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Afghan Hound

The Afghan Hound is a hound distinguished by its thick, fine, silky coat, and a tail with a ring curl at the end. The breed originates in the cold mountains of Afghanistan. Its local name is  or  (تاژي سپی). Other names for this breed are Tāzī, Balkh Hound, Baluchi Hound, and Barakzai Hound.

The American Kennel Club (AKC) describes the breed as among the most eye-catching of all. The Afghan Hound is an "aloof and dignified aristocrat of sublime beauty." Despite their regal appearance, the Afghan possesses an "endearing streak of silliness and a profound loyalty."

Admired since ancient times for their beauty, the Afghan Hound's distinctive coat was developed as protection from the harsh mountain climate. Their huge paw-pads served as shock absorbers on the rocky terrain.

History
The Afghan Hound has been identified as a basal breed that predates the emergence of the modern breeds in the 19th century. It is most closely related to the Saluki.

Connections with other types and breeds from the same area may provide clues to the history. A name for a desert coursing Afghan Hound, Tazi (Sag-e-Tazi), suggests a shared ancestry with the very similar Tazy breed from the Caspian Sea area of Russia and Turkmenistan. Other types or breeds of similar appearance are the Taigan from the mountainous Tian Shan region on the Chinese border of Afghanistan, and the Barakzay, or Kurram Valley Hound.

Once out of Afghanistan, the history of the Afghan Hound breed became entwined with that of the very earliest dog shows and the Kennel Club (UK). Various sighthounds were brought to England in the 1800s by army officers returning from British India and were exhibited at dog shows, which were then just becoming popular, under various names, such as Barukzy hounds. They were also called "Persian Greyhounds" by the English, in reference to their own indigenous sighthound.

One dog in particular, Zardin, was brought in 1907 from India by Captain John Barff. Zardin became the early ideal for the breed type still referred to as the Persian Greyhound. Zardin was the basis of the writing for the first breed standard in 1912, but this breeding cycle was stopped by World War I.

Out of the longhaired sighthound types known in Afghanistan, two main strains make up the modern Afghan Hound breed. The first were a group of hounds brought to Scotland from Balochistan by Major and Mrs. G. Bell-Murray and Miss Jean C. Manson in 1920, and they are known as the Bell-Murray strain. These dogs were of the "steppe" or “desert” type and were less heavily coated.

The second strain was a group of dogs from a kennel in Kabul owned by Mrs. Mary Amps, which she shipped to England in 1925. She and her husband came to Kabul after the Afghan war in 1919, and the foundation sire of her kennel (named Ghazni) in Kabul was a dog that closely resembled Zardin. Her Ghazni strain were the more heavily coated mountain type. Most of the Afghans in the United States were developed from the Ghazni strain from England. The first Afghans in Australia were imported from the United States in 1934, also of the Ghazni strain. The mountain and steppe strains became mixed into the modern Afghan Hound breed, and a new standard was written in 1948, which is still used today.

The Afghan Hound can also come with a much more "patterned" coat. This descends from the Bell-Murray's and the Ghazni lines, and is displayed in much lighter feathering of coat, deeper saddle (often actually looking like a saddle) and much shorter hair on the face and neck. It is believed that these particular Afghan Hounds were a product of much hotter parts of the country.

The beauty of Afghan Hound dogs caused them to become highly desirable show dogs and pets, and they are recognised by all of the major kennel clubs in the English-speaking world. One of the Amps Ghazni, Sirdar, won best in show at Crufts in 1928 and 1930. An Afghan Hound was featured on the cover of Life Magazine on November 26, 1945. Afghan Hounds were the most popular in Australia in the 1970s, and won most of the major shows. An Afghan Hound won Best in Show (BIS) at the 1996 World Dog Show in Budapest. Afghan Hounds were BIS at the Westminster Kennel Club Dog Show in 1957 and again in 1983.

The Afghan Hound breed is no longer used for hunting, although it can be seen in the sport of lure coursing.

On August 3, 2005, Korean scientist Hwang Woo-Suk announced that his team of researchers had become the first team to successfully clone a dog, an Afghan Hound named Snuppy. In 2006 Hwang Woo-Suk was dismissed from his university position for fabricating data in his research. Snuppy, nonetheless, was a genuine clone, and thus the first cloned dog in history.

Description

The dogs in this breed occur in many different coat colors. A study that mapped the genes of Afghan Hounds and discussed the effect of genes on coat colour in the breed was published in the Journal of Heredity in 2010.

The Afghan Hound is tall, standing in height 61 – and weighing 20 –. The coat may be any colour, but white markings, particularly on the head, are discouraged; many individuals have a black facial mask. A specimen may have a beard on the lower jaw, known as a "mandarin". Some Afghan Hounds are almost white, but parti-color hounds (white with islands of red or black) are penalized in the AKC standard, but not by the FCI.

Their long, fine-textured coat requires considerable care and grooming. The long topknot and the shorter-haired saddle on the back of the dog are distinctive features of the Afghan Hound coat. The high hipbones and unique small ring on the end of the tail are also characteristics of the breed.

