Math 2.0

Mechanics of particles and systems Constraints and generalized coordinates Theorem

Udwadia–Kalaba equation

Builds on Generalized coordinatesGeneralized forceHolonomic constraintNonholonomic systemPositive definite matrixMoore–Penrose inverseGauss's principle of least constraint

Statement

Consider a mechanical system with generalized coordinates q∈ℝnq \in \R^n. Let M(q,t)M(q, t) be its mass matrix, a symmetric positive definite n×nn \times n matrix, and let Q(q,q˙,t)Q(q, \dot q, t) collect the given generalized forces. Without constraints the motion obeys Mq¨=QM \ddot q = Q. Suppose the system also obeys constraints, holonomic or nonholonomic. After differentiation in time, let them take the form A(q,q˙,t) q¨=b(q,q˙,t)A(q, \dot q, t)\, \ddot q = b(q, \dot q, t), with AA an m×nm \times n matrix and b∈ℝmb \in \R^m. Assume this linear system in q¨\ddot q is consistent. Assume the constraint forces are ideal, that is, they do no work in any virtual displacement vv with Av=0A v = 0. Write a=M−1Qa = M^{-1} Q for the unconstrained acceleration, M1/2M^{1/2} for the positive definite square root of MM, and B+B^{+} for the Moore–Penrose inverse of a matrix BB. Then the constrained motion satisfies q¨=a+M−1/2(AM−1/2)+(b−Aa),\ddot q = a + M^{-1/2} \big(A M^{-1/2}\big)^{+} (b - A a), and the force of constraint is Qc=M1/2(AM−1/2)+(b−Aa)Q_c = M^{1/2} \big(A M^{-1/2}\big)^{+} (b - A a).

Remarks

The proof uses Gauss's principle of least constraint: the actual q¨\ddot q minimizes (q¨−a)TM(q¨−a)(\ddot q - a)^T M (\ddot q - a) subject to Aq¨=bA\ddot q = b. With z=M1/2(q¨−a)z = M^{1/2}(\ddot q - a) this asks for the solution of (AM−1/2)z=b−Aa(A M^{-1/2}) z = b - A a of least Euclidean norm, which is z=(AM−1/2)+(b−Aa)z = (A M^{-1/2})^{+}(b - A a). The formula needs no independence of the constraints: AA may have dependent rows, and the Moore–Penrose inverse handles this. Example: take a particle of mass mm in the plane under gravity, so M=mIM = m I and a=(0,−g)a = (0, -g), constrained to the circle x2+y2=ℓ2x^2 + y^2 = \ell^2. Differentiating the constraint twice gives xx¨+yy¨=−(x˙2+y˙2)x\ddot x + y\ddot y = -(\dot x^2 + \dot y^2). The formula then returns the pendulum equation, with the rod tension as constraint force. If the constraints are not ideal, an extra term for the work of the constraint forces must be added.

Going deeper

For a student

Unlike Lagrange's method with multipliers, the formula treats holonomic and nonholonomic constraints in one way and does not require the constraints to be independent. It is a closed-form version of Gauss's principle of least constraint and of Appell's equations. It is used in multibody dynamics and in control, where constraints may be imposed as desired trajectories.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Derivative, concept, no evidence levelDerivativeconceptEuclidean vector, concept, no evidence levelEuclidean vectorconceptEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptHolonomic constraint, concept, no evidence levelHolonomic constraintconceptMoore–Penrose inverse, concept, no evidence levelMoore–Penrose inverseconceptPositive definite matrix, concept, no evidence levelPositive definite mat…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptVelocity, concept, no evidence levelVelocityconceptd'Alembert's principle, statement, no evidence leveld'Alembert's principlestatementConfiguration space, concept, no evidence levelConfiguration spaceconceptGauss's principle of least constraint, statement, no evidence levelGauss's principle of …statementGeneralized coordinates, concept, no evidence levelGeneralized coordinat…conceptGeneralized force, concept, no evidence levelGeneralized forceconceptNonholonomic system, concept, no evidence levelNonholonomic systemconceptUdwadia–Kalaba equation, statement, no evidence levelUdwadia–Kalaba equati…statement
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. Euler–Lagrange equation concept
  4. Holonomic constraint concept
  5. Moore–Penrose inverse concept
  6. Positive definite matrix concept
  7. Smooth manifold concept
  8. Velocity concept
  9. d'Alembert's principle statement
  10. Configuration space concept
  11. Gauss's principle of least constraint statement
  12. Generalized coordinates concept
  13. Generalized force concept
  14. Nonholonomic system concept
  15. Udwadia–Kalaba equation statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Udwadia%E2%80%93Kalaba_formulation
About this page
TagTXMWJ9R9RG
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AboutNonholonomic system, Holonomic constraint
UsesGeneralized coordinates, Generalized force, Holonomic constraint, Nonholonomic system, Positive definite matrix, Moore–Penrose inverse, Gauss's principle of least constraint
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Changed3 November 2026

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