Math 2.0

Mechanics of particles and systems Foundations of particle mechanics Concept

Mechanical similarity

Builds on Potential energyNewton's laws of motion

Summary

Mechanical similarity is a scaling symmetry of systems whose potential energy is a homogeneous function of the positions. Let the potential be homogeneous of degree kk. Enlarge a trajectory by a factor α\alpha and stretch its time scale by the factor α1−k/2\alpha^{1 - k/2}; the result is again a trajectory. This yields relations between sizes, times and energies of motions without solving the equations of Newton's laws. Kepler's third law and the independence of the period of a harmonic oscillator from its amplitude are two consequences.

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Derivative, concept, no evidence levelDerivativeconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptFunction, concept, no evidence levelFunctionconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptTotal 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 functionconceptClassical mechanics, concept, no evidence levelClassical mechanicsconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptMomentum, concept, no evidence levelMomentumconceptInertial frame of reference, concept, no evidence levelInertial frame of ref…conceptGradient, concept, no evidence levelGradientconceptNewton's laws of motion, statement, no evidence levelNewton's laws of moti…statementScalar potential, concept, no evidence levelScalar potentialconceptConservative vector field, concept, no evidence levelConservative vector f…conceptPotential energy, concept, no evidence levelPotential energyconceptMechanical similarity, concept, no evidence levelMechanical similarityconcept
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.
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  1. Derivative concept
  2. Euclidean space object
  3. Euclidean vector concept
  4. Function concept
  5. Ordinary differential equation concept
  6. Total derivative concept
  7. Velocity concept
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Classical mechanics concept
  11. Differentiable function of several variables concept
  12. Partial derivative concept
  13. Vector field concept
  14. Momentum concept
  15. Inertial frame of reference concept
  16. Gradient concept
  17. Newton's laws of motion statement
  18. Scalar potential concept
  19. Conservative vector field concept
  20. Potential energy concept
  21. Mechanical similarity concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Mechanical_similarity
About this page
TagVZXCAFQFW9
URIm2://VZXCAFQFW9
Kindconcept
WritingW0 stub
MSC70F10, 70A05
UsesPotential energy, Newton's laws of motion
Same as
Changed3 November 2026

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