The temperament of the typical Afghan Hound can be aloof and dignified, but happy and clownish when playing. This breed, as tends to be the case with sighthounds, has a high prey drive and may kill small animals and livestock. Genomic studies have pointed to the Afghan Hound as one of the oldest of dog breeds.

The breed has a reputation among dog trainers of having a relatively slow "obedience intelligence"; Stanley Coren, in his book The Intelligence of Dogs, ranked the breed last among 138 breeds mentioned in ability to understand and obey commands, requiring more than 80 repetitions to understand a new command and obeying on the first command less than 25% of the time. Coren noted that Afghan Hounds were consistently ranked among the least obedient dog breeds among all of the trainers he consulted, with a majority (121 out of 199) ranking the Afghan Hound in the lowest ten breeds out of 133 listed.

Khalag Tazi
The Khalag Tazi is a variety of the Afghan Hound introduced to Europe in 1920, when an Indian Army officer, Major G Bell-Murray, brought some animals back from Afghanistan.

Bakhmull
Bakhmull (also Bakhmull Tazi or Tazi Bakhmull, also called the Aboriginal Afghan Hound) is a long-haired variety of sighthound. It has been bred mostly in Russia and claimed to represent an Afghan Hound aboriginal to Afghanistan. In Pashto the word means "velvet", applied in reference to the dog's silky coat, which is rather abundant and long on the whole body, except the "saddle" (middle to lower back), front parts of all four legs, and the muzzle. Its color is always fawn, ivory, or white, with a darker "saddle", thus it produces an impression of a (yellowish) dog whose coat color matches the khaki sandstone and limestone of the Hindu Kush mountain landscape and deserts. The following colors are not permissible: red, red with white spots, black, and black with white spots.

Since the 1980s, the centre of Bakhmull breeding has been Russia, beginning in Moscow, then spreading to various other places in the CIS. The foundation stock was brought to Russia in the 1970s by military men returning from Afghanistan. Natalia Gherasiova (a breeder, of the Blue Dale el Bark Bakhmull kennel, and dog show judge) established the National Bakhmull Club, affiliated with the Russian Federation for Hunting Dogs (RFOS) and Russian Kynological Federation (RKF). A breed standard was first published in 1985, and a shared RFOS–RKF revision was produced in 1997.

Bakhmulls hunt solo and in couples. Although its coat is long, it does not require much grooming. Paws are well protected from injuries by "feathering" (thick additional paw fur). Its long, velvety coat and its stamina makes more suitable than many breeds for harsh weather. The breed standard calls for "aristocratic gait and a beautiful head with gazelle-like ... eyes". The eyes should be large, brown, slanting upwards, and of almond shape, with rims outlined black. Black coloration is required on the nose and lips for both white and fawn bakhmulls. The dog's height should be between 68 –, 65 - 70 cm for bitches. The height at the withers is 3 – higher than at the croup.

Lifespan
A UK study found a life expectancy of 11.1 years for the breed compared to an average of 12.7 for purebreeds and 12 for crossbreeds.

Health concerns
Major health issues are allergies, cancer, and hip dysplasia. Like other sighthounds, the Afghan Hound is sensitive to anesthesia, as sighthounds have relatively low levels of body fat. Afghan Hounds are also among the dog breeds most likely to develop chylothorax, a rare condition which causes the thoracic ducts to leak, allowing large quantities of chyle fluid to enter the dog's chest cavity. This condition commonly results in a lung-lobe torsion (in which the dog's lung twists within the chest cavity, requiring emergency surgery), due to the breed's typically deep, "barrel"-shaped chest. If not corrected through surgery, chylothorax can ultimately cause fibrosing pleuritis, or a hardening of the organs, due to scar tissue forming around the organs to protect them from the chyle fluid. Chylothorax is often fatal.

Among other health problems are laryngeal paralysis, dilated cardiomyopathy (twice as common in males as females), and dermatological issues such as testosterone-responsive dermatosis of male dogs (often seen in castrated males), nasal depigmentation (also known as Dudley nose), and skin tumours. Afghans are also prone to Central diabetes insipidus (CDI), hypothyroidism and tricholemmoma, a rare condition which mainly affects older dogs in the Middle Ages. Ocular conditions that can occur include medial canthal pocket syndrome (breed predisposition due to shape of head), corneal dystrophy, cataract and generalized progressive retinal atrophy (GPRA). Afghan myelopathy (causing pelvic limb ataxia) is sometimes reported.

In popular culture
Pablo Picasso said that his 1967 statue located in Chicago's Daley Plaza represented the head of an Afghan Hound named Kabul.

The Afghan hound has been represented in multiple animated feature films and TV shows, including Universal Pictures' Balto (Sylvie), Disney's Lady and the Tramp II: Scamp's Adventure (Ruby), Hasbro Studios's Pound Puppies (Twiggy) and ABC Kids' Bluey (Indy). An Afghan hound also appeared in the films One Hundred and One Dalmatians, 101 Dalmatians, 102 Dalmatians, and 101 Dalmatians II: Patch's London Adventure. Other examples include Prince Amir of Kinjan from What-a-Mess, Persia from Road Rovers, Burt from Foofur, Laila from Roadside Romeo, and Brainy Barker from Krypto the Superdog. Malory Archer in the show Archer also had an Afghan hound named Duchess at some point in her childhood.

