Nonholonomic constraints and Lagrange multipliers
A constraint that can be written as a relation between coordinates alone (holonomic) can simply be used to eliminate a coordinate before forming the Lagrangian, but a constraint on velocities that cannot be integrated into such a relation (nonholonomic, like rolling without slipping in general) cannot — it must instead be kept explicit and enforced with a Lagrange multiplier, which turns out to equal the constraint force itself.
Practise this conceptMost constraints you meet early in Lagrangian mechanics are the easy kind: a bead stuck on a wire, a pendulum bob at a fixed distance from its pivot. These can be written as a plain relation between coordinates — the bead's height is a function of how far along the wire it's travelled — so you can just substitute that relation in and end up with one fewer coordinate to solve for. This kind of constraint is called holonomic, and it disappears cleanly.
A rolling wheel is different. Rolling without slipping says the contact point's velocity is zero, which relates the wheel's linear velocity to its angular velocity. That's a relation between velocities, and in general it cannot be integrated back into a single relation between the position coordinates alone — there's no function of position whose derivative gives you that condition for every possible path the wheel might take. A constraint like this is nonholonomic, and it can't be used to quietly eliminate a coordinate.
The fix is to keep the constraint explicit and add it to the problem with an extra unknown: a Lagrange multiplier, one per constraint. You end up with the ordinary Euler–Lagrange equations plus a multiplier term, plus the constraint equation itself, and you solve them all together. The reward for this extra bookkeeping is not just correctness — the multiplier you solve for turns out to equal the actual constraint force (the friction force enforcing the rolling condition, say) up to a geometric factor you can read off directly. Ordinary unconstrained Lagrangian mechanics throws constraint forces away as irrelevant; this method hands them back.
The traps are procedural. Plugging the constraint in early, the way you would for a holonomic one, gives equations of motion that are simply wrong for a nonholonomic constraint — the order of operations (differentiate first, constrain after, via the multiplier) is not optional here the way it is for the holonomic case.
Key ideas
Requires: Lagrangian mechanics
Common misconceptions
- Believing any constraint involving velocities is automatically nonholonomic. Many velocity constraints are secretly holonomic — they're the time-derivative of a coordinate relation in disguise and can still be integrated back into one, eliminating a coordinate the ordinary way. A constraint only earns the name nonholonomic when no such integration exists.
- Treating the Lagrange multiplier as a bookkeeping trick with no physical content, rather than recognising that it equals the actual constraint force (e.g. the normal force, or the friction force enforcing rolling without slipping) up to a known geometric factor — which is exactly why the method lets you solve for constraint forces that plain unconstrained Lagrangian mechanics discards.
- Believing you can just plug the constraint equation into the Lagrangian before differentiating (eliminate the coordinate first, then apply the Euler–Lagrange equation as usual) the same way you would for a holonomic constraint. For a nonholonomic one this gives the wrong equations of motion — the constraint has to be added with its own multiplier term and enforced afterward, alongside the unconstrained Euler–Lagrange equations, not substituted in beforehand.
Resources
Part of these UT courses
This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.
- Analytical mechanicsLTFY.04.016