Generalised Coordinates and Constraints

A pendulum bob swings on the end of a rigid rod. Where is it? You could reach for the map you already know and write down its Cartesian coordinates (x, y) — but that is two numbers, and they are a lie, because they are not free to be anything they like. The rod pins them together: x^2 + y^2 = \ell^2 at every instant. The bob really has just one way to move — around the arc — and the honest way to say where it is needs just one number: the swing angle \theta.

This is the first big idea of analytical mechanics, and it is quietly radical: stop describing a system by the positions of its pieces in space, and describe it instead by the smallest set of numbers that can move freely. Those numbers are the generalised coordinates q_1, q_2, \dots, q_n. They need not be lengths; they can be angles, arc-lengths, anything convenient. Choosing them well is half the art of solving a mechanics problem, and it is what lets Lagrange's method sail past the tangle of forces that Newton has to fight through.

Configuration space: one point that carries the whole system

Here is the mental shift. Instead of tracking a cloud of particles moving around 3-D space, imagine a single abstract point living in a space of its own — the configuration space. Each axis of that space is one generalised coordinate. A single point in it fixes the position of every particle in the system at once. As the system evolves, that one point traces a curve, and that curve is the motion.

A bead sliding on a straight wire needs one number (how far along) — its configuration space is a line.
Our pendulum needs one angle — its configuration space is a circle.
A double pendulum needs two angles — its configuration space is a torus (a doughnut: angle-around times angle-around).
A single free particle drifting in space needs three numbers — its configuration space is ordinary 3-D space itself.

The dimension of the configuration space — the number of coordinates you genuinely need — is the system's number of degrees of freedom. It is the single most important number to pin down before you write a line of physics.

Watch the constraint at work

Drag the angle slider. The bob is free to move — but only along the dashed circle of radius \ell, never off it. Two Cartesian numbers (x, y) are being spent to describe something that only ever needs one: the angle \theta. The circle is the configuration space, and \theta is the natural coordinate that runs around it.

The bob's Cartesian position, in terms of the one real coordinate, is x = \ell\sin\theta, y = -\ell\cos\theta. Feeding those into the kinetic and potential energies will, on the next few pages, hand us the equation of motion with none of the rod's tension force ever appearing. That is the payoff of counting coordinates properly.

Constraints: the rules that cut coordinates away

A constraint is any rule that restricts where the parts of a system may go. The rigid rod, a bead threaded on a wire, a ball rolling inside a bowl, two masses joined by an inextensible string — each is a constraint. Constraints are what make the honest coordinate count smaller than the naive one.

The friendly, well-behaved kind are called holonomic: a constraint is holonomic if it can be written as an equation among the coordinates (and possibly time) of the form

f(x_1, y_1, z_1, \dots, x_N, y_N, z_N, t) = 0.

The rod is holonomic: x^2 + y^2 - \ell^2 = 0. A bead on a wire shaped like y = g(x) is holonomic: y - g(x) = 0. Each such equation is one relation you can solve, and every relation you solve removes one coordinate from the tally.

The rule. For N particles in d-dimensional space subject to c independent holonomic constraints, the number of degrees of freedom is n = dN - c.
Coordinates match freedoms. With holonomic constraints you can always choose exactly n generalised coordinates q_1, \dots, q_n that are free to vary independently — no leftover relations to remember.

So the plane pendulum: one particle in d = 2 dimensions (N = 1), one constraint (c = 1), giving n = 2\cdot 1 - 1 = 1. The double pendulum: two particles in the plane (2\cdot 2 = 4) with two rod constraints, so n = 4 - 2 = 2 — the two swing angles.

Worked examples: count before you calculate

Example 1 — a free particle. One particle, no constraints, in ordinary space: d = 3, N = 1, c = 0, so n = 3. Three coordinates, and every one of them free. Good old (x, y, z) is already a perfectly efficient choice.

Example 2 — a bead on a fixed wire. A bead threaded on a bent wire in the plane is one particle (N=1, d=2) with one constraint (it must lie on the wire), so n = 2 - 1 = 1. The natural coordinate is the arc-length s measured along the wire — a single number that slides the bead wherever the wire goes.

Example 3 — a rigid body. A brick tumbling freely has, no matter how many atoms it contains, exactly n = 6 degrees of freedom: three to say where its centre is, and three angles to say how it is oriented. Rigidity (every inter-atom distance fixed) is a colossal pile of constraints, and it collapses 3N down to a tidy six.

Example 4 — two masses on a taut string over a pulley. Atwood's machine: two particles that would have two vertical coordinates, but the inextensible string ties them (y_1 + y_2 = \text{const}), one constraint, so n = 2 - 1 = 1. When one mass drops by h the other rises by h — a single number tracks them both.

No — and this is the classic trap. The clean rule n = dN - c only works for holonomic constraints, ones you can write as an equation purely among the coordinates. Some constraints refuse to be written that way. A coin rolling without slipping on a table is the famous example: the rolling condition links the coin's velocity to its orientation, and that relationship cannot be integrated into a plain equation among the positions alone. Such constraints are called non-holonomic.

The tell-tale sign: a rolling coin can be manoeuvred to any position on the table with any final orientation — so no equation forbids any configuration, yet at each instant its motion is genuinely restricted. It reduces the number of allowed velocities without reducing the number of reachable positions. When you meet a non-holonomic constraint, do not just subtract it from the coordinate count — that shortcut silently gives the wrong answer.

Because Newton makes you draw every force — including the ones you do not care about. To do the pendulum by Newton you must introduce the rod's tension, resolve it along and across the motion, and then eliminate it, even though tension does no work and never appears in the final answer. The generalised-coordinate approach sidesteps the whole thing: pick \theta, and because \theta already respects the constraint, the constraint force never has to be named. A well-chosen coordinate bakes the geometry in for free. Multiply that saving across a machine with dozens of rods and joints, and you see why the eighteenth century fell in love with the method.