Nonholonomic Systems and the Lagrange-d'Alembert Method

Nonholonomic mechanics is quietly one of the most industrial corners of analytical dynamics. Every autonomous vehicle that parks itself is solving a nonholonomic motion-planning problem: the "no sideways velocity" Pfaffian constraint of each axle defines exactly the distribution of allowed directions, and the parking maneuver is a Lie-bracket excursion — allowed moves cycled to produce forbidden displacement, the mathematics formalized from the observation in this article's second figure. Planners in 2026 driving stacks still bottom out in curvature-bounded nonholonomic path families (Reeds–Shepp curves and their optimized descendants), while the vehicle-dynamics layer underneath solves Lagrange–d'Alembert-type equations and monitors the multipliers: tire-force estimates checked against the friction ellipse, the grown-up version of this article's $|\lambda|$ versus $\mu N$ test, deciding in real time whether "rolling" is still a valid constraint or the car has begun to slide. The other living branch is locomotion theory. A falling cat, a snake robot, a micro-swimmer, an astronaut reorienting in free fall — all exploit nonintegrability: cyclic shape changes that produce net rotation or translation, governed by the same geometry (connections on principal bundles) that grew out of the vakonomic/d'Alembert distinction. And the vakonomic branch itself thrives under its true name, optimal control: the shortest admissible parking path is a sub-Riemannian geodesic, computed by exactly the augmented-action principle that fails for rolling dynamics — the twin machines of this article, each finally employed on the problem it actually solves.

Picture an ice skate on a frozen pond. At every instant the blade can move only along its own length — sideways velocity is forbidden. That is one velocity constraint, one equation. Yet by alternately gliding and turning, a skater reaches any point on the ice with any heading: the three-dimensional configuration space (position plus orientation) is fully accessible. The constraint lives in velocity space, not in configuration space. That mismatch — fewer velocity directions than accessible configuration dimensions — is the essence of nonholonomy.

Why does this defeat Hamilton's principle? Because that principle compares nearby paths, all of which must satisfy the constraint. For a holonomic constraint that is easy: the paths live on a lower-dimensional surface, and you vary freely within it. But a nonintegrable velocity constraint defines admissible directions pointwise, and the family of constraint-respecting neighbor paths sharing both endpoints is too rigid — in general the variations you need do not exist. d'Alembert's principle sidesteps the whole difficulty: it never compares paths. It asks one question per instant — do the applied-plus-inertial forces do zero work against a virtual displacement taken at frozen time? — and constraining those frozen-time probes is always possible. The best intuition for nonintegrability is the parking maneuver. Drive forward with wheels turned, back up with wheels straight, and repeat: each individual motion respects "no sideways slip", yet the cycle produces a net sideways displacement. The forbidden direction is manufactured out of allowed ones. Whenever cycling through allowed directions generates a new direction, no coordinate function can be constant along the motion — the constraint cannot be integrated into $f(q)=\text{const}$, and the system is genuinely nonholonomic. (In control theory this is exactly the Lie-bracket test; mechanics knows it as the Frobenius condition.)

One more piece of intuition, because it pays rent later: the Lagrange multiplier $\lambda_l$ in this formalism is not bookkeeping — it is the reaction. For a rolling wheel, $\sum_l \lambda_l a_{lk}$ is the static friction at the contact enforcing "no slip". Solving for $\lambda_l$ tells you the force the contact must supply, and comparing it against the available budget $\mu N$ tells you whether the wheel actually rolls or breaks loose. A formalism born of eighteenth-century rationalism turns out to be a slip detector.

