Math 2.0

Mechanics of particles and systems Definition

Central force

Builds on Conservative vector fieldPotential energyAngular momentum

Definition

Let c∈ℝ3c \in \R^3 and let f ⁣:(0,∞)→ℝf \colon (0, \infty) \to \R be a function. The central force with centre cc and magnitude ff is the force field F ⁣:ℝ3∖{c}→ℝ3F \colon \R^3 \setminus \set{c} \to \R^3 given by F(x)=f(∣x−c∣) x−c∣x−c∣.F(x) = f(\abs{x - c})\, \frac{x - c}{\abs{x - c}}. The force is attractive at xx if f(∣x−c∣)<0f(\abs{x - c}) < 0 and repulsive if f(∣x−c∣)>0f(\abs{x - c}) > 0.

Remarks

Example: Newtonian gravity of a mass MM at the origin on a mass mm is central with c=0c = 0 and f(r)=−GMm/r2f(r) = -G M m / r^2, where GG is the gravitational constant. Non-example: F(x)=(−x2,x1,0)F(x) = (-x_2, x_1, 0) is not central about 00, since F(x)F(x) is orthogonal to xx and not parallel to it. If ff is continuous, FF is a conservative vector field with potential energy V(x)=−∫r0∣x−c∣f(s) dsV(x) = -\int_{r_0}^{\abs{x - c}} f(s)\, ds for any fixed r0>0r_0 > 0. Since (x−c)×F(x)=0(x - c) \times F(x) = 0, the angular momentum about cc of a particle moving under FF alone is constant, and the motion stays in a plane through cc. Some texts call a force central whenever F(x)F(x) is parallel to x−cx - c, without asking that its size depend only on ∣x−c∣\abs{x - c}.

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Derivative, concept, no evidence levelDerivativeconceptDifferential equation, concept, no evidence levelDifferential equationconceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuclidean vector, concept, no evidence levelEuclidean vectorconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptVelocity, concept, no evidence levelVelocityconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionCross product, definition, no evidence levelCross productdefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionGradient, definition, no evidence levelGradientdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionPicard–Lindelöf hypotheses, definition, no evidence levelPicard–Lindelöf hypot…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionPicard operator, definition, no evidence levelPicard operatordefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementIntegral equation form of an initial value problem, statement, no evidence levelIntegral equation for…statementAn iterate of the Picard operator is a contraction, statement, no evidence levelAn iterate of the Pic…statementVector field, definition, no evidence levelVector fielddefinitionPicard–Lindelöf theorem, statement, no evidence levelPicard–Lindelöf theor…statementConservative vector field, definition, no evidence levelConservative vector f…definitionClassical mechanics, definition, no evidence levelClassical mechanicsdefinitionMomentum, definition, no evidence levelMomentumdefinitionPotential energy, definition, no evidence levelPotential energydefinitionAngular momentum, definition, no evidence levelAngular momentumdefinitionCentral force, definition, no evidence levelCentral forcedefinition
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. Differential equation concept
  3. Euclidean space object
  4. Euclidean vector concept
  5. Function concept
  6. Interval concept
  7. Ordinary differential equation concept
  8. Velocity concept
  9. Fréchet derivative definition
  10. Ordinary differential equation definition
  11. Cross product definition
  12. Function of several variables concept
  13. Vector-valued function concept
  14. Lipschitz condition definition
  15. Limit of a function of several variables concept
  16. Solution of an ordinary differential equation definition
  17. Partial derivative concept
  18. Continuity in R^n definition
  19. Differentiable function of several variables definition
  20. Initial value problem definition
  21. Partial derivative definition
  22. Differentiable on an open set definition
  23. Gradient definition
  24. Higher partial derivatives definition
  25. Picard–Lindelöf hypotheses definition
  26. Iterated derivative definition
  27. Picard operator definition
  28. C^k function characterised by continuous derivatives statement
  29. Integral equation form of an initial value problem statement
  30. An iterate of the Picard operator is a contraction statement
  31. Vector field definition
  32. Picard–Lindelöf theorem statement
  33. Conservative vector field definition
  34. Classical mechanics definition
  35. Momentum definition
  36. Potential energy definition
  37. Angular momentum definition
  38. Central force definition

Where this leads

76 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Central_force
About this page
TagTHX3FXH5C1
URIm2://THX3FXH5C1
Kinddefinition
WritingW0 stub
MSC70F05, 70A05
DefinesCentral force
UsesConservative vector field, Potential energy, Angular momentum
Changed3 November 2026

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