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Mechanics of particles and systems Foundations of particle mechanics Fact

Newton's laws of motion

Builds on Classical mechanicsInertial frame of referenceMomentum

Statement

Consider point particles described in an inertial frame of reference, in the setting of classical mechanics. Write mm for the mass of a particle, x(t)∈ℝ3x(t) \in \R^3 for its position at time tt, and p=mx˙p = m \dot x for its momentum.

  1. First law: a particle on which the net force is zero moves with constant velocity.
  2. Second law: if F(t)F(t) is the net force on a particle at time tt, then dpdt=F(t)\dfrac{dp}{dt} = F(t); for constant mass this reads mx¨(t)=F(t)m \ddot x(t) = F(t).
  3. Third law: if FijF_{ij} is the force that particle jj exerts on particle ii, then Fji=−FijF_{ji} = -F_{ij}.

Remarks

The three laws are the axioms of Newtonian mechanics, not theorems; they are confirmed by experiment within their range of validity. The first law singles out the inertial frames, and the second law then gives the equations of motion, a system of second-order differential equations. The third law, together with the second, gives conservation of the total momentum of an isolated system. The strong form of the third law adds that FijF_{ij} is parallel to xi−xjx_i - x_j; it gives conservation of angular momentum as well. Example: a ball of mass mm near the ground feels the force (0,0,−mg)(0, 0, -mg), so the second law gives x¨=(0,0,−g)\ddot x = (0, 0, -g). The ball then moves on a parabola, or on a vertical line if it is thrown straight up or down. In a rotating frame the second law fails unless fictitious forces are added, which is why the frame must be inertial.

Going deeper

For a student

The second law turns a mechanical problem into an initial value problem for an ordinary differential equation. Under mild conditions on the forces, the positions and velocities at one time determine the motion. From the three laws one derives the conservation laws of momentum, angular momentum and, for conservative forces, energy. Lagrangian and Hamiltonian mechanics reformulate the same laws in coordinates adapted to constraints and symmetries.

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  1. Derivative concept
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Where this leads

122 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion
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Kindstatement, fact
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AboutClassical mechanics, Inertial frame of reference
UsesClassical mechanics, Inertial frame of reference, Momentum
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Changed3 November 2026

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