Start with the constraint inventory from the classification article. The velocity constraints we can handle systematically are the Pfaffian (linear-in-velocity) ones. There are $m$ of them, on $n$ generalized coordinates:

where: $q_k$ — generalized coordinates, $k=1,\dots,n$; $a_{lk}$ — constraint coefficients (functions of $q,t$); $a_{l0}$ — the inhomogeneous term, present when the constraint itself moves (rheonomic case). Read each row as a plane in velocity space: at every configuration, the allowed velocities lie in an $(n-m)$-dimensional slice. If the whole row happens to be the differential of some function $f_l(q,t)$, the constraint silently integrates to $f_l=\text{const}$ and is holonomic in disguise; the interesting case is when no such $f_l$ exists. The virtual displacements of d'Alembert's machinery are frozen-time probes, and freezing time changes the constraint they must obey. Multiply (1) by $\mathrm{d}t$ to get $\sum_k a_{lk}\,\mathrm{d}q_k + a_{l0}\,\mathrm{d}t = 0$; a virtual displacement replaces $\mathrm{d}q_k \to \delta q_k$ while $\mathrm{d}t \to 0$, since time is held fixed during the variation. The inhomogeneous term dies, leaving

where: $\delta q_k$ — virtual displacements at fixed time, $\delta t = 0$. This little asymmetry between (1) and (2) is physical, not pedantic: on a moving constraint the real motion has a compulsory component (the $a_{l0}$ part) along which reactions do real work, while the virtual probes see only the instantaneous, frozen geometry — the same distinction that let d'Alembert's principle survive moving constraints unchanged. Now transplant the whole d'Alembert argument into generalized coordinates. Using $\delta\mathbf r_i=\sum_k(\partial\mathbf r_i/\partial q_k)\,\delta q_k$ and the standard kinetic-energy identity, the principle $\sum_i(\mathbf F_i^{\text{app}}-\dot{\mathbf p}_i)\cdot\delta\mathbf r_i=0$ becomes its Lagrangian form:

where: $L = T - V$ — the Lagrangian of the unconstrained problem, written in all $n$ coordinates. For a holonomic system the $\delta q_k$ would be independent, each bracket would vanish separately, and we would be done — that was the punchline of the Euler–Lagrange article. Here the probes are chained together by (2), and the correct extraction of equations of motion needs one more idea: multipliers. The result (derived below) is

where: $\lambda_l(t)$ — one Lagrange multiplier per constraint, an unknown function of time determined alongside the motion. The right-hand side is exactly the generalized force the constraints exert:

where: $Q_k^{(c)}$ — the reaction conjugate to $q_k$. Note the geometry: each constraint's force points along its own coefficient row $a_{lk}$, i.e. perpendicular (in the virtual-work sense) to every allowed displacement — the workless-reaction postulate made algebraic. Counting closes the system:

where: nothing was eliminated — unlike the holonomic strategy of absorbing constraints into fewer coordinates, the nonholonomic strategy keeps all $n$ coordinates and pays with $m$ extra unknowns. The payment is refunded with interest: the $\lambda_l$ are the contact forces you usually want anyway. Finally, energy. Define the energy (Jacobi) function and differentiate along the motion:

where: the reaction-power term evaluates, using (5) and then (1), to $\sum_k Q_k^{(c)}\dot q_k=\sum_l\lambda_l\sum_k a_{lk}\dot q_k=-\sum_l\lambda_l\,a_{l0}$. So for scleronomic constraints ($a_{l0}=0$) with a time-independent Lagrangian, $E$ is exactly conserved — a conservation law delivered with no variational principle in sight. The rolling disk below conserves energy; a ball rolling inside a spinning drum ($a_{l0}\neq 0$) does not. The full story of $E$ lives in the Jacobi-integral article.

Which statement answers which question

The cleanest laboratory for the machinery is a thin disk of mass $M$ and radius $R$ rolling upright on a horizontal plane. Four coordinates describe it: the contact point $(x,y)$, the heading angle $\phi$ (the direction the disk is rolling toward), and the spin angle $\theta$ about its own axle. Rolling without slipping welds the contact velocity to the spin:

where: the factor $R\dot\theta$ is the rim speed and $(\cos\phi,\sin\phi)$ the rolling direction. Two Pfaffian constraints on four velocities: at each instant only two velocity directions survive (roll, and turn the heading), yet — Frobenius will confirm — all four configuration dimensions remain reachable. A coin can end up anywhere on the table, facing any way, with any mark on its rim touching the ground: exactly the parking phenomenon.

