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	{{Plebeian\|Hamilton's principle/Explainers/Multipliers}}		{{Plebeian\|Hamilton's principle/Explainers/Multipliers}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/Multipliers}}

The '''method of multipliers''' restores the independence that a constraint
destroys. It buys back the select-one-index argument by introducing one new
unknown function per constraint, and it returns the constraint forces as a
by-product. This page derives the method for the holonomic constraints defined on
[[Hamilton's principle/Mathematics/Holonomic constraints]].

== Statement ==

[[File:OPL-HP-F-19.png|thumb|right|400px|Tension in a swinging load.]]

'''Theorem.''' Let <math>y = (y_1, \ldots, y_n)</math> be stationary for

:<math>J[y] = \int_a^b L(x, y_i, y_i')\, dx
\qquad\qquad\mathrm{(1)}</math>

among admissible paths satisfying the <math>r</math> independent holonomic
constraints <math>g_\beta(x, y) = 0</math>. Then there exist functions
<math>\lambda_1(x), \ldots, \lambda_r(x)</math> such that <math>y</math> is
stationary for the '''augmented functional'''

:<math>J^*[y,\lambda] = \int_a^b \left[ L
+ \sum_{\beta=1}^r \lambda_\beta(x)\, g_\beta(x,y) \right] dx
\qquad\qquad\mathrm{(2)}</math>

with <math>y</math> and <math>\lambda</math> now varied independently.

== Derivation for one constraint ==

Take <math>r = 1</math> and drop the index. From
[[Hamilton's principle/Mathematics/Holonomic constraints]], stationarity gives

:<math>\int_a^b \sum_{i=1}^n E_i\, \eta_i \, dx = 0
\qquad\qquad\mathrm{(3)}</math>

for every variation obeying the admissibility condition

:<math>\sum_{i=1}^n \frac{\partial g}{\partial y_i}\, \eta_i = 0
\qquad\qquad\mathrm{(4)}</math>

'''Step 1. Pick an index the constraint actually constrains.''' By the rank
assumption there is some index — relabel it <math>n</math> — with
<math>\partial g/\partial y_n \neq 0</math> on the interval. Solve (4) for
<math>\eta_n</math>:

:<math>\eta_n = -\left( \frac{\partial g}{\partial y_n} \right)^{-1}
\sum_{i=1}^{n-1} \frac{\partial g}{\partial y_i}\, \eta_i
\qquad\qquad\mathrm{(5)}</math>

The first <math>n-1</math> variations are now free, and the last is determined by
them.

'''Step 2. Define the multiplier so as to kill the <math>n</math>-th term.''' Set

:<math>\lambda(x) = -E_n \left( \frac{\partial g}{\partial y_n} \right)^{-1}
\qquad\qquad\mathrm{(6)}</math>

This is a definition, not an assumption. <math>\lambda</math> is whatever (6)
says it is, and it is continuous because <math>E_n</math> and the partial are.
By construction,

:<math>E_n + \lambda\, \frac{\partial g}{\partial y_n} = 0
\qquad\qquad\mathrm{(7)}</math>

so the <math>n</math>-th equation of the target system holds automatically.

'''Step 3. Substitute and use the freedom.''' Add
<math>\int \lambda \sum_i (\partial g/\partial y_i)\eta_i\, dx</math> to (3),
which is legitimate because that integral is zero by (4):

:<math>\int_a^b \sum_{i=1}^n
\left[ E_i + \lambda\, \frac{\partial g}{\partial y_i} \right] \eta_i \, dx = 0
\qquad\qquad\mathrm{(8)}</math>

The <math>i = n</math> term vanishes identically by (7), leaving a sum over
<math>i = 1, \ldots, n-1</math> whose variations are ''free'' by Step 1. The
select-one-index argument now applies, and the fundamental lemma gives

:<math>E_i + \lambda\, \frac{\partial g}{\partial y_i} = 0,
\qquad i = 1, \ldots, n-1
\qquad\qquad\mathrm{(9)}</math>

Together with (7), the relation holds for all <math>n</math> indices.
<math>\blacksquare</math>

The pattern is worth naming: '''one index is spent buying freedom for the
rest.''' The multiplier is defined precisely so that the spent index costs
nothing, and the remaining indices are then unconstrained. With <math>r</math>
constraints, <math>r</math> indices are spent and the same argument runs with an
invertible <math>r \times r</math> block in place of the single nonzero partial.

== Why the augmented functional works ==

Varying (2) with respect to <math>y_i</math> gives (9) directly, since
<math>g</math> contains no <math>y_i'</math> and contributes only
<math>\lambda\,\partial g/\partial y_i</math> to the <math>y_i</math> equation.
Varying with respect to <math>\lambda_\beta</math> gives

:<math>\frac{\partial J^*}{\partial \lambda_\beta}:
\qquad g_\beta(x,y) = 0
\qquad\qquad\mathrm{(10)}</math>

which returns the constraint. So the augmented problem is unconstrained in
<math>n + r</math> unknowns, and its stationarity conditions are the
<math>n</math> equations (9) plus the <math>r</math> constraints. The count
balances, and the constraint is enforced by a variation rather than by
restricting the admissible set.

That last point is the conceptual content of the method. A restriction on the
''domain'' has been traded for an addition to the ''integrand''.

== The multiplier is the constraint force ==

Equation (9) reads <math>E_i = -\lambda\,\partial g/\partial y_i</math>. The
right side is normal to the constraint surface, with <math>\lambda</math> setting
its magnitude. So <math>\lambda</math> is not bookkeeping: it is the force the
constraint exerts, in the units the coordinates imply, and it is usually the
answer the engineer wanted.

