Math 2.0

Mechanics of particles and systems Foundations of particle mechanics Theorem

Conservation of momentum

Builds on Newton's laws of motionMomentum

Statement

Let NN point particles have masses mi>0m_i > 0 and positions xi(t)∈ℝ3x_i(t) \in \R^3, and let their motion obey Newton's second law. Let Fij(t)F_{ij}(t) be the force of particle jj on particle ii. Suppose that no external force acts, so that the net force on particle ii is ∑j≠iFij(t)\sum_{j \ne i} F_{ij}(t). Suppose that these forces obey Newton's third law, Fji=−FijF_{ji} = -F_{ij} for all i≠ji \ne j. Then the total momentum P(t)=∑i=1Nmix˙i(t)P(t) = \sum_{i=1}^{N} m_i \dot x_i(t) is constant in tt.

Remarks

The proof: P˙=∑i∑j≠iFij\dot P = \sum_i \sum_{j \ne i} F_{ij}, and the terms cancel in pairs, since Fij+Fji=0F_{ij} + F_{ji} = 0. As a consequence, the centre of mass X=1M∑imixiX = \frac{1}{M}\sum_i m_i x_i, with M=∑imiM = \sum_i m_i, moves with constant velocity P/MP/M. Example: a gun of mass MM at rest fires a bullet of mass mm at velocity vv; the gun recoils at velocity −mv/M-m v / M. If external forces act, the same computation gives P˙=∑iFiext\dot P = \sum_i F_i^{\text{ext}}, so PP changes; for instance a falling system gains downward momentum. Without the third law the conclusion fails: two particles with F12=F21=(1,0,0)F_{12} = F_{21} = (1, 0, 0) give P˙=(2,0,0)\dot P = (2, 0, 0).

Going deeper

For a student

Momentum conservation is the main tool for collisions, where the forces are unknown and brief but act in equal and opposite pairs. In Lagrangian mechanics it follows from invariance of the Lagrangian under translations, an instance of Noether's theorem. It also lets one pass to the centre-of-mass frame, which separates the motion of the system as a whole from its internal motion.

Reading path

To understand this page from the ground up, read these 24 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…definitionClassical mechanics, concept, no evidence levelClassical mechanicsconceptLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionMomentum, concept, no evidence levelMomentumconceptInertial frame of reference, concept, no evidence levelInertial frame of ref…conceptSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionNewton's laws of motion, statement, no evidence levelNewton's laws of moti…statementInitial value problem, definition, no evidence levelInitial value problemdefinitionPicard–Lindelöf hypotheses, definition, no evidence levelPicard–Lindelöf hypot…definitionPicard operator, definition, no evidence levelPicard operatordefinitionAn iterate of the Picard operator is a contraction, statement, no evidence levelAn iterate of the Pic…statementIntegral equation form of an initial value problem, statement, no evidence levelIntegral equation for…statementPicard–Lindelöf theorem, statement, no evidence levelPicard–Lindelöf theor…statementClassical mechanics, definition, no evidence levelClassical mechanicsdefinitionMomentum, definition, no evidence levelMomentumdefinitionConservation of momentum, statement, no evidence levelConservation of momen…statement
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. Classical mechanics concept
  11. Lipschitz condition definition
  12. Momentum concept
  13. Inertial frame of reference concept
  14. Solution of an ordinary differential equation definition
  15. Newton's laws of motion statement
  16. Initial value problem definition
  17. Picard–Lindelöf hypotheses definition
  18. Picard operator definition
  19. An iterate of the Picard operator is a contraction statement
  20. Integral equation form of an initial value problem statement
  21. Picard–Lindelöf theorem statement
  22. Classical mechanics definition
  23. Momentum definition
  24. Conservation of momentum statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Momentum
About this page
TagT1SHG5MD9R
URIm2://T1SHG5MD9R
Kindstatement, theorem
WritingW0 stub
MSC70A05, 70F10
StatusThis copy of the site has no build data. Run m2 status T1SHG5MD9R.
AboutMomentum
UsesNewton's laws of motion, Momentum
Changed3 November 2026

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