Math 2.0

Mechanics of particles and systems Definition

Potential energy

Builds on Conservative vector fieldGradient

Definition

Let U⊆ℝnU \subseteq \R^n be open and let F ⁣:U→ℝnF \colon U \to \R^n be a continuous force field. A potential energy for FF is a function V ⁣:U→ℝV \colon U \to \R of class C1C^1 such that F(x)=−∇V(x)for all x∈U,F(x) = -\nabla V(x) \quad \text{for all } x \in U, where ∇V\nabla V is the gradient of VV. For NN particles in ℝ3\R^3, take n=3Nn = 3N, so that V(x1,…,xN)V(x_1, \dots, x_N) and the force on particle ii is Fi=−∇xiVF_i = -\nabla_{x_i} V.

Remarks

A potential energy exists exactly when FF is a conservative vector field, and then −V-V is a scalar potential of FF. On a connected set UU, the potential energy is unique up to an additive constant. Example: uniform gravity F(x)=(0,0,−mg)F(x) = (0, 0, -mg) on ℝ3\R^3 has the potential energy V(x)=mgx3V(x) = m g x_3, since −∇V=(0,0,−mg)-\nabla V = (0, 0, -mg). A spring force F(x)=−kxF(x) = -k x with k>0k > 0 has V(x)=k2∣x∣2V(x) = \frac{k}{2}\abs{x}^2. Non-example: the field F(x)=(−x2,x1,0)F(x) = (-x_2, x_1, 0) has no potential energy on ℝ3\R^3, since its line integral around the unit circle in the plane x3=0x_3 = 0 is 2π≠02\pi \ne 0.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial 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…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionGradient, definition, no evidence levelGradientdefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementVector field, definition, no evidence levelVector fielddefinitionConservative vector field, definition, no evidence levelConservative vector f…definitionPotential energy, definition, no evidence levelPotential energydefinition
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  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Limit of a function of several variables concept
  7. Partial derivative concept
  8. Continuity in R^n definition
  9. Differentiable function of several variables definition
  10. Partial derivative definition
  11. Gradient definition
  12. Differentiable on an open set definition
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. C^k function characterised by continuous derivatives statement
  16. Vector field definition
  17. Conservative vector field definition
  18. Potential energy definition

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84 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Potential_energy
About this page
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URIm2://5T5GAKQD3S
Kinddefinition
WritingW0 stub
MSC70A05, 70F10
DefinesPotential energy
UsesConservative vector field, Gradient
Changed3 November 2026

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