Bicycles and motorcycles, flying discs,[1] rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum.

Conservation of angular momentum is also why hurricanes[2] form spirals and neutron stars have high rotational rates.

However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar (more precisely, a pseudoscalar).

Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation.

This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day,[23] and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year.

The fact that the physics of a system is unchanged if it is rotated by any angle about an axis implies that angular momentum is conserved.

[26] Note, however, that this is no longer true in quantum mechanics, due to the existence of particle spin, which is angular momentum that cannot be described by the cumulative effect of point-like motions in space.

As an example, consider decreasing of the moment of inertia, e.g. when a figure skater is pulling in their hands, speeding up the circular motion.

In classical mechanics it can be shown that the rotational invariance of action functionals implies conservation of angular momentum.

Thus, assuming the potential energy does not depend on ωz (this assumption may fail for electromagnetic systems), we have the angular momentum of the ith object:

This caveat is reflected in quantum mechanics in the non-trivial commutation relations of the different components of the angular momentum operator.

Angular momentum is also an extremely useful concept for describing rotating rigid bodies such as a gyroscope or a rocky planet.

This has the advantage of a clearer geometric interpretation as a plane element, defined using the vectors x and p, and the expression is true in any number of dimensions.

In quantum mechanics, angular momentum (like other quantities) is expressed as an operator, and its one-dimensional projections have quantized eigenvalues.

Angular momentum is subject to the Heisenberg uncertainty principle, implying that at any time, only one projection (also called "component") can be measured with definite precision; the other two then remain uncertain.

The uncertainty is closely related to the fact that different components of an angular momentum operator do not commute, for example

When describing the motion of a charged particle in an electromagnetic field, the canonical momentum P (derived from the Lagrangian for this system) is not gauge invariant.

Tropical cyclones and other related weather phenomena involve conservation of angular momentum in order to explain the dynamics.

If the low pressure intensifies and the slowly circulating air is drawn toward the center, the molecules must speed up in order to conserve angular momentum.

The invention of rifled firearms and cannons gave their users significant strategic advantage in battle, and thus were a technological turning point in history.

The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

[46]He did not further investigate angular momentum directly in the Principia, saying:From such kind of reflexions also sometimes arise the circular motions of bodies about their own centers.

[47]However, his geometric proof of the law of areas is an outstanding example of Newton's genius, and indirectly proves angular momentum conservation in the case of a central force.

Newton derived a unique geometric proof, and went on to show that the attractive force of the Sun's gravity was the cause of all of Kepler's laws.

Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve.

Another areal proof of conservation of angular momentum for any central force uses Mamikon's sweeping tangents theorem.

[48][49] Leonhard Euler, Daniel Bernoulli, and Patrick d'Arcy all understood angular momentum in terms of conservation of areal velocity, a result of their analysis of Kepler's second law of planetary motion.

[51] Bernoulli wrote in a 1744 letter of a "moment of rotational motion", possibly the first conception of angular momentum as we now understand it.

William J. M. Rankine's 1858 Manual of Applied Mechanics defined angular momentum in the modern sense for the first time:... a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius-vector of the body seems to have right-handed rotation.In an 1872 edition of the same book, Rankine stated that "The term angular momentum was introduced by Mr. Hayward,"[53] probably referring to R.B.

Hayward's article On a Direct Method of estimating Velocities, Accelerations, and all similar Quantities with respect to Axes moveable in any manner in Space with Applications,[54] which was introduced in 1856, and published in 1864.