Newton's laws of motion

Importance and range of validity

Newton's laws were verified by experiment and observation for over 200 years, and they are excellent approximations at the scales and speeds of everyday life. Newton's laws of motion, together with his law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena.

These three laws hold to a good approximation for macroscopic objects under everyday conditions. Together with Newton's law of universal gravitation, they are often adequate to calculate the motion of celestial bodies, but sometimes the exclusion of general relativistic effects make the calculations verifiably inaccurate (see Mercury). However, Newton's laws (combined with Universal Gravitation and Classical Electrodynamics) are inappropriate for use in certain circumstances, most notably at very small scales, very high speeds (in special relativity, the Lorentz factor must be included in the expression for momentum along with rest mass and velocity) or very strong gravitational fields. Therefore, the laws cannot be used to explain phenomena such as conduction of electricity in a semiconductor, optical properties of substances, errors in non-relativistically corrected GPS systems and superconductivity. Explanation of these phenomena requires more sophisticated physical theory, including General Relativity and Relativistic Quantum Mechanics.

According to the principle of relativity, there is no preferred frame of reference. The laws of physics are equally valid in all frames of reference. Motion can only be measured relative to a frame of reference. According to the equivalence principle, an observer on the surface of the Earth could not find any difference between the gravitational attraction of earth and the inertial force that he feels when he is in a rocket in outer space that accelerates upwards (from the standpoint of the observer) at g. In other words, he may regard any inertial force as a gravitational force. Consequently, Newton's laws of motion are only valid in an inertial frame of reference. Notice that the surface of the Earth does not define an inertial frame of reference because it is rotating and orbiting and because of Earth's gravity. However, since the rotation and revolution are relatively slow, the inertial force is tiny. Therefore, Newton's laws of motion remain a good approximation on earth. In a non-inertial frame of reference, inertial forces must be considered for Newton's laws to remain valid.

In quantum mechanics concepts such as force, momentum, and position are defined by linear operators that operate on the quantum state; at speeds that are much lower than the speed of light, Newton's laws are just as exact for these operators as they are for classical objects. At speeds comparable to the speed of light, the second law holds in the original form

which says that the force is the derivative of the momentum of the object with respect to time, but some of the newer versions of the second law (such as the constant mass approximation above) do not hold at relativistic velocities.