Here is the trap this article exists to spring before it catches you. For a holonomic constraint $f(q,t)=0$ you learned that adjoining $\lambda f$ to the Lagrangian and extremizing works perfectly. It is deeply tempting to do the same with a velocity constraint: write $L + \lambda(-\dot x\sin\phi + \dot y\cos\phi)$ and turn the Euler–Lagrange crank. The result — called vakonomic ("variational axiomatic kind") dynamics — is a perfectly consistent mathematical theory. It is also, for rolling and sliding mechanical contact, experimentally false.

1 · Disk rolling down an incline — multipliers as a slip detector (confidence builder)

A uniform solid disk, $M = 2.0\ \mathrm{kg}$, $R = 0.30\ \mathrm{m}$, rolls without slipping straight down a plane inclined at $\alpha = 20°$. The static friction coefficient is $\mu_s = 0.25$. Find the acceleration, find the friction force via the multiplier, and decide whether rolling actually holds. (This one-dimensional rolling constraint is secretly integrable — $x = R\theta + \text{const}$ — but the multiplier machinery neither knows nor cares, and the warm-up shows every gear turning.)

Step 1. Coordinates and constraint. Let $x$ run down the slope and $\theta$ be the roll angle; rolling means $\dot x - R\dot\theta = 0$, a single Pfaffian row $a = (1, -R)$ in coordinates $(x,\theta)$.

Step 2. Lagrangian, keeping both coordinates: $L = \tfrac12 M\dot x^2 + \tfrac12\big(\tfrac12 MR^2\big)\dot\theta^2 + Mgx\sin\alpha$. The multiplier equations (4):

Step 3. Close with the constraint, $\ddot x = R\ddot\theta$. The second equation gives $\lambda = -\tfrac12 M\ddot x$; substituting into the first, $M\ddot x - Mg\sin\alpha = -\tfrac12 M\ddot x$, so

Step 4. The reaction. $\lambda = -\tfrac12 M \ddot x = -\tfrac13 Mg\sin\alpha = -\tfrac13 \times 2.0 \times 9.81 \times 0.342 = -2.24\ \mathrm{N}$: the constraint force on the $x$ coordinate is 2.24 N up the slope — static friction, as it must be to retard sliding and spin up the disk.

Step 5. The slip check the formalism hands you for free. Available budget: $\mu_s N = \mu_s Mg\cos\alpha = 0.25 \times 2.0 \times 9.81 \times \cos 20° = 4.61\ \mathrm{N}$. Required: $2.24\ \mathrm{N} < 4.61\ \mathrm{N}$ — the disk rolls. The general threshold is $\tan\alpha \le 3\mu_s$: this disk would break loose above $\alpha = \arctan(0.75) = 36.9°$.

2 · The turning skater — a genuinely nonholonomic orbit (exam level)

A skater of mass $m = 70\ \mathrm{kg}$ glides on the blade constraint of Derivation C with along-blade speed $u_0 = 2.0\ \mathrm{m/s}$, holding a steady heading rate $\dot\phi = \Omega = 0.5\ \mathrm{rad/s}$ (the free equation $I\ddot\phi = 0$ preserves whatever heading rate the push established). Find the trajectory, the constraint force, and check it against the ice's grip, $\mu_s = 0.15$.

Step 1. Kinematics from the constraint. No side-slip means the velocity lies along the blade: $\dot x = u\cos\phi$, $\dot y = u\sin\phi$, with $\phi = \Omega t$.

