Hamilton's principle/Mathematics/Lagrange multipliers

Plain-language version: 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 Holonomic constraints.

Statement

Tension in a swinging load.

Theorem. Let $y=(y_1,\dots ,y_n)$ be stationary for

$J[y]={\int }_a^{b}L(x,y_i,{y_i}^{\prime })dx\mathrm{(}\mathrm{1}\mathrm{)}$

among admissible paths satisfying the $r$ independent holonomic constraints $g_{\beta }(x,y)=0$. Then there exist functions ${\lambda }_1(x),\dots ,{\lambda }_r(x)$ such that $y$ is stationary for the augmented functional

$J^{\ast }[y,\lambda ]={\int }_a^b[L+\sum _{\beta =1}^r{\lambda }_{\beta }(x)g_{\beta }(x,y)]dx\mathrm{(}\mathrm{2}\mathrm{)}$

with $y$ and $\lambda$ now varied independently.

Derivation for one constraint

Take $r=1$ and drop the index. From Holonomic constraints, stationarity gives

${\int }_a^b\sum _{i=1}^n{E}_i{\eta }_{i}dx=0\mathrm{(}\mathrm{3}\mathrm{)}$

for every variation obeying the admissibility condition

$\sum _{i=1}^n\frac{\partial g}{\partial y_i}{\eta }_i=0\mathrm{(}\mathrm{4}\mathrm{)}$

Step 1. Pick an index the constraint actually constrains. By the rank assumption there is some index — relabel it $n$ — with $\partial g/\partial y_n\ne 0$ on the interval. Solve (4) for ${\eta }_n$:

${\eta }_n=-{\left(\frac{\partial g}{\partial y_n}\right)}^{-1}\sum _{i=1}^{n-1}\frac{\partial g}{\partial y_i}{\eta }_i\mathrm{(}\mathrm{5}\mathrm{)}$

The first $n-1$ variations are now free, and the last is determined by them.

Step 2. Define the multiplier so as to kill the $n$-th term. Set

$\lambda (x)=-E_n{\left(\frac{\partial g}{\partial y_n}\right)}^{-1}\mathrm{(}\mathrm{6}\mathrm{)}$

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

$E_n+\lambda \frac{\partial g}{\partial y_n}=0\mathrm{(}\mathrm{7}\mathrm{)}$

so the $n$-th equation of the target system holds automatically.

Step 3. Substitute and use the freedom. Add $\int \lambda \sum _i(\partial g/\partial y_i){\eta }_{i}dx$ to (3), which is legitimate because that integral is zero by (4):

${\int }_a^b\sum _{i=1}^n[E_i+\lambda \frac{\partial g}{\partial y_i}]{\eta }_{i}dx=0\mathrm{(}\mathrm{8}\mathrm{)}$

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

$E_i+\lambda \frac{\partial g}{\partial y_i}=0,i=1,\dots ,n-1\mathrm{(}\mathrm{9}\mathrm{)}$

Together with (7), the relation holds for all $n$ indices. $◼$

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 $r$ constraints, $r$ indices are spent and the same argument runs with an invertible $r\times r$ block in place of the single nonzero partial.

Why the augmented functional works

Varying (2) with respect to $y_i$ gives (9) directly, since $g$ contains no ${y_i}^{\prime }$ and contributes only $\lambda \partial g/\partial y_i$ to the $y_i$ equation. Varying with respect to ${\lambda }_{\beta }$ gives

$\frac{\partial J^{\ast }}{\partial {\lambda }_{\beta }}:g_{\beta }(x,y)=0\mathrm{(}\mathrm{10}\mathrm{)}$

which returns the constraint. So the augmented problem is unconstrained in $n+r$ unknowns, and its stationarity conditions are the $n$ equations (9) plus the $r$ 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 $E_i=-\lambda \partial g/\partial y_i$. The right side is normal to the constraint surface, with $\lambda$ setting its magnitude. So $\lambda$ 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 Holonomic constraints:

$L=\frac{1}{2}m({\dot{y}}_1^2+{\dot{y}}_2^2)+m{g}_0{y}_2,g=y_1^2+y_2^2-{\ell }^2\mathrm{(}\mathrm{11}\mathrm{)}$

Then $\partial g/\partial y_1=2y_1$ and $\partial g/\partial y_2=2y_2$, and (9) gives

$m{\ddot{y}}_1=-2\lambda y_1,m{\ddot{y}}_2-m{g}_0=-2\lambda y_2\mathrm{(}\mathrm{12}\mathrm{)}$

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

$m{\ell }^2\ddot{\theta }+m{g}_0\ell \sin \theta =0\mathrm{(}\mathrm{13}\mathrm{)}$

as before. Combining the other way, so as to keep $\lambda$, gives

$2\lambda \ell =m{g}_0\cos \theta -m\ell {\dot{\theta }}^2\mathrm{(}\mathrm{14}\mathrm{)}$

and the rod tension is $T=2\lambda \ell$, 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 $x$, 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 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 ${\lambda }_1,\dots ,{\lambda }_k$ to adjoin the equations of condition before decomposing the variation.

Provenance: the attribution to Lagrange is editorial. The Mécanique analytique rows of the survey on 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.

Next

Isoperimetric problems treats the constant-multiplier case.