The force a coordinate cannot see

A pendulum has one degree of freedom and two coordinates, and the first thing anybody is taught to do about that is to throw one of the coordinates away.

The bob is on a string of fixed length, so its two Cartesian coordinates satisfy $x^2 + y^2 = L^2$ and only one number is really free. Choose the angle from the vertical, write the kinetic and potential energies in terms of it, and the stationary-action condition hands back a single second-order equation with no string in it whatever.

That is the standard advertisement for the method, and it is entirely true. It is also the reason the method has to be extended almost immediately, because the string is still there and it can still snap.

What the good choice of coordinate actually did #

The constraint was not solved. It was made unnecessary.

An equation of motion written in the angle produces motion that satisfies $x^2 + y^2 = L^2$ automatically, at every step, to every precision, because every configuration the coordinate can describe already lies on the circle. Nothing has to enforce the constraint, so nothing in the equations does — and a force that enforces nothing is a force that does not appear.

In Cartesian coordinates the situation is the reverse. There the constraint is a genuine extra condition, the motion has to be held on the circle by something, and that something is a force with a magnitude. The two descriptions give the same trajectory and only one of them contains the tension.

The recovery is a device of Lagrange’s, and it is arranged so that the constraint and the force that maintains it are two halves of one object. Keep both Cartesian coordinates, write the constraint as $g(x, y) = x^2 + y^2 - L^2 = 0$, and add to the equations a term $\lambda \nabla g$ — a force pointing along the gradient of the constraint, which for a circle is along the radius, with an unknown strength $\lambda$.

The unknown is then fixed by the requirement that the motion actually stays on the circle. Differentiating the constraint twice with respect to time and substituting the equations of motion gives one equation for $\lambda$ at every instant, and that number is the tension.

Two routes, and nothing shared between them #

The claim that the multiplier is the physical force is worth more than an assertion, because it is the whole justification for calling the method mechanics rather than bookkeeping.

The curve is the elementary calculation. Resolve the forces along the radius: the string pulls inward with $T$, gravity contributes $mg\cos\theta$ inward, and the net must supply the centripetal requirement $mv^2/L$. With the speed taken from conservation of energy this gives

$$\frac{T}{mg} = 3\cos\theta - 2\cos\theta_0,$$

where $\theta_0$ is the amplitude. Nothing in that derivation is Lagrangian.

The points are the other route. A computer is handed the two Cartesian coordinates, the constraint, and the multiplier prescription, and integrates. At each step it solves for $\lambda$, multiplies by the gradient, and steps forward. It has no idea what an angle is.

The two agree to about $3 \times 10^{-8}$ of a weight across the swing, which is the accuracy of the integration rather than a disagreement about physics — and the integration’s own constraint residual over the same run is $10^{-8}$ of the length, which is the number that says whether the multiplier was read off the circle or off a trajectory quietly leaving it.

Two things in that figure are worth staring at. The tension at the ends of the swing is less than the weight of the bob, because there the string only has to hold up the component of gravity along itself. And at the bottom it is three weights, not one.

Three weights is the number a rope is chosen against, and it is nowhere in the equation of motion. A swing designed by somebody who had solved the pendulum perfectly in the angle and never asked for the multiplier would be built to hold a child’s weight and would fail at the bottom of the first full swing.

The sign is a physical condition #

The multiplier carries something the magnitude alone does not: a direction, and therefore a sign.

A string can pull and cannot push. A surface can push and cannot pull. A rigid rod can do both. Each of those is a statement about which sign of $\lambda$ the apparatus is capable of supplying, and the point at which the required sign flips is the point at which the constraint stops describing the world.

The hero figure is that condition drawn. A bead slides from rest at the top of a smooth sphere. The surface’s push is $N = mg(3\cos\theta - 2\cos\theta_0)$ — the same expression as the tension, because the geometry is the same and only the sign convention differs — and it decreases as the bead speeds up, because more and more of the available inward force is being spent on turning.

At $\cos\theta = \tfrac{2}{3}\cos\theta_0$ it reaches zero. For a bead released at the very top that is $48.19°$, and past it the sphere would have to pull the bead inward to keep it on the surface. It cannot, so the bead leaves, and everything after that is a projectile problem.

The equation written in the angle does not know this. It continues to produce a solution — the dashed continuation in the figure — in which the bead circles happily round the underside of a sphere it separated from a third of the way down. The solution is a correct solution of the wrong problem, and the only thing that could have flagged it is the sign of a quantity the angular formulation does not contain.

This is the site’s recurring hazard in an unusually clean form: not a wrong answer but a silent one. Every gate one might apply to the angular solution — energy conserved, constraint satisfied, equation obeyed — passes at every instant after the bead has left.

Where the constraint force becomes the whole story #

There is a case where the multiplier is not a supplement to the motion but the thing that decides what the motion is.