In the 1941 novel Between the Acts, Virginia Woolf uses an Afghan hound named Sohrab to represent aspects of one of the book's human characters.

The Afghan Hound features prominently in the avant-garde music video of French band M83's, "Set in Stone (M83 Remix)".

title: Afghan Hound


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Azawakh

The Azawakh is a breed of dog from West Africa. With ancient origins, it is raised throughout the Sahelian zone of Mali, Niger, and Burkina Faso. This region includes the Azawagh Valley for which the breed is named. While commonly associated with the nomadic Tuareg people, the dogs are also bred and owned by other ethnic groups, such as the Peulh, Bella, and Hausa. The Azawakh is more related to the Sloughi than it is to the Saluki.

Appearance
Slim and elegant, with bone structure and muscles showing through thin skin. Eyes are almond-shaped.

The coat is very short and almost absent on the belly. Its bone structure shows clearly through the skin and musculature. Its muscles are "dry", meaning that they are quite flat, unlike the Greyhound and Whippet. In this respect it is similar in type to the Saluki.

Colours
Colours permitted by the Fédération Cynologique Internationale (FCI) breed standard are clear sand to dark fawn/brown, red and brindle (with or without a dark mask), with white bib, tail tip, and white on all feet (which can be tips of toes to high stockings). Since 2015 white stockings that go above the elbow joint are considered disqualifying features in the FCI member countries, as is a white collar or half collar (Irish marked).

Some conservationists support the idea that in Africa, Azawakhs are still found in a variety of colours such as red, blue fawn (that is, with a lilac cast), grizzle, and, rarely, blue and black with various white markings including Irish marked (white collar) and particolour (mostly white). Because of this wide color variation in the native population, the American standard used by the AKC and UKC allows any color combination found in Africa.

Movement

The Azawakh's light, supple, lissome gait is a notable breed characteristic, as is an upright double suspension gallop.

Temperament
Bred by the Tuareg, Fula and various other nomads of the Sahara and sub-Saharan Sahel in the countries of Mali, Niger, Burkina Faso, and southern Algeria, the breed known by the tuaregs as ”Oska” was used there as a guard dog and to hunt gazelle and hare at speeds up to 40 mph. The austerity of the Sahel environment has ensured that only the most fit dogs survive and has accentuated the breed's ruggedness and independence. Unlike some other sighthounds, the Azawakh is more of a pack hunter and they bump down the quarry with hindquarters when it has been tired out. In role of a guard dog, if an Azawakh senses danger it will bark to alert the other members of the pack, and they will gather together as a pack under the lead of the alpha dog, then chase off or attack the predator.

Unlike other sighthounds, the primary function of the Azawakh in its native land is that of a guard dog. It develops an intense bond with its owner, and tend to be reserved with strangers.

Azawakh have high energy and tremendous endurance. They are excellent training companions for runners. Many Azawakh dislike rain and cold weather.

Azawakh are pack oriented and form complex social hierarchies. They have tremendous memories and are able to recognize each other after long periods of separation. They can often be found sleeping on top of each other for warmth and companionship.

Breed history

The breed is relatively uncommon in Europe and North America but there is a growing band of devotees. Azawakh may be registered with the FCI in the USA via the Federación Canófila de Puerto Rico (FCPR). European FCI clubs and the AKC recognize the FCPR as an acceptable registry. The AKC recognized the Azawakh a member of the Hound group in 2019. The American Azawakh Association (AAA) is the AKC Parent Club for the Azawakh. Azawakh may be registered with the UKC and ARBA. The breed is not yet registered by CKC. Azawakh are eligible for ASFA and AKC lure coursing and NOFCA open field coursing events.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Azawakh


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Acrylic paint

Acrylic paint is a fast-drying paint made of pigment suspended in acrylic polymer emulsion and plasticizers, silicone oils, defoamers, stabilizers, or metal soaps. Most acrylic paints are water-based, but become water-resistant when dry. Depending on how much the paint is diluted with water, or modified with acrylic gels, mediums, or pastes, the finished acrylic painting can resemble a watercolor, a gouache, or an oil painting, or it may have its own unique characteristics not attainable with other media.

Water-based acrylic paints are used as latex house paints, as latex is the technical term for a suspension of polymer microparticles in water. Interior latex house paints tend to be a combination of binder (sometimes acrylic, vinyl, PVA, and others), filler, pigment, and water. Exterior latex house paints may also be a co-polymer blend, but the best exterior water-based paints are 100% acrylic, because of its elasticity and other factors. Vinyl, however, costs half of what 100% acrylic resins cost, and polyvinyl acetate (PVA) is even cheaper, so paint companies make many different combinations of them to match the market.

