Math 2.0

Mechanics of particles and systems Definition

Rayleigh dissipation function

Builds on Generalized coordinatesGeneralized forceLagrange's equations of motionPositive definite matrix

Definition

Let a mechanical system have generalized coordinates q=(q1,…,qn)q = (q_1, \ldots, q_n) and Lagrangian L(q,q˙,t)L(q, \dot q, t). A Rayleigh dissipation function is a function of the form F(q,q˙)=12∑i,j=1ncij(q) q˙iq˙j,F(q, \dot q) = \frac{1}{2} \sum_{i, j = 1}^n c_{ij}(q) \, \dot q_i \dot q_j, where C(q)=(cij(q))C(q) = (c_{ij}(q)) is a symmetric positive semidefinite n×nn \times n matrix for each qq. The dissipative generalized forces are Qi=−∂F/∂q˙iQ_i = -\partial F / \partial \dot q_i, and the Lagrange equations become ddt∂L∂q˙i−∂L∂qi=−∂F∂q˙i,i=1,…,n.\frac{d}{dt} \frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = -\frac{\partial F}{\partial \dot q_i}, \qquad i = 1, \ldots, n.

Remarks

Since FF is homogeneous of degree 22 in q˙\dot q, the power of the dissipative forces is ∑iQiq˙i=−2F≤0\sum_i Q_i \dot q_i = -2F \le 0. For L=T−VL = T - V with TT quadratic in q˙\dot q and V=V(q)V = V(q), the energy E=T+VE = T + V therefore satisfies E˙=−2F\dot E = -2F.

Example: L=12mx˙2−12kx2L = \frac{1}{2} m \dot x^2 - \frac{1}{2} k x^2 and F=12cx˙2F = \frac{1}{2} c \dot x^2 with c≥0c \ge 0. The matrix (c)(c) is positive semidefinite, and the equation is mx¨+cx˙+kx=0m \ddot x + c \dot x + k x = 0, the damped harmonic oscillator.

Non-example: F=q˙1q˙2F = \dot q_1 \dot q_2 has the matrix with c12=c21=1c_{12} = c_{21} = 1 and c11=c22=0c_{11} = c_{22} = 0, which is not positive semidefinite, since its eigenvalues are ±1\pm 1 (compare positive definite matrices). For q˙1=1\dot q_1 = 1, q˙2=−1\dot q_2 = -1 it gives F=−1F = -1, so these forces would feed energy into the system.

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Derivative, concept, no evidence levelDerivativeconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptEuler–Lagrange equation, concept, no evidence levelEuler–Lagrange equati…conceptFunction, concept, no evidence levelFunctionconceptHolonomic constraint, concept, no evidence levelHolonomic constraintconceptMatrix, definition, no evidence levelMatrixdefinitionMechanical equilibrium, concept, no evidence levelMechanical equilibriumconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTotal derivative, concept, no evidence levelTotal derivativeconceptVelocity, concept, no evidence levelVelocityconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptd'Alembert's principle, statement, no evidence leveld'Alembert's principlestatementClassical mechanics, concept, no evidence levelClassical mechanicsconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionConfiguration space, concept, no evidence levelConfiguration spaceconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptKinetic energy, concept, no evidence levelKinetic energyconceptGeneralized coordinates, concept, no evidence levelGeneralized coordinat…conceptSymmetric matrix, definition, no evidence levelSymmetric matrixdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionGradient, concept, no evidence levelGradientconceptGeneralized force, concept, no evidence levelGeneralized forceconceptHermitian matrix, definition, no evidence levelHermitian matrixdefinitionScalar potential, concept, no evidence levelScalar potentialconceptPositive definite matrix, definition, no evidence levelPositive definite mat…definitionPrinciple of virtual work, statement, no evidence levelPrinciple of virtual …statementConservative vector field, concept, no evidence levelConservative vector f…conceptPotential energy, concept, no evidence levelPotential energyconceptLagrange's equations of motion, statement, no evidence levelLagrange's equations …statementRayleigh dissipation function, definition, no evidence levelRayleigh dissipation …definition
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  1. Derivative concept
  2. Euclidean space object
  3. Euclidean vector concept
  4. Euler–Lagrange equation concept
  5. Function concept
  6. Holonomic constraint concept
  7. Matrix definition
  8. Mechanical equilibrium concept
  9. Ordinary differential equation concept
  10. Smooth manifold concept
  11. Total derivative concept
  12. Velocity concept
  13. Function of several variables concept
  14. Vector-valued function concept
  15. d'Alembert's principle statement
  16. Classical mechanics concept
  17. Square matrix definition
  18. Transpose definition
  19. Configuration space concept
  20. Differentiable function of several variables concept
  21. Partial derivative concept
  22. Vector field concept
  23. Kinetic energy concept
  24. Generalized coordinates concept
  25. Symmetric matrix definition
  26. Conjugate transpose definition
  27. Gradient concept
  28. Generalized force concept
  29. Hermitian matrix definition
  30. Scalar potential concept
  31. Positive definite matrix definition
  32. Principle of virtual work statement
  33. Conservative vector field concept
  34. Potential energy concept
  35. Lagrange's equations of motion statement
  36. Rayleigh dissipation function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Rayleigh_dissipation_function
About this page
TagX5ZH6CS7QF
URIm2://X5ZH6CS7QF
Kinddefinition
WritingW0 stub
MSC70H03, 70F40
DefinesRayleigh dissipation function
UsesGeneralized coordinates, Generalized force, Lagrange's equations of motion, Positive definite matrix
Changed3 November 2026

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