again admissible. Equation (3) is the whole device. It turns a question about
a set of paths into a question about a single real variable
<math>\varepsilon</math>, and every derivation in this track runs through it.

The relation between <math>\mathcal{A}</math> and <math>\mathcal{V}</math> is
that <math>\mathcal{A}</math> is an ''affine'' set: pick any one admissible path
as an origin and every other admissible path is that one plus a member of
<math>\mathcal{V}</math>.

== The functional ==

Let <math>L(x, y, y')</math> be a function of three real arguments, continuous
with continuous partial derivatives up to second order. Define

:<math>J[y] = \int_a^b L\bigl(x,\, y(x),\, y'(x)\bigr)\, dx
\qquad\qquad\mathrm{(4)}</math>

for <math>y \in \mathcal{A}</math>. The map <math>J</math> takes a path and
returns a number. <math>L</math> is the '''integrand''', or, when the independent
variable is time and the functional is action, the '''Lagrangian'''.

Note what the three slots of <math>L</math> mean. Inside the integral,
<math>y</math> and <math>y'</math> are not independent — the second is the
derivative of the first. But <math>L</math> is defined as a function of three
free arguments, and its partial derivatives <math>\partial L/\partial y</math>
and <math>\partial L/\partial y'</math> are taken with all three held
independent. This distinction is the commonest source of error in the subject.
When <math>\partial L/\partial y'</math> is written below, the slot is
differentiated, not the function <math>y'(x)</math>.

== Distance between paths ==

"Nearby path" needs a metric, and there is more than one reasonable choice. Two
are used throughout this track.

The '''weak norm''' is

:<math>\Vert \eta \Vert_1 = \max_{[a,b]} \vert \eta(x) \vert
+ \max_{[a,b]} \vert \eta'(x) \vert \qquad\qquad\mathrm{(5)}</math>

and the '''strong norm''' is

:<math>\Vert \eta \Vert_0 = \max_{[a,b]} \vert \eta(x) \vert
\qquad\qquad\mathrm{(6)}</math>

A path is close to <math>y</math> in the strong sense if it stays close in
position. It is close in the weak sense only if it also stays close in slope.

Every <math>\Vert\cdot\Vert_1</math>-neighbourhood is contained in a
<math>\Vert\cdot\Vert_0</math>-neighbourhood, so a strong neighbourhood is the
''larger'' set and a strong minimum is the ''stronger'' claim. The naming is
unfortunate and standard; it is worth reading twice.

The distinction is not decoration. A path can minimise <math>J</math> against
every weakly nearby competitor and lose to a strongly nearby one that wanders
close in position while oscillating wildly in slope. Sorting out which claim
holds is the business of
[[Hamilton's principle/Mathematics/The Weierstrass excess function]].

== Extrema and stationarity ==

<math>y \in \mathcal{A}</math> is a '''weak local minimum''' of <math>J</math> if
there is <math>\delta > 0</math> with <math>J[y + \eta] \geq J[y]</math> for
every <math>\eta \in \mathcal{V}</math> satisfying
<math>\Vert\eta\Vert_1 < \delta</math>. Replacing
<math>\Vert\cdot\Vert_1</math> by <math>\Vert\cdot\Vert_0</math> gives a strong
local minimum, and reversing the inequality gives the maxima.

'''Stationary''' is a different and weaker condition, defined on the next page.
A stationary path need be neither a minimum nor a maximum. Hamilton was explicit
about this from the start: reviewing what Lagrange and others had drawn from the
vanishing of the variation, he called it the 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.</ref>

The gap between "stationary" and "minimum" is the subject of
[[Hamilton's principle/Mathematics/The second variation]] and
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]],
and it is the reason this volume does not call the subject the principle of least
action.

== Notation ==

This volume writes <math>L</math> for a general integrand and reserves
<math>T - V</math> for the mechanical Lagrangian, introduced on
[[Hamilton's principle/Applications/Central force and the Kepler problem]].
Hamilton's own <math>V</math> is the characteristic function, an entirely
different object, treated on
[[Hamilton's principle/Mathematics/Action as a function of its endpoints]]. The
concordance on [[Hamilton's principle/Notation]] tabulates the collision.

== Next ==

[[Hamilton's principle/Mathematics/The first variation]] differentiates (4) with
respect to <math>\varepsilon</math> along the family (3).

== References ==

<references />

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	brachistochrone with a broken extremal is the standard reason to care. The		brachistochrone with a broken extremal is the standard reason to care. The
	Weierstrass theory on		Weierstrass theory on
	[[Hamilton's principle/Mathematics/The Weierstrass excess function]] works in the		[[Hamilton's principle/Mathematics/The Weierstrass excess function\|the Weierstrass excess function]] works in the
	wider class.		wider class.

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	close in position while oscillating wildly in slope. Sorting out which claim		close in position while oscillating wildly in slope. Sorting out which claim
	holds is the business of		holds is the business of
	[[Hamilton's principle/Mathematics/The Weierstrass excess function]].		[[Hamilton's principle/Mathematics/The Weierstrass excess function\|the Weierstrass excess function]].

	== Extrema and stationarity ==		== Extrema and stationarity ==
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	The gap between "stationary" and "minimum" is the subject of		The gap between "stationary" and "minimum" is the subject of
	[[Hamilton's principle/Mathematics/The second variation]] and		[[Hamilton's principle/Mathematics/The second variation\|the second variation]] and
	[[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 it is the reason this volume does not call the subject the principle of least		and it is the reason this volume does not call the subject the principle of least
	action.		action.
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	This volume writes <math>L</math> for a general integrand and reserves		This volume writes <math>L</math> for a general integrand and reserves
	<math>T - V</math> for the mechanical Lagrangian, introduced on		<math>T - V</math> for the mechanical Lagrangian, introduced on
	[[Hamilton's principle/Applications/Central force and the Kepler problem]].		[[Hamilton's principle/Applications/Central force and the Kepler problem\|Central force and the Kepler problem]].
	Hamilton's own <math>V</math> is the characteristic function, an entirely		Hamilton's own <math>V</math> is the characteristic function, an entirely
	different object, treated on		different object, treated on
	[[Hamilton's principle/Mathematics/Action as a function of its endpoints]]. The		[[Hamilton's principle/Mathematics/Action as a function of its endpoints\|Action as a function of its endpoints]]. The
	concordance on [[Hamilton's principle/Notation]] tabulates the collision.		concordance on [[Hamilton's principle/Notation\|Notation]] tabulates the collision.

	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/The first variation]] differentiates (4) with		[[Hamilton's principle/Mathematics/The first variation\|the first variation]] differentiates (4) with
	respect to <math>\varepsilon</math> along the family (3).		respect to <math>\varepsilon</math> along the family (3).