History
Otto Röhm invented acrylic resin, which was quickly transformed into acrylic paint. As early as 1934, the first usable acrylic resin dispersion was developed by German chemical company BASF, and patented by Rohm and Haas. The synthetic paint was first used in the 1940s, combining some of the properties of oil and watercolor. Between 1946 and 1949, Leonard Bocour and Sam Golden invented a solution acrylic paint under the brand Magna paint. These were mineral spirit-based paints.

Water-based acrylic paints were subsequently sold as latex house paints.

Soon after the water-based acrylic binders were introduced as house paints, artists and companies alike began to explore the potential of the new binders. Diego Rivera, David Alfaro Siqueiros, and José Clemente Orozco were the first ones who experimented with acrylic paint. This is because they were very impressed with the durability of the acrylic paint. Because of this, artists and companies alike began to produce Politec Acrylic Artists' Colors in Mexico in 1953. According to The Times newspaper, Lancelot Ribeiro pioneered the use of acrylic paints in the UK because of his "increasing impatience" by the 1960s over the time it took for oil paints to dry, as also its "lack of brilliance in its colour potential." He took to the new synthetic plastic bases that commercial paints were beginning to use and soon got help from manufacturers like ICI, Courtaulds, and Geigy. The companies supplied him samples of their latest paints in quantities that he was using three decades later, according to the paper. Initially, the firms thought the PVA compounds would not be needed in commercially viable quantities. But they quickly recognised the potential demand and "so Ribeiro became the godfather of generations of artists using acrylics as an alternative to oils."

In 1956, José L. Gutiérrez produced Politec Acrylic Artists' Colors in Mexico, and Henry Levison of Cincinnati-based Permanent Pigments Co. produced Liquitex colors. These two product lines were the first acrylic emulsion artists' paints, with modern high-viscosity paints becoming available in the early 1960s. Meanwhile on the other side of the globe, 1958 saw the inception of Vynol Paints Pty Ltd (now Derivan) in Australia, who started producing a water-based artist acrylic called Vynol Colour, followed by Matisse Acrylics in the 1960s. Following that development, Golden came up with a waterborne acrylic paint called "Aquatec". In 1963, George Rowney (part of Daler-Rowney since 1983) was the first manufacturer to introduce artists' acrylic paints in Europe, under the brand name "Cryla".

Painting with acrylics
Acrylic painters can modify the appearance, hardness, flexibility, texture, and other characteristics of the paint surface by using acrylic mediums or simply by adding water. Watercolor and oil painters also use various mediums, but the range of acrylic mediums is much greater. Acrylics have the ability to bond to many different surfaces, and mediums can be used to modify their binding characteristics. Acrylics can be used on paper, canvas, and a range of other materials; however, their use on engineered woods such as medium-density fiberboard can be problematic because of the porous nature of those surfaces. In these cases, it is recommended that the surface first be sealed with an appropriate sealer. The process of sealing acrylic painting is called varnishing. Artists use removable varnishes over isolation coat to protect paintings from dust, UV, scratches, etc. This process is similar to varnishing an oil painting.

Acrylics can be applied in thin layers or washes to create effects that resemble watercolors and other water-based mediums. They can also be used to build thick layers of paint — gel and molding paste are sometimes used to create paintings with relief features. Acrylic paints are also used in hobbies such as trains, cars, houses, DIY projects, and human models. People who make such models use acrylic paint to build facial features on dolls or raised details on other types of models. Wet acrylic paint is easily removed from paintbrushes and skin with water, whereas oil paints require the use of a hydrocarbon.

Acrylics are the most common paints used in grattage, a surrealist technique that began to be used with the advent of this type of paint. Acrylics are used for this purpose because they easily scrape or peel from a surface.

Painting techniques
Acrylic artists' paints may be thinned with water or acrylic medium and used as washes in the manner of watercolor paints, but unlike watercolor the washes are not rehydratable once dry. For this reason, acrylics do not lend themselves to the color lifting techniques of gum arabic-based watercolor paints. Instead, the paint is applied in layers, sometimes diluting with water or acrylic medium to allow layers underneath to partially show through. Using an acrylic medium gives the paint more of a rich and glossy appearance, whereas using water makes the paint look more like watercolor and have a matte finish.

Acrylic paints with gloss or matte finishes are common, although a satin (semi-matte) sheen is most common. Some brands exhibit a range of finishes (e.g. heavy-body paints from Golden, Liquitex, Winsor & Newton and Daler-Rowney); Politec acrylics are fully matte. As with oils, pigment amounts and particle size or shape can affect the paint sheen. Matting agents can also be added during manufacture to dull the finish. If desired, the artist can mix different media with their paints and use topcoats or varnishes to alter or unify sheen.

When dry, acrylic paint is generally non-removable from a solid surface if it adheres to the surface. Water or mild solvents do not re-solubilize it, although isopropyl alcohol can lift some fresh paint films off. Toluene and acetone can remove paint films, but they do not lift paint stains very well and are not selective. The use of a solvent to remove paint may result in removal of all of the paint layers (acrylic gesso, et cetera). Oils and warm, soapy water can remove acrylic paint from skin. Acrylic paint can be removed from nonporous plastic surfaces such as miniatures or models using cleaning products such as Dettol (containing chloroxylenol 4.8% v/w).

