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{{Plebeian|Hamilton's principle/Explainers/Free ends}}

The derivation on
[[Hamilton's principle/Mathematics/The first variation]] discarded a boundary
term because the endpoints were held fixed. When an endpoint is free, that term
survives and imposes its own condition on the extremal. These are the
'''natural boundary conditions''': conditions the variational problem generates
by itself, rather than conditions imposed on it.

== The term that was discarded ==

[[File:OPL-HP-F-04.png|thumb|right|400px|A free endpoint and its natural boundary condition.]]

Integration by parts gave

:<math>\delta J[y;\eta] = \Bigl[ L_{y'}\,\eta \Bigr]_a^b
+ \int_a^b \left[ L_y - \frac{d}{dx} L_{y'} \right] \eta \, dx
\qquad\qquad\mathrm{(1)}</math>

With <math>\eta(a) = \eta(b) = 0</math> the first term vanishes. Now enlarge the
admissible set. Fix only the left endpoint:

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

<math>\eta(b)</math> is now unconstrained. Equation (1) becomes

:<math>\delta J[y;\eta] = L_{y'}\bigl(b, y(b), y'(b)\bigr)\,\eta(b)
+ \int_a^b \left[ L_y - \frac{d}{dx} L_{y'} \right] \eta \, dx
\qquad\qquad\mathrm{(3)}</math>

== Extracting two conditions from one requirement ==

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

Stationarity requires (3) to vanish for every
<math>\eta \in \mathcal{V}^*</math>. The argument splits in two stages, and the
order matters.

'''Stage 1.''' <math>\mathcal{V}^*</math> contains <math>\mathcal{V}</math> as a
subset. Restrict attention to those <math>\eta</math> that do vanish at
<math>b</math>. For these the boundary term is absent and (3) reduces to the
fixed-endpoint case, so the fundamental lemma applies and

:<math>L_y - \frac{d}{dx} L_{y'} = 0 \qquad\qquad\mathrm{(4)}</math>

on <math>[a,b]</math>. The Euler-Lagrange equation is unchanged. A free endpoint
does not alter the differential equation.

'''Stage 2.''' With (4) established, the integral in (3) vanishes for
''every'' <math>\eta</math>, so stationarity now demands

:<math>L_{y'}\bigl(b, y(b), y'(b)\bigr)\,\eta(b) = 0
\qquad\qquad\mathrm{(5)}</math>

for every <math>\eta \in \mathcal{V}^*</math>. Choosing any <math>\eta</math>
with <math>\eta(b) \neq 0</math> forces

:<math>L_{y'}\bigl(b, y(b), y'(b)\bigr) = 0
\qquad\qquad\mathrm{(6)}</math>

Equation (6) is the natural boundary condition at the free end.

The two-stage structure is worth stating plainly: a larger class of variations
yields the ''same'' equation plus an ''extra'' condition. Nothing is lost by
freeing an endpoint; something is gained.

== The condition count still balances ==

[[File:OPL-HP-F-41.png|thumb|right|400px|An endpoint free to lie on a curve.]]

The Euler-Lagrange equation is second order and needs two conditions. With both
endpoints fixed, the admissible set supplies both. With the right endpoint free,
it supplies only <math>y(a) = y_a</math>, and (6) supplies the second. With both
endpoints free, (6) holds at each end and again the count is two.

This is why the term "natural" is apt. The conditions are not an extra
assumption; they are what the problem produces when the modeller declines to
impose one.

== Interpretation ==

[[File:OPL-HP-F-42.png|thumb|right|400px|A corner in an extremal.]]

<math>L_{y'}</math> is the conjugate momentum of
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]. Condition (6)
therefore reads: '''the conjugate momentum vanishes at a free end.'''

In mechanics, a free end carries no momentum transfer. In elasticity, where
<math>L</math> is a strain energy and <math>L_{y'}</math> a stress, (6) is the
statement that an unloaded edge is stress free. The same algebra covers both, and
the interpretation follows from what <math>L</math> was taken to be.

== Transversality: endpoints on a curve ==

A more general case fixes the right endpoint not to a point but to a curve. Let
the right end be required to lie on <math>y = g(x)</math>, with the crossing
abscissa <math>b</math> itself free. Varying both <math>y</math> and
<math>b</math>, the boundary contribution becomes

:<math>\Bigl[ L + (g' - y')\,L_{y'} \Bigr]_{x = b} = 0
\qquad\qquad\mathrm{(7)}</math>

Equation (7) is the '''transversality condition'''. It reduces to (6) when
<math>g</math> is a vertical line, and to <math>L = 0</math> when
<math>g' = y'</math>.

For the arc-length integrand <math>L = \sqrt{1 + y'^2}</math> of
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]], substituting
into (7) gives <math>1 + y'g' = 0</math>, so <math>y'g' = -1</math> and the
extremal meets the curve at a right angle. Transversality is orthogonality for
this integrand, and only for integrands of this kind; the general condition (7)
is not a statement about angles.

