Math 2.0

Mechanics of particles and systems Definition

Angular momentum

Builds on MomentumCross product

Definition

Let a particle of mass m>0m > 0 have position x(t)∈ℝ3x(t) \in \R^3 and momentum p(t)=mx˙(t)p(t) = m \dot x(t) at time tt, and let c∈ℝ3c \in \R^3 be a fixed point. The angular momentum of the particle about cc at time tt is L(t)=(x(t)−c)×p(t),L(t) = \big(x(t) - c\big) \times p(t), where ×\times is the cross product of ℝ3\R^3. For NN particles with positions xix_i and momenta pip_i, the total angular momentum about cc is L=∑i=1N(xi−c)×piL = \sum_{i=1}^{N} (x_i - c) \times p_i.

Remarks

Example: let x(t)=(Rcos⁡ωt,Rsin⁡ωt,0)x(t) = (R \cos \omega t, R \sin \omega t, 0) with R>0R > 0 and c=0c = 0. Then x˙=(−Rωsin⁡ωt,Rωcos⁡ωt,0)\dot x = (-R\omega \sin \omega t, R \omega \cos \omega t, 0) and L=(0,0,mR2ω)L = (0, 0, m R^2 \omega), a constant vector along the axis of rotation. Non-example: a particle moving on a line through cc, say x(t)=c+tux(t) = c + t u, has L=tu×mu=0L = t u \times m u = 0; it has momentum but no angular momentum about cc. When no point is named, cc is the origin. The angular momentum depends on the choice of cc; for the total angular momentum, moving cc to c′c' adds (c−c′)×P(c - c') \times P, where PP is the total momentum.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

Show the path as a graph
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 levelVelocityconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionCross product, definition, no evidence levelCross productdefinitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionPicard–Lindelöf hypotheses, definition, no evidence levelPicard–Lindelöf hypot…definitionPicard operator, definition, no evidence levelPicard operatordefinitionIntegral 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…statementPicard–Lindelöf theorem, statement, no evidence levelPicard–Lindelöf theor…statementClassical mechanics, definition, no evidence levelClassical mechanicsdefinitionMomentum, definition, no evidence levelMomentumdefinitionAngular momentum, definition, no evidence levelAngular momentumdefinition
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. 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. Ordinary differential equation definition
  10. Cross product definition
  11. Lipschitz condition definition
  12. Solution of an ordinary differential equation definition
  13. Initial value problem definition
  14. Picard–Lindelöf hypotheses definition
  15. Picard operator definition
  16. Integral equation form of an initial value problem statement
  17. An iterate of the Picard operator is a contraction statement
  18. Picard–Lindelöf theorem statement
  19. Classical mechanics definition
  20. Momentum definition
  21. Angular momentum definition

Where this leads

77 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Angular_momentum
About this page
TagF5W4PMJF8V
URIm2://F5W4PMJF8V
Kinddefinition
WritingW0 stub
MSC70A05, 70F10
DefinesAngular momentum
UsesMomentum, Cross product
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by