Take a bead threaded on a circular wire and spin the wire about a vertical diameter. In the rotating frame the bead has one coordinate, an angle from the bottom, and an effective potential built from gravity and the centrifugal term that the rotation introduces.

Below a critical spin rate that potential has a single minimum at the bottom. Above $\omega^2 = g/R$ the bottom becomes a maximum and a pair of minima opens at $\cos\theta = g/\omega^2 R$, so the bead climbs the wire and sits at an angle that grows with the spin. A single system with a single parameter turned smoothly produces a qualitative change in the number of equilibria, which is what a bifurcation is — the same shape of transition a parameter turned past a threshold produces elsewhere in mechanics — and it appears here in a laboratory object made of a wire and a bead. The effective potential it happens in is the same construction a radial coordinate always invites: fold the ignorable motion into the potential and read the equilibria off the shape.

The generalised coordinate handles all of that beautifully. What it does not report is that the wire is pushing on the bead with $m\omega^2 R$ once the bead has left the bottom — independent of the angle, a number that grows as the square of the spin, and the number that decides whether the apparatus can be built. The two branches of that force meet at the bifurcation, where the reaction is $mg$ from below and $m\omega^2 R = mg$ from above.

What the constraint costs the integration #

The two formulations differ in a way that has nothing to do with physics and everything to do with what a computer does with them.

The multiplier is determined by differentiating the constraint twice, so what the equations enforce exactly is $\ddot{g} = 0$ rather than $g = 0$. In continuous time those are equivalent given the right initial conditions. Once discretised they are not: a small error in $\ddot{g}$ integrates twice into a growing error in $g$, and the bob drifts slowly off its own circle.

The drift is the integrator’s rather than the formulation’s — quadruple the number of steps and it falls by two orders of magnitude, which is what a fourth-order method does — but it never becomes zero, and on a long run it has to be stabilised deliberately.

The angular formulation has no such line to draw. Its constraint is satisfied identically, at every step, at every step size, for ever, because there is no configuration in its vocabulary that violates it. That is the practical payment for what the coordinate choice bought, and it runs in both directions: the description that cannot express the constraint force also cannot get the constraint wrong.

The pattern, stated generally #

Everything above is one theorem wearing three costumes.

A constraint that can be written as $g(q) = 0$ restricts the motion to a surface. The force that maintains it must be perpendicular to that surface — because a component along the surface would do work on the constrained motion, and constraint forces of the ideal kind do not. Perpendicular to the surface means along $\nabla g$. So the constraint force is $\lambda \nabla g$ with exactly one unknown per constraint, and the constraint itself is exactly one condition per constraint. The count works out, and it works out for any number of constraints and any number of coordinates.

The counting is worth doing once. A system of $N$ particles in three dimensions has $3N$ coordinates; $k$ independent holonomic constraints reduce the free ones to $3N - k$. Written in generalised coordinates there are $3N - k$ equations and no constraint forces. Written in Cartesian coordinates with multipliers there are $3N$ equations of motion plus $k$ constraint equations, for $3N + k$ unknowns — the $3N$ coordinates and the $k$ multipliers — which is the same problem with $2k$ more of everything and $k$ physical quantities recovered. Neither is a trick; they are the two ways of spending the same information.

The direction of the constraint force is the part that is doing the real work, and it deserves stating without the calculus. Consider any displacement that keeps the constraint satisfied — a virtual displacement, in the eighteenth-century phrase, meaning one the apparatus permits rather than one the motion performs. An ideal constraint force does no work under such a displacement, by definition: a frictionless track pushes sideways and the bead moves along, a rigid rod pulls along itself and the ends move perpendicular to it. Since the permitted displacements are exactly those tangent to the constraint surface, a force doing no work on all of them must be perpendicular to all of them — and the perpendicular direction to the surface $g = 0$ is $\nabla g$. The multiplier is then the only thing left undetermined, and one constraint equation is exactly enough to determine it. Nothing about that argument mentions the particular constraint, which is why the method has no special cases.

That is why the method scales to things no free-body diagram survives. A robot arm with seven joints, a molecule with fixed bond lengths, a rigid body treated as a cloud of points whose separations never change — each is a large number of coordinates and a large number of constraints, and each is solved by writing one multiplier per constraint and letting the linear algebra find them all at once. Molecular dynamics does exactly this several billion times a second, and the multipliers are what tell it the tension in every chemical bond.

Choosing coordinates that satisfy the constraints is the alternative, and for a molecule with rings in it there may be no such choice that anybody can write down.

Where it stops #

The constraint has to be a function of the coordinates alone. Everything above assumes $g(q, t) = 0$ — a holonomic constraint, which reduces the number of degrees of freedom by one. A rolling wheel is not of this kind: its constraint relates velocities and cannot be integrated into a relation between positions, so the wheel keeps all its coordinates and merely restricts how it may move between them. Multipliers still work, and the reduction in degrees of freedom does not.

