Constrained motion on surfaces — constraints and Lagrange multipliers

A bead lives on the hoop and nowhere else — how a constraint trims the degrees of freedom, and how a Lagrange multiplier reveals the constraint force itself.

Opening

When we tidied up Newtonian mechanics in chapter 1, we assumed a particle drifts freely through three-dimensional space. Most real problems are not like that. A bead is stuck on a wire hoop, a pendulum bob hangs at the end of a string of fixed length, and a coin rolls along a floor without slipping. In this chapter we split such constraints into two flavours and then spend most of our time on the more common kind — holonomic constraints — using motion on the sphere $S^2$ as a working picture. Along the way we get a preview of the Lagrange multiplier, which chapter 7 will treat formally, and we will see that it is not merely a bookkeeping trick: it measures the constraint force itself.

Main 1 — Two flavours of constraint

Constraints fall into two formal families.

A holonomic constraint is an equation that involves only the coordinates (and possibly time),

f(q, t) = 0,

where $q = (q^1, \dots, q^n)$ is the bundle of generalized coordinates. One such equation removes one dimension from the configuration space — the set of allowed positions. A bead on a planar circular hoop of radius $R$ , for example, satisfies $f(x, y) = x^2 + y^2 - R^2 = 0$ , which collapses the plane down to a one-dimensional curve. Likewise a point on the unit sphere $S^2$ satisfies $x^2 + y^2 + z^2 = 1$ , sending three dimensions down to a two-dimensional surface.

A nonholonomic constraint is instead a linear condition on the velocities,

\sum_i a_i(q)\, \dot q^i = 0,

that cannot be integrated into a relation among the coordinates alone. There is no $f(q) = 0$ that captures it. The textbook example is a coin rolling without slipping on a plane: the coin can reach any point of the plane (the configuration space is not trimmed), but the direction of its velocity at each instant is locked to the way it is rolling. We will treat only holonomic constraints in this chapter. Most constraints you meet in a mechanics textbook are of that kind, and nonholonomic systems read more cleanly as a separate section after chapter 7.

Main 2 — Bead on a hoop, from start to finish

A bead of mass $m$ slides on a vertical circular hoop of radius $R$ under gravity of magnitude $g$ . Pick the angle measured from the bottom of the hoop, $\theta$ (theta), as the single coordinate. Then

x = R\sin\theta, \qquad y = R(1 - \cos\theta),

and the kinetic and potential energies fall out immediately:

T = \tfrac{1}{2} m R^2 \dot\theta^2, \qquad V = m g R (1 - \cos\theta).

Chapter 7 introduces the Lagrangian $L = T - V$ properly; here we borrow the result. The constant $mgR$ does not affect the equation of motion, so dropping it leaves

L = \tfrac{1}{2} m R^2 \dot\theta^2 + m g R \cos\theta.

Substituting this into the Euler–Lagrange equation

\frac{d}{dt}\!\left( \frac{\partial L}{\partial \dot\theta} \right) - \frac{\partial L}{\partial \theta} = 0

gives $m R^2 \ddot\theta + m g R \sin\theta = 0$ , that is,

\ddot\theta + \frac{g}{R} \sin\theta = 0.

This is just the pendulum equation. What began as a three-DOF problem in $(x, y, z)$ with two constraint equations attached has, by a single well-chosen generalized coordinate, become a one-DOF problem with no constraints in sight. This is the scene that first hooks students on the Lagrangian formalism — the constraints are solved algebraically and made to vanish.

Main 3 — What a Lagrange multiplier actually points to

Switching to a generalized coordinate makes the constraint force disappear from the equations. But sometimes you want exactly that force — for example, the normal force the hoop exerts on the bead, so you can ask whether the hoop will hold.

In that case you keep the redundant coordinates $(x, y)$ and impose the constraint $f(x, y) = x^2 + y^2 - R^2 = 0$ separately. You add a single term to the equation of motion,

m \ddot{\mathbf{r}} = -\nabla V + \lambda\, \nabla f,

where $\lambda$ (lambda) is the Lagrange multiplier. Since $\nabla f$ is the vector normal to the constraint surface, the added term $\lambda \nabla f$ points along the surface normal — exactly the direction of the normal force we were trying to extract. Its magnitude, signed, is precisely $\lambda$ .

In other words, the multiplier is not just a mathematical aide brought in to enforce $f = 0$ ; it is the amplitude of the constraint force itself, a physical quantity in its own right. The formal derivation lives in chapter 7, Variational principles and constraints. For now keep one line in mind: a Lagrange multiplier is the size of the constraint force.

In Python

# Bead on a hoop: integrate theta_ddot = -(g/R) sin(theta) with hand-rolled RK4.
import numpy as np
import matplotlib.pyplot as plt

g, R = 9.81, 0.2
dt, T_end = 1e-3, 4.0
N = int(T_end / dt)

def f(state):
    th, om = state
    return np.array([om, -(g / R) * np.sin(th)])

state = np.array([np.pi / 3, 0.0])   # theta0 = 60 deg, released from rest
ts = np.linspace(0, T_end, N + 1)
ths = np.empty(N + 1)
oms = np.empty(N + 1)
ths[0], oms[0] = state

for i in range(N):
    k1 = f(state)
    k2 = f(state + 0.5 * dt * k1)
    k3 = f(state + 0.5 * dt * k2)
    k4 = f(state + dt * k3)
    state = state + (dt / 6) * (k1 + 2*k2 + 2*k3 + k4)
    ths[i + 1], oms[i + 1] = state

fig, ax = plt.subplots(2, 1, sharex=True)
ax[0].plot(ts, ths); ax[0].set_ylabel(r'$\theta$ [rad]')
ax[1].plot(ts, oms); ax[1].set_ylabel(r'$\dot\theta$ [rad/s]')
ax[1].set_xlabel('t [s]')
plt.tight_layout(); plt.show()

The trajectory closes one period and returns to the same amplitude. For a small initial angle the curve would look almost sinusoidal, but at $\theta_0 = \pi/3$ you can see a faint anharmonicity — the peaks flatten slightly. That is the nonlinearity of $\sin\theta$ making itself felt.

To the next chapter

Chapter 3: tensors and the covariant derivative gives the surface we have been drawing — a smooth space of reduced dimension — its proper name (manifold) and then asks how to differentiate a vector on it in a way that does not depend on the choice of coordinates. The fact that a Lagrangian gives the same equations of motion under any change of coordinates will turn out to be no accident.