A nonholonomic system in classical mechanics is a physical system with some constraints that are impossible to be cast into the form of holonomic constraints. That is, a nonholonomic system is a system that is not a holonomic system. Intuitively stated, they are mechanical systems with constraints on their velocity that are not derivable from position constraints. They are contrasted with classical Lagrangian and Hamiltonian systems, in which there are only constraints on position.

A system is a holonomic system if and only if all its constraints are Pfaffian and integrable. A system is a nonholonomic system, if and only if all its constraints are Pfaffian, but some are not integrable. A system with non-Pfaffian constraint does not have a standard name.

In general, consider a system whose state is fully specified by ${\displaystyle q=(q_{1},\dots ,q_{n})}$. Its evolution over time is constrained, in that only certain velocities are allowed, and others are disallowed.

Consider an upright wheel on a plane. Let ${\displaystyle \theta }$ is the steering angle relative to the ${\displaystyle x}$-axis, and ${\displaystyle x}$ and ${\displaystyle y}$ be the location where the wheel touches the plane. Since the wheel can only move in the direction it is pointing towards, we obtain the constraint ${\displaystyle {\dot {x}}\sin \theta -{\dot {y}}\cos \theta =0}$. The constraint is rewritten into a 1-form as ${\displaystyle \omega :=dx\sin \theta -dy\cos \theta }$.

In general, a constraint that can be written as a 1-form in the ${\displaystyle (n+1)}$-dimensional space of ${\displaystyle (t,q_{1},\dots ,q_{n})}$ is called a Pfaffian constraint. Otherwise it is a non-Pfaffian constraint.

Nonholonomic Pfaffian constraints are given by nonintegrable distributions; i.e., taking the Lie bracket of two vector fields in such a distribution may give rise to a vector field not contained in this distribution.

Geometrically, a system of integrable Pfaffian constraints is integrable: one can foliate the whole configuration space into submanifolds of maximal dimension, such that a trajectory satisfies all constraints if and only if the trajectory stays within a submanifold. See integrability conditions for differential systems for how to decide whether a system of Pfaffian constraints.

In the special case of upright wheel on a plane, the single constraint is not integrable, because it is a contact form:$${\displaystyle \omega \wedge d\omega =-\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} \theta \neq 0}$$For a system with holonomic constraints, its dynamics is restricted to a submanifold of the full configuration space. Therefore, we can make a coordinate chart only on an individual submanifold, which then allows us to eliminate the constraints, since any trajectory within an individual submanifold automatically satisfy all constraints. The coordinate chart on that is called generalized coordinates and is the foundation of Lagrangian mechanics. Nonholonomic constraints cannot be eliminated by using generalized coordinates.