Math 2.0

Mechanics of particles and systems Definition

Nonholonomic system

Builds on Holonomic constraintGeneralized coordinatesVelocityFrobenius theorem

Definition

Consider a mechanical system with generalized coordinates q=(q1,…,qn)q = (q_1, \dots, q_n) in an open set V⊆ℝnV \subseteq \R^n and time tt. Let ajia_{ji} and bjb_j be smooth functions of (q,t)(q, t), for 1≤j≤m1 \le j \le m and 1≤i≤n1 \le i \le n, such that the m×nm \times n matrix (aji)(a_{ji}) has rank mm everywhere. Suppose the motion obeys the mm constraints, linear in the generalized velocities, ∑i=1naji(q,t) q˙i+bj(q,t)=0,j=1,…,m.\sum_{i=1}^n a_{ji}(q, t)\, \dot q_i + b_j(q, t) = 0, \qquad j = 1, \dots, m. The system is nonholonomic if this family is not equivalent to a family of holonomic constraints. This means: there are no smooth functions f1(q,t),…,fm(q,t)f_1(q, t), \dots, f_m(q, t) such that, near each point, a motion obeys the constraints exactly when each fj(q(t),t)f_j(q(t), t) is constant.

Remarks

Example: a skate on the plane has coordinates (x,y,θ)(x, y, \theta), and the blade forbids sideways motion: x˙sin⁡θ−y˙cos⁡θ=0\dot x \sin\theta - \dot y \cos\theta = 0. The allowed velocities at each point are spanned by X=(cos⁡θ,sin⁡θ,0)X = (\cos\theta, \sin\theta, 0) and Y=(0,0,1)Y = (0, 0, 1), and their Lie bracket [Y,X]=(−sin⁡θ,cos⁡θ,0)[Y, X] = (-\sin\theta, \cos\theta, 0) is not in that span. By the Frobenius theorem the constraint is not integrable, so the skate is nonholonomic. Non-example: the constraint x˙cos⁡y−xsin⁡y y˙=0\dot x \cos y - x \sin y\, \dot y = 0 on the plane is ddt(xcos⁡y)=0\frac{d}{dt}(x \cos y) = 0, equivalent to the holonomic constraint xcos⁡y=cx \cos y = c. For scleronomous constraints with bj=0b_j = 0, nonholonomic means that the distribution of allowed velocities is not involutive.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Derivative, concept, no evidence levelDerivativeconceptEuclidean vector, concept, no evidence levelEuclidean vectorconceptHolonomic constraint, concept, no evidence levelHolonomic constraintconceptLie bracket of vector fields, concept, no evidence levelLie bracket of vector…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTangent bundle, concept, no evidence levelTangent bundleconceptVelocity, concept, no evidence levelVelocityconceptConfiguration space, concept, no evidence levelConfiguration spaceconceptFrobenius theorem, statement, no evidence levelFrobenius theoremstatementGeneralized coordinates, concept, no evidence levelGeneralized coordinat…conceptNonholonomic system, definition, no evidence levelNonholonomic systemdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Derivative concept
  2. Euclidean vector concept
  3. Holonomic constraint concept
  4. Lie bracket of vector fields concept
  5. Smooth manifold concept
  6. Tangent bundle concept
  7. Velocity concept
  8. Configuration space concept
  9. Frobenius theorem statement
  10. Generalized coordinates concept
  11. Nonholonomic system definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Nonholonomic_system
About this page
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Kinddefinition
WritingW0 stub
MSC70F25, 70G45
DefinesNonholonomic system
UsesHolonomic constraint, Generalized coordinates, Velocity, Frobenius theorem
Changed3 November 2026

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