Newton's three dynamical equations describe the motion of a body with mass m under the influence of a force F. They form the heart of Newton's principal work, the Principia Mathematica, where for the first time motion is defined and described in precise mathematical terms using derivatives. These equations were the starting point for the mathematical modeling of dynamical systems in the most general sense. They form the cradle of quantitative science. The fourth equation gives the explicit expression for the gravitational force. Together the equations provided the explanation of the gravitational phenomena – both celestial and terrestrial – known in Newton's time. These laws furnished the solid theoretical foundation for the findings of Copernicus, Brahe, Kepler, and Galileo concerning the elliptic orbital motion of the planets around the sun, but they also accounted for why an apple falls from a tree. They explain how different forces balance each other to guarantee the stability of bridges and buildings, and on the other hand why constructions sometimes collapse.

These equations involve time derivatives, because the velocity v = dr/dt is the change in time of position and the acceleration a = dv/dt is the change in time of velocity. Position, velocity and acceleration are all vectors: they have a magnitude and point in a certain direction.

The first equation defines what the momentum p is; this is sometimes called ‘the amount of motion’.

The second, most celebrated equation determines the motion in terms of the force exerted on a body. If we apply a force F, this causes an acceleration a in the same direction as the force. The equation implies in particular that if no force is exerted, there will be no change in velocity, and the momentum will remain constant. In the early days of mechanics, it was quite a revolutionary idea that motion would persist in the absence of a force – a rather disturbing, counter-intuitive thought, moreover in conflict with the commonly held views of Aristotle. If there are many forces working on a particle – for example friction, air resistance, and an electric force – then F is the net resulting force that the particle experiences.

The third equation states that if we exert a force on a body, then that body will exert an equal but opposite force on us; in short, ‘action is reaction’.