Step 2. The speed is constant — and the formalism says so before any integration. Project the equations of motion $m\ddot x = -\lambda\sin\phi$, $m\ddot y = \lambda\cos\phi$ along the blade: $m\dot u = m(\ddot x\cos\phi + \ddot y\sin\phi) = -\lambda\sin\phi\cos\phi + \lambda\cos\phi\sin\phi = 0$. So $u = u_0$. (This is equation (7) at work: scleronomic constraint, time-independent $L$, energy conserved, and here energy is pure kinetic.)

Step 3. The trajectory. $\dot x = u_0\cos\Omega t$, $\dot y = u_0\sin\Omega t$ integrate to a circle of radius

traversed once every $2\pi/\Omega = 12.6\ \mathrm{s}$.

Step 4. The constraint force. Project across the blade: $m\,u_0\Omega = \lambda$, so $\lambda = 70 \times 2.0 \times 0.5 = \boxed{70\ \mathrm{N}}$, directed along $(-\sin\phi,\cos\phi)$ — the blade's sideways normal. Sanity check: a circle of radius $4.0\ \mathrm{m}$ at speed $2.0\ \mathrm{m/s}$ demands centripetal force $mu_0^2/\rho = 70\times4.0/4.0 = 70\ \mathrm{N}$ ✓ — the "abstract" multiplier is the ice's lateral push on the blade, nothing more mysterious.

Step 5. Grip check. The ice can supply up to $\mu_s m g = 0.15 \times 70 \times 9.81 = 103\ \mathrm{N}$ laterally before the blade skids: $70\ \mathrm{N} < 103\ \mathrm{N}$, so the carve holds. Tighten the turn to $\Omega = 0.8\ \mathrm{rad/s}$ at the same speed and the demand rises to $112\ \mathrm{N} > 103\ \mathrm{N}$ — the blade washes out, and every skater has felt precisely this multiplier exceed precisely this budget.

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §1.3–1.4, §2.4 (constraints and Lagrange multipliers) · Landau & Lifshitz, Mechanics, 3rd ed., §1–2 (variational basis) · Morin, Introduction to Classical Mechanics, Ch. 6 (Lagrangian method, constraints) · Taylor, Classical Mechanics, Ch. 7 (Lagrangian mechanics with constraints) · Bloch, Nonholonomic Mechanics and Control, Ch. 1–5

Provenance. Constraint forms, multiplier equations, and counting recomputed against Goldstein, Poole & Safko (3rd ed.) §1.3–1.4 and §2.4; energy result and the d'Alembert derivation cross-checked with Taylor Ch. 7 and Landau–Lifshitz Mechanics §1–2. The vakonomic-vs-Lagrange–d'Alembert distinction and rolling-disk/skate examples follow Bloch, Nonholonomic Mechanics and Control, Ch. 1–5. Inertia values ($I_3=\tfrac12 MR^2$ for a thin disk about its symmetry axis) are standard textbook results. Verifier should recompute: (1) the nonintegrability (Frobenius) test for the disk and skate constraints, (2) the multiplier equations from the boxed d'Alembert derivation, (3) the energy-rate identity for scleronomic constraints. Worked-example numbers on this page ($\ddot x = 2.24\ \mathrm{m/s^2}$, $|\lambda| = 2.24\ \mathrm{N}$ vs $\mu_s N = 4.61\ \mathrm{N}$, slip threshold $36.9°$; skater $\rho = 4.0\ \mathrm{m}$, $\lambda = 70\ \mathrm{N}$ vs $103\ \mathrm{N}$) are illustrative arithmetic recomputed from the boxed formulas.

Builds on: Constraints and Their Classification · d'Alembert's Principle · Virtual Displacements and the Principle of Virtual Work · The Euler-Lagrange Equations · Generalized Coordinates and Configuration Space Leads to: Generalized Forces and Generalized Momenta · Cyclic Coordinates and Conservation of Conjugate Momenta · The Energy Function and Jacobi Integral · Degrees of Freedom