An acrylic sizing should be used to prime canvas in preparation for painting with acrylic paints, to prevent Support Induced Discoloration (SID). Acrylic paint contains surfactants that can pull up discoloration from a raw canvas, especially in transparent glazed or translucent gelled areas. Gesso alone will not stop SID; a sizing must be applied before using a gesso.

The viscosity of acrylic can be successfully reduced by using suitable extenders that maintain the integrity of the paint film. There are retarders to slow drying and extend workability time, and flow releases to increase color-blending ability.

Grades
Commercial acrylic paints come in two grades by manufacturers:
 * Artist acrylics (professional acrylics) are created and designed to resist chemical reactions from exposure to water, ultraviolet light, and oxygen. Professional-grade acrylics have the most pigment, which allows for more medium manipulation and limits the color shift when mixed with other colors or after drying.
 * Student acrylics have working characteristics similar to artist acrylics, but with lower pigment concentrations, less-expensive formulas, and fewer available colors. More expensive pigments are generally replicated by hues. Colors are designed to be mixed even though color strength is lower. Hues may not have exactly the same mixing characteristics as full-strength colors.

Varieties

 * Heavy body acrylics are typically found in the Artist and Student Grade paints. "Heavy Body" refers to the viscosity or thickness of the paint. They are the best choice for impasto or heavier paint applications and will hold a brush or knife stroke and even a medium stiff peak. Gel Mediums ("pigment-less paints") are also available in various viscosities and used to thicken or thin paints, as well as extend paints and add transparency.
 * Examples of Heavy Body Acrylics are Matisse Structure Acrylic Colors, Lukas Pastos Acrylics, Liquitex Heavy Body Acrylics and Golden Heavy Body Acrylics.
 * Medium viscosity acrylics – Fluid acrylics, Soft body acrylics, or High Flow acrylics – have a lower viscosity but generally the same pigmentation as the Heavy Body acrylics. Available in either Artist quality or Craft quality, the cost and quality vary accordingly. These paints are good for watercolor techniques, airbrush application, or when smooth coverage is desired. Fluid acrylics can be mixed with any medium to thicken them for impasto work, or to thin them for glazing applications.
 * Examples of fluid acrylics include Lukascryl Liquid, Lukascryl Studio, Liquitex Soft Body and Golden Fluid acrylics.
 * Open acrylics were created to address the one major difference between oil and acrylic paints: the shortened time it takes acrylic paints to dry. Designed by Golden Artist Colors, Inc. with a hydrophilic acrylic resin, these paints can take anywhere from a few hours to a few days, or even weeks, to dry completely, depending on paint thickness, support characteristics, temperature, and humidity.
 * Iridescent, pearl and interference acrylic colors combine conventional pigments with powdered mica (aluminium silicate) or powdered bronze to achieve complex visual effects. Colors have shimmering or reflective characteristics, depending on the coarseness or fineness of the powder. Iridescent colors are used in fine arts and crafts.
 * Acrylic gouache is like traditional gouache because it dries to a matte, opaque finish. However, unlike traditional gouache, the acrylic binder makes it water-resistant once it dries. Like craft paint, it will adhere to a variety of surfaces, not only canvas and paper. This paint is typically used by water-colorists, cartoonists, or illustrators, and for decorative or folk art applications.
 * Examples of acrylic gouache are Lascaux Gouache and Turner Acryl Gouache.
 * Craft acrylics can be used on surfaces besides canvas, such as wood, metal, fabrics, and ceramics. They are used in decorative painting techniques and faux finishes to decorate objects of ordinary life. Although colors can be mixed, pigments are often not specified. Each color line is formulated instead to achieve a wide range of premixed colors. Craft paints usually employ vinyl or PVA resins to increase adhesion and lower cost.
 * Interactive acrylics are all-purpose acrylic artists' colors which have the characteristic fast-drying nature of artists' acrylics, but are formulated to allow artists to delay drying when they need more working time, or re-wet their work when they want to do more wet blending.
 * Exterior acrylics are paints that can withstand outdoor conditions. Like craft acrylics, they adhere to many surfaces. They are more resistant to both water and ultraviolet light. This makes them the acrylic of choice for architectural murals, outdoor signs, and many faux-finishing techniques.
 * Acrylic glass paint is water-based and semi-permanent, making it a suitable paint for temporary displays on glass windows.
 * Acrylic enamel paint creates a smooth, hard shell. It can be oven-baked or air dried. It can be permanent if kept away from harsh conditions such as dishwashing.

Differences between acrylic and oil paint
The vehicle and binder of oil paints is linseed oil (or another drying oil), whereas acrylic paint has water as the vehicle for an emulsion (suspension) of acrylic polymer, which serves as the binder. Thus, oil paint is said to be "oil-based", whereas acrylic paint is "water-based" (or sometimes "water-borne").

The main practical difference between most acrylics and oil paints is the inherent drying time. Oils allow for more time to blend colors and apply even glazes over underpaintings. This slow-drying aspect of oil can be seen as an advantage for certain techniques, but it impedes an artist trying to work quickly. The fast evaporation of water from regular acrylic paint films can be slowed with the use of acrylic retarders. Retarders are generally glycol or glycerin-based additives. The addition of a retarder slows the evaporation rate of the water.