== Worked example: the pendulum, with tension ==

Return to the system of
[[Hamilton's principle/Mathematics/Holonomic constraints]]:

:<math>L = \tfrac{1}{2}m(\dot y_1^2 + \dot y_2^2) + m g_0 y_2,
\qquad
g = y_1^2 + y_2^2 - \ell^2
\qquad\qquad\mathrm{(11)}</math>

Then <math>\partial g/\partial y_1 = 2y_1</math> and
<math>\partial g/\partial y_2 = 2y_2</math>, and (9) gives

:<math>m\ddot y_1 = -2\lambda y_1,
\qquad
m\ddot y_2 - m g_0 = -2\lambda y_2
\qquad\qquad\mathrm{(12)}</math>

together with <math>y_1^2 + y_2^2 = \ell^2</math>. Substituting
<math>y_1 = \ell\sin\theta</math>, <math>y_2 = \ell\cos\theta</math> into (12)
and combining so as to eliminate <math>\lambda</math> reproduces

:<math>m\ell^2\ddot\theta + m g_0\ell\sin\theta = 0
\qquad\qquad\mathrm{(13)}</math>

as before. Combining the other way, so as to keep <math>\lambda</math>, gives

:<math>2\lambda\ell = m g_0\cos\theta - m\ell\dot\theta^2
\qquad\qquad\mathrm{(14)}</math>

and the rod tension is <math>T = 2\lambda\ell</math>, containing both the weight
component and the centripetal term. The elimination method on the previous page
produced (13) and could not produce (14). That is the practical case for
multipliers.

== Constant multipliers ==

In (2) the multipliers are functions of <math>x</math>, because the constraint is
imposed at every point. A different kind of constraint fixes a single number —
the total length of a curve, the total mass of a distribution — and is imposed
once rather than pointwise. Its multiplier is then a single constant. Those are
'''isoperimetric''' constraints, and they are the subject of
[[Hamilton's principle/Mathematics/Isoperimetric problems]].

Confusing the two is a common error. The test is whether the constraint is one
equation per point or one equation for the whole path.

== Historical note ==

The method is Lagrange's, and multipliers appear in the ''Mécanique analytique''
as the device by which the equations of constrained systems are written without
first eliminating coordinates. Hamilton reproduces the technique in the 1834
essay, introducing factors <math>\lambda_1, \ldots, \lambda_k</math> to adjoin
the equations of condition before decomposing the variation.<ref>Hamilton, ''On a
General Method in Dynamics'' (1834), equations (31.) to (D1.). The multipliers are
Hamilton's <math>\lambda</math> and <math>\Lambda</math>. ESTABLISHED source.</ref>

<blockquote>
Provenance: the attribution to Lagrange is editorial. The
''Mécanique analytique'' rows of the survey on
[[Hamilton's principle/Sources]] are PROVISIONAL, and the 1788 and 1811 editions
are tracked separately because they differ substantially. No equation on this
page is attributed to either edition. The Hamilton citation is ESTABLISHED.
</blockquote>

== Next ==

[[Hamilton's principle/Mathematics/Isoperimetric problems]] treats the
constant-multiplier case.

== References ==

<references />

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	unknown function per constraint, and it returns the constraint forces as a		unknown function per constraint, and it returns the constraint forces as a
	by-product. This page derives the method for the holonomic constraints defined on		by-product. This page derives the method for the holonomic constraints defined on
	[[Hamilton's principle/Mathematics/Holonomic constraints]].		[[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]].

	== Statement ==		== Statement ==
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	Take <math>r = 1</math> and drop the index. From		Take <math>r = 1</math> and drop the index. From
	[[Hamilton's principle/Mathematics/Holonomic constraints]], stationarity gives		[[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]], stationarity gives

	:<math>\int_a^b \sum_{i=1}^n E_i\, \eta_i \, dx = 0		:<math>\int_a^b \sum_{i=1}^n E_i\, \eta_i \, dx = 0
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	Return to the system of		Return to the system of
	[[Hamilton's principle/Mathematics/Holonomic constraints]]:		[[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]]:

	:<math>L = \tfrac{1}{2}m(\dot y_1^2 + \dot y_2^2) + m g_0 y_2,		:<math>L = \tfrac{1}{2}m(\dot y_1^2 + \dot y_2^2) + m g_0 y_2,
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	once rather than pointwise. Its multiplier is then a single constant. Those are		once rather than pointwise. Its multiplier is then a single constant. Those are
	'''isoperimetric''' constraints, and they are the subject of		'''isoperimetric''' constraints, and they are the subject of
	[[Hamilton's principle/Mathematics/Isoperimetric problems]].		[[Hamilton's principle/Mathematics/Isoperimetric problems\|Isoperimetric problems]].

	Confusing the two is a common error. The test is whether the constraint is one		Confusing the two is a common error. The test is whether the constraint is one
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	Provenance: the attribution to Lagrange is editorial. The		Provenance: the attribution to Lagrange is editorial. The
	''Mécanique analytique'' rows of the survey on		''Mécanique analytique'' rows of the survey on
	[[Hamilton's principle/Sources]] are PROVISIONAL, and the 1788 and 1811 editions		[[Hamilton's principle/Sources\|Sources]] are PROVISIONAL, and the 1788 and 1811 editions
	are tracked separately because they differ substantially. No equation on this		are tracked separately because they differ substantially. No equation on this
	page is attributed to either edition. The Hamilton citation is ESTABLISHED.		page is attributed to either edition. The Hamilton citation is ESTABLISHED.
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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/Isoperimetric problems]] treats the		[[Hamilton's principle/Mathematics/Isoperimetric problems\|Isoperimetric problems]] treats the
	constant-multiplier case.		constant-multiplier case.

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