Hamilton's principle/Mathematics/Holonomic constraints

Plain-language version: Constraints

Every derivation in the machinery track rested on one assumption: that the variations of the several unknowns could be chosen independently. Real systems violate it. A bead threaded on a wire, a pendulum on a rigid rod, two masses joined by an inextensible string — in each the coordinates are tied together, and the select-one-index argument of Several dependent variables fails at its first step. This page states the failure exactly. The repair is on Lagrange multipliers.

Definition

A constraint force normal to the permitted motion.

A holonomic constraint on $n$ coordinates $y_1,\dots ,y_n$ is a relation

$g(x,y_1,\dots ,y_n)=0\mathrm{(}\mathrm{1}\mathrm{)}$

involving the coordinates and possibly the independent variable, but not their derivatives. A system of $r$ such relations $g_{\beta }=0$, $\beta =1,\dots ,r$, is holonomic if each has this form and the $r\times n$ matrix $\partial g_{\beta }/\partial y_i$ has rank $r$ everywhere on the constraint set. That rank condition says the constraints are independent and non-degenerate; it is assumed throughout.

A constraint that cannot be written in the form (1) is non-holonomic. The distinction is not cosmetic, and it is drawn at the end of this page.

Two ways to proceed

Varying one degree of freedom at a time.

Eliminate. With $r$ independent relations, $r$ of the coordinates can in principle be solved for and substituted out, leaving $n-r$ free coordinates $q_1,\dots ,q_{n-r}$. These are generalised coordinates. In terms of them the problem is unconstrained, the variations are again independent, and Several dependent variables applies verbatim.

Adjoin. Keep all $n$ coordinates and carry the constraints alongside, paying for them with $r$ new unknowns. This is the multiplier method.

Elimination is cheaper when it is possible. It is often not possible in closed form, it usually destroys symmetry, and it discards the constraint forces, which are frequently the quantities of interest. The multiplier method keeps all three and is the general tool.

Why the independence argument fails

Let $y$ be an admissible path satisfying (1), and let $y+\epsilon \eta$ be a competitor. For the competitor to be admissible it must satisfy the constraint too:

$g(x,y+\epsilon \eta )=0\mathrm{(}\mathrm{2}\mathrm{)}$

Expanding to first order in $\epsilon$ and using $g(x,y)=0$,

$\sum _{i=1}^n\frac{\partial g}{\partial y_i}{\eta }_i=0\mathrm{(}\mathrm{3}\mathrm{)}$

Equation (3) is the admissibility condition on the variations. It is a linear relation among the ${\eta }_i$ at each point $x$. Choosing ${\eta }_k$ arbitrary and all others zero violates it, unless $\partial g/\partial y_k$ happens to vanish there.

So the step that produced $n$ separate equations is unavailable. What survives from Several dependent variables is only the integral statement

${\int }_a^b\sum _{i=1}^n{E}_i{\eta }_{i}dx=0,E_i=\frac{\partial L}{\partial y_i}-\frac{d}{dx}\frac{\partial L}{\partial {y_i}^{\prime }}\mathrm{(}\mathrm{4}\mathrm{)}$

holding for those $\eta$ obeying (3), and no longer for all $\eta$. The fundamental lemma does not apply, because its hypothesis is that the test class is unrestricted.

This is the whole difficulty, and it is a genuine one rather than a technicality. Equation (4) with (3) says the vector $E$ is orthogonal to every admissible variation. It does not say $E$ is zero. It says $E$ lies in the space spanned by the constraint gradients, and turning that observation into a usable equation is what multipliers do.

Constraint forces

Physically, $E_i\ne 0$ is expected. The unconstrained Euler-Lagrange expression measures the mismatch between the applied forces and the motion, and for a constrained system that mismatch is exactly the force the constraint exerts. The wire pushes on the bead. The rod pulls on the bob.

So the constrained problem should be expected to read

$E_i=-\sum _{\beta =1}^r{\lambda }_{\beta }\frac{\partial g_{\beta }}{\partial y_i}\mathrm{(}\mathrm{5}\mathrm{)}$

with the right side the constraint force. Equation (5) is what Lagrange multipliers derives; it is stated here only to name the target.

Note the structure: constraint forces are normal to the constraint surface, so they do no work under any admissible variation. That is d'Alembert's principle, and it is why (4) can hold with $E\ne 0$ without contradiction.

Worked example: the simple pendulum

A bob of mass $m$ on a rigid rod of length $\ell$, in the plane, under gravity $g_0$. Cartesian coordinates $(y_1,y_2)$ measured from the pivot, with $y_2$ downward.

$L=\frac{1}{2}m({\dot{y}}_1^2+{\dot{y}}_2^2)+m{g}_0{y}_2,g=y_1^2+y_2^2-{\ell }^2=0\mathrm{(}\mathrm{6}\mathrm{)}$

By elimination. Put $y_1=\ell \sin \theta$, $y_2=\ell \cos \theta$. The constraint is satisfied identically, one free coordinate remains, and

$L=\frac{1}{2}m{\ell }^2{\dot{\theta }}^2+m{g}_0\ell \cos \theta \mathrm{(}\mathrm{7}\mathrm{)}$

The single Euler-Lagrange equation is

$m{\ell }^2\ddot{\theta }+m{g}_0\ell \sin \theta =0\mathrm{(}\mathrm{8}\mathrm{)}$

which is the pendulum equation. Nothing about the rod tension appears, and it cannot be recovered from (8) alone.

What was given up. The elimination worked because the constraint was solvable in closed form and the parametrisation was obvious. Neither is generally true. The multiplier treatment of the same system, on the next page, returns (8) together with the tension.

Non-holonomic constraints

A constraint involving derivatives,

$\sum _{i=1}^n{a}_i(x,y){y_i}^{\prime }+a_0(x,y)=0\mathrm{(}\mathrm{9}\mathrm{)}$

is holonomic in disguise if the left side is an exact derivative or has an integrating factor, in which case it integrates to the form (1). Otherwise it is genuinely non-holonomic. The rolling coin on a plane is the standard example: it can reach any position and orientation, so the constraint removes no configurations, yet it restricts every motion.

Warning. Naively adjoining a non-holonomic constraint with a multiplier and varying gives the vakonomic equations, which are not the equations of the rolling body. The correct dynamics comes from d'Alembert's principle applied to the velocity constraint, and the two disagree. This volume treats holonomic constraints only, and records the distinction here so that the multiplier method of the next page is not over-applied.

Provenance: the vakonomic-versus-dynamic distinction is editorial summary of the modern literature. No primary source is cited for it and none is claimed. See Sources.

Next

Lagrange multipliers converts the restricted statement (4) with (3) into the unrestricted system (5).