where e is the electric charge of the particle and A the magnetic vector potential of the electromagnetic field. The gauge-invariant angular momentum, that is kinetic angular momentum, is given by The interplay with quantum mechanics is discussed further in the article on canonical commutation relations. Angular momentum in optics In classical Maxwell electrodynamics the Poynting vector is a linear momentum density of electromagnetic field. The angular momentum density vector is given by a vector product as in classical mechanics: The above identities are valid locally, i.e. in each space point in a given moment . Angular momentum in nature and the cosmos Tropical cyclones and other related weather phenomena involve conservation of angular momentum in order to explain the dynamics. Winds revolve slowly around low pressure systems, mainly due to the coriolis effect. 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. By the time they reach the center, the speeds become destructive. Johannes Kepler determined the laws of planetary motion without knowledge of conservation of momentum. However, not long after his discovery their derivation was determined from conservation of angular momentum. Planets move more slowly the further they are out in their elliptical orbits, which is explained intuitively by the fact that orbital angular momentum is proportional to the radius of the orbit. Since the mass does not change and the angular momentum is conserved, the velocity drops. Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite (e.g. the Moon) and the primary planet that it orbits (e.g. Earth). The gravitational torque between the Moon and the tidal bulge of Earth causes the Moon to be constantly promoted to a slightly higher orbit and Earth to be decelerated in its rotation. The Earth loses angular momentum which is transferred to the Moon such that the overall angular momentum is conserved. Angular momentum in engineering and technology Examples of using conservation of angular momentum for practical advantage are abundant. In engines such as steam engines or internal combustion engines, a flywheel is needed to efficiently convert the lateral motion of the pistons to rotational motion. Inertial navigation systems explicitly use the fact that angular momentum is conserved with respect to the inertial frame of space. Inertial navigation is what enables submarine trips under the polar ice cap,