== Corner conditions ==

If the admissible class is widened to piecewise smooth paths, a path may have a
corner at an interior point <math>c</math>. The variation then produces a
boundary term at <math>c</math> from each side, and stationarity requires both

:<math>\bigl[ L_{y'} \bigr]_{c^-}^{c^+} = 0
\qquad \text{and} \qquad
\bigl[ L - y'L_{y'} \bigr]_{c^-}^{c^+} = 0
\qquad\qquad\mathrm{(8)}</math>

These are the '''Weierstrass-Erdmann corner conditions'''. Both bracketed
quantities are continuous across a corner even though <math>y'</math> is not.
The second is the Beltrami quantity of
[[Hamilton's principle/Mathematics/The Beltrami identity]], and the pair
reappears on
[[Hamilton's principle/Mathematics/The Weierstrass excess function]].

Note that when <math>L_{y'y'} \neq 0</math>, the first condition in (8) forces
<math>y'</math> to be continuous, so no corner can occur. Corners require
degeneracy, which is the connection back to
[[Hamilton's principle/Mathematics/The Legendre condition]].

== Why this matters for Hamilton's principle ==

The endpoints in Hamilton's principle are configurations at two fixed times, and
they are held fixed. So natural boundary conditions play no role in the basic
mechanical statement. They matter in two places this volume reaches later: in
continuum problems on
[[Hamilton's principle/Applications/Continuum systems: the vibrating string]],
where a free string end generates its own condition, and in field theory on
[[Hamilton's principle/Applications/Classical field theory]], where the
surface term must be argued away rather than assumed away. The
Einstein-Hilbert action on
[[Hamilton's principle/Applications/The Einstein-Hilbert action]] is the case
where it cannot be argued away and a compensating surface term must be added by
hand.

== Next ==

[[Hamilton's principle/Mathematics/The Beltrami identity]] treats the first
integral available when <math>L</math> does not contain the independent variable.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	The derivation on		The derivation on
	[[Hamilton's principle/Mathematics/The first variation]] discarded a boundary		[[Hamilton's principle/Mathematics/The first variation\|the first variation]] discarded a boundary
	term because the endpoints were held fixed. When an endpoint is free, that term		term because the endpoints were held fixed. When an endpoint is free, that term
	survives and imposes its own condition on the extremal. These are the		survives and imposes its own condition on the extremal. These are the
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	<math>L_{y'}</math> is the conjugate momentum of		<math>L_{y'}</math> is the conjugate momentum of
	[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]. Condition (6)		[[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]]. Condition (6)
	therefore reads: '''the conjugate momentum vanishes at a free end.'''		therefore reads: '''the conjugate momentum vanishes at a free end.'''

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	For the arc-length integrand <math>L = \sqrt{1 + y'^2}</math> of		For the arc-length integrand <math>L = \sqrt{1 + y'^2}</math> of
	[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]], substituting		[[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]], substituting
	into (7) gives <math>1 + y'g' = 0</math>, so <math>y'g' = -1</math> and the		into (7) gives <math>1 + y'g' = 0</math>, so <math>y'g' = -1</math> and the
	extremal meets the curve at a right angle. Transversality is orthogonality for		extremal meets the curve at a right angle. Transversality is orthogonality for
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	quantities are continuous across a corner even though <math>y'</math> is not.		quantities are continuous across a corner even though <math>y'</math> is not.
	The second is the Beltrami quantity of		The second is the Beltrami quantity of
	[[Hamilton's principle/Mathematics/The Beltrami identity]], and the pair		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]], and the pair
	reappears on		reappears on
	[[Hamilton's principle/Mathematics/The Weierstrass excess function]].		[[Hamilton's principle/Mathematics/The Weierstrass excess function\|the Weierstrass excess function]].

	Note that when <math>L_{y'y'} \neq 0</math>, the first condition in (8) forces		Note that when <math>L_{y'y'} \neq 0</math>, the first condition in (8) forces
	<math>y'</math> to be continuous, so no corner can occur. Corners require		<math>y'</math> to be continuous, so no corner can occur. Corners require
	degeneracy, which is the connection back to		degeneracy, which is the connection back to
	[[Hamilton's principle/Mathematics/The Legendre condition]].		[[Hamilton's principle/Mathematics/The Legendre condition\|the Legendre condition]].

	== Why this matters for Hamilton's principle ==		== Why this matters for Hamilton's principle ==
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	mechanical statement. They matter in two places this volume reaches later: in		mechanical statement. They matter in two places this volume reaches later: in
	continuum problems on		continuum problems on
	[[Hamilton's principle/Applications/Continuum systems: the vibrating string]],		[[Hamilton's principle/Applications/Continuum systems: the vibrating string\|Continuum systems: the vibrating string]],
	where a free string end generates its own condition, and in field theory on		where a free string end generates its own condition, and in field theory on
	[[Hamilton's principle/Applications/Classical field theory]], where the		[[Hamilton's principle/Applications/Classical field theory\|Classical field theory]], where the
	surface term must be argued away rather than assumed away. The		surface term must be argued away rather than assumed away. The
	Einstein-Hilbert action on		Einstein-Hilbert action on
	[[Hamilton's principle/Applications/The Einstein-Hilbert action]] is the case		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|the Einstein-Hilbert action]] is the case
	where it cannot be argued away and a compensating surface term must be added by		where it cannot be argued away and a compensating surface term must be added by
	hand.		hand.
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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/The Beltrami identity]] treats the first		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]] treats the first
	integral available when <math>L</math> does not contain the independent variable.		integral available when <math>L</math> does not contain the independent variable.