Math 2.0

Mechanics of particles and systems Definition

Momentum

Builds on Classical mechanicsVelocity

Definition

Let a particle of mass m>0m > 0 move with velocity x˙(t)∈ℝ3\dot x(t) \in \R^3 at time tt. Its momentum at time tt is the vector p(t)=m x˙(t).p(t) = m\, \dot x(t). For a Newtonian system of NN point particles with masses mim_i and velocities x˙i(t)\dot x_i(t), the total momentum is P(t)=∑i=1Nmix˙i(t)P(t) = \sum_{i=1}^{N} m_i \dot x_i(t).

Remarks

Example: a particle of mass 22 with position x(t)=(3t,t2,0)x(t) = (3t, t^2, 0) has velocity (3,2t,0)(3, 2t, 0) and momentum p(t)=(6,4t,0)p(t) = (6, 4t, 0). Non-example: the quantity m∣x˙∣2/2m \abs{\dot x}^2/2 is a scalar, the kinetic energy, and not the momentum. With M=∑imiM = \sum_i m_i and the centre of mass X=1M∑imixiX = \frac{1}{M}\sum_i m_i x_i, the total momentum is P=MX˙P = M \dot X. Some texts call pp the linear momentum, to separate it from angular momentum. Newton's second law reads p˙=F\dot p = F, where FF is the net force.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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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 levelVelocityconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz 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 levelMomentumdefinition
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. Ordinary differential equation definition
  10. Lipschitz condition definition
  11. Solution of an ordinary differential equation definition
  12. Initial value problem definition
  13. Picard–Lindelöf hypotheses definition
  14. Picard operator definition
  15. Integral equation form of an initial value problem statement
  16. An iterate of the Picard operator is a contraction statement
  17. Picard–Lindelöf theorem statement
  18. Classical mechanics definition
  19. Momentum definition

Where this leads

93 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Momentum
About this page
TagHFDS8NCJ1A
URIm2://HFDS8NCJ1A
Kinddefinition
WritingW0 stub
MSC70A05, 70F10
DefinesMomentum
UsesClassical mechanics, Velocity
Changed3 November 2026

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