The assumption that constraint forces do no work is an assumption. It is what fixes the direction to be along $\nabla g$, and it is false for a surface with friction, where the reaction has a component along the motion by construction. Friction has to be put in by hand, as a force rather than as a constraint, and the elegance is lost exactly where the physics gets interesting.

And the multiplier is not always determined. Two constraints whose gradients are parallel give one condition for two unknowns, and the individual forces are then not fixed by the equations at all.

A table on three legs has its leg forces decided by statics alone; add a fourth and it does not. The constraints then outnumber what rigid-body balance can determine, and the answer depends on how the legs and the floor deform — so the problem stops being one of mechanics and becomes one of materials. That is where the whole approach stops: constraint forces are computable exactly when the constraints are independent, and a redundant one leaves an underdetermined system that no formulation repairs.

That is the table statics cannot settle, and it is the standing limitation of the whole apparatus. Multipliers give the constraint forces when the constraints are independent. When they are not — a table on four legs, a beam on three supports, a body clamped at both ends — the rigid idealisation has run out of information and the real answer depends on elasticity, which means abandoning the constraint picture and letting things bend.

The stationary condition picks out the trajectory and says nothing whatever about the forces holding it on its surface. That is the trade this essay is about, stated at its most extreme: the action principle gets the path right, in generalised coordinates, without ever computing a normal reaction — and if the normal reaction is the quantity wanted, it has to be got another way.

The history, which is the other way round #

Lagrange did not add multipliers to mechanics as an afterthought. The Mécanique analytique of 1788 develops the whole subject from the principle of virtual work — the statement that the constraint forces do no work under any displacement consistent with the constraints — and the multipliers appear as the natural bookkeeping for that principle rather than as a repair to a method that had lost something.

Read in that order the priorities invert. What is fundamental is the geometry of the allowed motions; the forces that hold a system in that geometry are a derived quantity, computed when they are wanted and ignored when they are not. Newton’s formulation puts the forces first and derives the geometry, which is why a free-body diagram is the more natural place to begin and the harder place to continue.

The Newtonian starting point draws every force, including the normal reaction, and hides nothing. That is both the strength of the picture and the reason it does not scale: for a bead on a wire it is clearer than any Lagrangian, and for a robot arm with six joints it is unusable. The history runs the other way round from the teaching — the general method came later and was built to avoid exactly the bookkeeping the elementary picture makes so vivid.

Both formulations are complete, and the choice between them is a choice about what one wants to be told without asking. The Lagrangian is silent about constraint forces and voluble about symmetries; a symmetry of the Lagrangian is a conserved quantity, read off by inspection, which the force picture has no comparable way of seeing.

What the Lagrangian formulation gives away for free is a conserved quantity read off a symmetry rather than found by integrating. That is the other half of the trade: constraint forces are lost and conservation laws arrive without work, and which of those matters more is a question about the problem rather than about the formulation.

What the pictures cannot show #

Nothing here is a measurement. Every number in these figures comes out of two assumptions — that the string is inextensible and that the surface is smooth — and both are approximations to objects that stretch and rub.

A real string has a stiffness, so a real pendulum has a second degree of freedom at a frequency far above the swinging one, and the tension oscillates slightly about the value drawn. A real surface has friction, so a real bead leaves a sphere slightly later than 48.19° because it is going slightly slower when it gets there. The constraint picture is the limit in which the stiff degree of freedom is infinitely stiff and infinitely fast, and like every such limit it is excellent until something asks a question about the thing that was removed.

The constraint has already been used by the time a one-dimensional potential can be drawn. Reducing a two-dimensional problem to a single radial coordinate is what makes the picture drawable at all — the constraint force has been eliminated rather than computed, and the effective potential that appears contains its effect without naming it. Every such picture in this collection has that elimination behind it.

The honest summary is that the choice of coordinates is a choice about which questions are easy. The angle makes the motion easy and the tension invisible. The Cartesian pair makes the tension explicit and the motion harder. Neither is more correct, and a method that reports only one of the two while looking complete is the one to watch, because it will keep looking complete when the string breaks.

The ladder from here #

Later rungs on this anchor: the Hamiltonian formulation, where the constraint forces reappear in a different guise as restrictions on the phase space itself; non-holonomic constraints treated properly, with the Chaplygin sleigh and the rolling coin as the worked examples; the Gauss principle of least constraint, which characterises the true motion as the one minimising a weighted departure from the unconstrained one; and constrained numerical integration in earnest, where the drift above is removed by projection or by a formulation that conserves the constraint by construction.

The neighbouring ladders are least action, and what it does not minimise, the conservation law a symmetry hands over, and the pendulum’s exact period, where the same swing is asked a different question and answered with an elliptic integral.