Oil paints may require the use of solvents such as mineral spirits or turpentine to thin the paint and clean up. These solvents generally have some level of toxicity and can be found objectionable. Relatively recently, water-miscible oil paints have been developed for artists' use. Oil paint films can gradually yellow and lose their flexibility over time creating cracks in the paint film; the "fat over lean" rule must be observed to ensure its durability.

Oil paint has a higher pigment load than acrylic paint. As linseed oil contains a smaller molecule than acrylic paint, oil paint is able to absorb substantially more pigment. Oil provides a refractive index that is less clear than acrylic dispersions, which imparts a unique "look and feel" to the resultant paint film. Not all the pigments of oil paints are available in acrylics and vice versa, as each medium has different chemical sensitivities. Some historical pigments are alkali sensitive, and therefore cannot be made in an acrylic emulsion; others are just too difficult to formulate. Approximate "hue" color formulations, that do not contain the historical pigments, are typically offered as substitutes.

Because of acrylic paint's more flexible nature and more consistent drying time between layers, an artist does not have to follow the same rules of oil painting, where more medium must be applied to each layer to avoid cracking. It usually takes 10–20 minutes for one to two layers of acrylic paint to dry, depending on the brand, quality, and humidity levels of the surrounding environment. Some professional grades of acrylic paint can take 20-30 minutes or even more than an hour. Although canvas needs to be properly primed before painting with oils to prevent the paint medium from eventually rotting the canvas, acrylic can be safely applied straight to the canvas. The rapid drying of acrylic paint tends to discourage blending of color and use of wet-in-wet technique as in oil painting. Even though acrylic retarders can slow drying time to several hours, it remains a relatively fast-drying medium and adding too much acrylic retarder can prevent the paint from ever drying properly.

Meanwhile, acrylic paint is very elastic, which prevents cracking from occurring. Acrylic paint's binder is acrylic polymer emulsion – as this binder dries, the paint remains flexible.

Another difference between oil and acrylic paints is the versatility offered by acrylic paints. Acrylics are very useful in mixed media, allowing the use of pastel (oil and chalk), charcoal and pen (among others) on top of the dried acrylic painted surface. Mixing other bodies into the acrylic is possible—sand, rice, and even pasta may be incorporated in the artwork. Mixing artist or student grade acrylic paint with household acrylic emulsions is possible, allowing the use of premixed tints straight from the tube or tin, and thereby presenting the painter with a vast color range at their disposal. This versatility is also illustrated by the variety of additional artistic uses for acrylics. Specialized acrylics have been manufactured and used for linoblock printing (acrylic block printing ink has been produced by Derivan since the early 1980s), face painting, airbrushing, watercolor-like techniques, and fabric screen printing.

Another difference between oil and acrylic paint is the cleanup. Acrylic paint can be cleaned out of a brush with any soap, while oil paint needs a specific type to be sure to get all the oil out of the brushes. Also, it is easier to let a palette with oil paint dry and then scrape the paint off, whereas one can easily clean wet acrylic paint with water.

Difference between acrylic and watercolor paint
The biggest difference is that acrylic paint is opaque, whereas watercolor paint is translucent in nature. Watercolors take about 5 to 15 minutes to dry while acrylics take about 10 to 20 minutes. In order to change the tone or shade of a watercolor pigment, you change the percentage of water mixed in to the color. For brighter colors, add more water. For darker colors, add less water. In order to create lighter or darker colors with acrylic paints, you add white or black.

Another difference is that watercolors must be painted onto a porous surface, primarily watercolor paper. Acrylic paints can be used on many different surfaces.

Both acrylic and watercolor are easy to clean up with water. Acrylic paint should be cleaned with soap and water immediately following use. Watercolor paint can be cleaned with just water.

title: Acrylic paint


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Angular momentum

In physics, angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.

The three-dimensional angular momentum for a point particle is classically represented as a pseudovector $L$, the cross product of the particle's position vector $L = Iω = r × p$ (relative to some origin) and its momentum vector; the latter is $r × p$ in Newtonian mechanics. Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle's position is measured from it.

Angular momentum is an extensive quantity; that is, the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous rigid body or a fluid, the total angular momentum is the volume integral of angular momentum density (angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body.

Similar to conservation of linear momentum, where it is conserved if there is no external force, angular momentum is conserved if there is no external torque. Torque can be defined as the rate of change of angular momentum, analogous to force. The net external torque on any system is always equal to the total torque on the system; the sum of all internal torques of any system is always 0 (this is the rotational analogue of Newton's third law of motion). Therefore, for a closed system (where there is no net external torque), the total torque on the system must be 0, which means that the total angular momentum of the system is constant.

The change in angular momentum for a particular interaction is called angular impulse, sometimes twirl. Angular impulse is the angular analog of (linear) impulse.

Examples
The trivial case of the angular momentum $$L$$ of a body in an orbit is given by $$L = 2 \pi M f r^2$$ where $$M$$ is the mass of the orbiting object, $$f$$ is the orbit's frequency and $$r$$ is the orbit's radius.

The angular momentum $$L$$ of a uniform rigid sphere rotating around its axis, instead, is given by

$$L = \frac{4}{5} \pi M f r^2 $$

where $$M$$ is the sphere's mass, $$f$$ is the frequency of rotation and $$r$$ is the sphere's radius.

Thus, for example, the orbital angular momentum of the Earth with respect to the Sun is about 2.66 × 1040 kg⋅m2⋅s−1, while its rotational angular momentum is about 7.05 × 1033 kg⋅m2⋅s−1.

In the case of a uniform rigid sphere rotating around its axis, if, instead of its mass, its density is known, the angular momentum $$L$$ is given by

$$L = \frac{16}{15} \pi^2 \rho f r^5$$

where $$\rho$$ is the sphere's density, $$f$$ is the frequency of rotation and $$r$$ is the sphere's radius.

In the simplest case of a spinning disk, the angular momentum $$L$$ is given by

$$L = \pi M f r^2$$

where $$M$$ is the disk's mass, $$f$$ is the frequency of rotation and $$r$$ is the disk's radius.

If instead the disk rotates about its diameter (e.g. coin toss), its angular momentum $$L$$ is given by

$$L = \frac{1}{2} \pi M f r^2$$

Definition in classical mechanics
Just as for angular velocity, there are two special types of angular momentum of an object: the spin angular momentum is the angular momentum about the object's centre of mass, while the orbital angular momentum is the angular momentum about a chosen center of rotation. The Earth has an orbital angular momentum by nature of revolving around the Sun, and a spin angular momentum by nature of its daily rotation around the polar axis. The total angular momentum is the sum of the spin and orbital angular momenta. In the case of the Earth the primary conserved quantity is the total angular momentum of the solar system because angular momentum is exchanged to a small but important extent among the planets and the Sun. The orbital angular momentum vector of a point particle is always parallel and directly proportional to its orbital angular velocity vector ω, where the constant of proportionality depends on both the mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector Ω, making the constant of proportionality a second-rank tensor rather than a scalar.

Orbital angular momentum in two dimensions

Angular momentum is a vector quantity (more precisely, a pseudovector) that represents the product of a body's rotational inertia and rotational velocity (in radians/sec) about a particular axis. 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). Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum $p$ is proportional to mass $m$ and linear speed $v$, p = mv,

angular momentum $L$ is proportional to moment of inertia $I$ and angular speed $ω$ measured in radians per second.

Unlike mass, which depends only on amount of matter, moment of inertia depends also on the position of the axis of rotation and the distribution of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, $L$ should be referred to as the angular momentum relative to that center.

In the case of circular motion of a single particle, we can use $$I = r^2m$$ and $$\omega = {v}/{r}$$ to expand angular momentum as $$L = r^2 m \cdot {v}/{r},$$ reducing to: $$L = rmv,$$

the product of the radius of rotation $r$ and the linear momentum of the particle $$p = mv$$, where $$v= r\omega$$ is the linear (tangential) speed.

This simple analysis can also apply to non-circular motion if one uses the component of the motion perpendicular to the radius vector: $$L = rmv_\perp,$$

where $$v_\perp = v\sin(\theta)$$ is the perpendicular component of the motion. Expanding, $$L = rmv\sin(\theta),$$ rearranging, $$L = r\sin(\theta)mv,$$ and reducing, angular momentum can also be expressed, $$L = r_\perp mv,$$

where $$r_\perp = r\sin(\theta)$$ is the length of the moment arm, a line dropped perpendicularly from the origin onto the path of the particle. It is this definition, $r$, to which the term moment of momentum refers.

Scalar angular momentum from Lagrangian mechanics
Another approach is to define angular momentum as the conjugate momentum (also called canonical momentum) of the angular coordinate $$\phi$$ expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass $$m$$ constrained to move in a circle of radius $$r$$ in the absence of any external force field. The kinetic energy of the system is

And the potential energy is $$U = 0.$$

Then the Lagrangian is $$\mathcal{L}\left(\phi, \dot{\phi}\right) = T - U = \tfrac{1}{2}mr^2 \dot{\phi}^2.$$

The generalized momentum "canonically conjugate to" the coordinate $$\phi$$ is defined by $$p_\phi = \frac{\partial \mathcal{L}}{\partial \dot{\phi}} = mr^2 \dot{\phi} = I\omega = L.$$

Orbital angular momentum in three dimensions

To completely define orbital angular momentum in three dimensions, it is required to know the rate at which the position vector sweeps out angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin can be expressed as: $$\mathbf{L} = I\boldsymbol{\omega},$$ where
 * $$I = r^2m$$ is the moment of inertia for a point mass,
 * is the orbital angular velocity of the particle about the origin,
 * $$\mathbf{r}$$ is the position vector of the particle relative to the origin, and $$r=\left\vert\mathbf{r}\right\vert$$,
 * $$\mathbf{v}$$ is the linear velocity of the particle relative to the origin, and
 * $$m$$ is the mass of the particle.

This can be expanded, reduced, and by the rules of vector algebra, rearranged: $$\begin{align} \mathbf{L} &= \left(r^2m\right)\left(\frac{\mathbf{r}\times\mathbf{v}}{r^2}\right) \\ &= m\left(\mathbf{r}\times\mathbf{v}\right) \\ &= \mathbf{r}\times m\mathbf{v} \\ &= \mathbf{r}\times\mathbf{p}, \end{align}$$ which is the cross product of the position vector $$\mathbf{r}$$ and the linear momentum $$\mathbf{p} = m\mathbf{v}$$ of the particle. By the definition of the cross product, the $$\mathbf{L}$$ vector is perpendicular to both $$\mathbf{r}$$ and $$\mathbf{p}$$. It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the $$\mathbf{L}$$ vector defines the plane in which $$\mathbf{r}$$ and $$\mathbf{p}$$ lie.

By defining a unit vector $$\mathbf{\hat{u}}$$ perpendicular to the plane of angular displacement, a scalar angular speed $$\omega$$ results, where $$\omega\mathbf{\hat{u}} = \boldsymbol{\omega},$$ and $$\omega = \frac{v_\perp}{r},$$ where $$v_\perp$$ is the perpendicular component of the motion, as above.

The two-dimensional scalar equations of the previous section can thus be given direction: $$\begin{align} \mathbf{L} &= I\boldsymbol{\omega}\\ &= I\omega\mathbf{\hat{u}}\\ &= \left(r^2m\right)\omega\mathbf{\hat{u}}\\ &= rmv_\perp \mathbf{\hat{u}}\\ &= r_\perp mv\mathbf{\hat{u}}, \end{align}$$ and $$\mathbf{L} = rmv\mathbf{\hat{u}}$$ for circular motion, where all of the motion is perpendicular to the radius $$r$$.

In the spherical coordinate system the angular momentum vector expresses as
 * $$\mathbf{L} =

m \mathbf{r} \times \mathbf{v} = m r^2 \left(\dot\theta\,\hat{\boldsymbol\varphi} - \dot\varphi \sin\theta\,\mathbf{\hat{\boldsymbol\theta}}\right). $$

Analogy to linear momentum
Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape.

Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product,

$$\begin{align} (\text{amount of inertia}) \times (\text{amount of displacement})&=\text{amount of (inertia⋅displacement)}\\ \text{mass} \times \text{velocity} &= \text{momentum}\\ m \times v &= p\\ \end{align} $$

is the matter's momentum. Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the moment arm. It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a moment. Hence, the particle's momentum referred to a particular point,

$$\begin{align} (\text{moment arm}) \times (\text{amount of inertia}) \times (\text{amount of displacement})&=\text{moment of (inertia⋅displacement)}\\ \text{length} \times \text{mass} \times \text{velocity} &= \text{moment of momentum}\\ r \times m \times v &= L\\ \end{align} $$

is the angular momentum, sometimes called, as here, the moment of momentum of the particle versus that particular center point. The equation $$L = rmv$$ combines a moment (a mass $$m$$ turning moment arm $$r$$) with a linear (straight-line equivalent) speed $$v$$. Linear speed referred to the central point is simply the product of the distance $$r$$ and the angular speed $$\omega$$ versus the point: $$v=r\omega,$$ another moment. Hence, angular momentum contains a double moment: $$L = rmr \omega.$$ Simplifying slightly, $$L = r^2 m\omega,$$ the quantity $$r^2m$$ is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia.

The above analogy of the translational momentum and rotational momentum can be expressed in vector form:
 * $\mathbf p = m\mathbf  v$  for linear motion
 * $$\mathbf L = I\boldsymbol\omega$$ for rotation

The direction of momentum is related to the direction of the velocity for linear movement. The direction of angular momentum is related to the angular velocity of the rotation.

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. Therefore, the total moment of inertia, and the angular momentum, is a complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits.

For a rigid body, for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass.

For a collection of objects revolving about a center, for instance all of the bodies of the Solar System, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random.

In brief, the more mass and the farther it is from the center of rotation (the longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by, I=k^2m,where $$k$$ is the radius of gyration, the distance from the axis at which the entire mass $$m$$ may be considered as concentrated.

Similarly, for a point mass $$m$$ the moment of inertia is defined as, $$I=r^2m$$where $$r$$ is the radius of the point mass from the center of rotation,

and for any collection of particles $$m_i$$ as the sum, $$\sum_i I_i = \sum_i r_i^2m_i$$

Angular momentum's dependence on position and shape is reflected in its units versus linear momentum: kg⋅m2/s or N⋅m⋅s for angular momentum versus kg⋅m/s or N⋅s for linear momentum. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity. (When performing dimensional analysis, it may be productive to use orientational analysis which treats radians as a base unit, but this is not done in the International system of units). The units if angular momentum can be interpreted as torque⋅time. An object with angular momentum of L N⋅m⋅s can be reduced to zero angular velocity by an angular impulse of L N⋅m⋅s.

The plane perpendicular to the axis of angular momentum and passing through the center of mass is sometimes called the invariable plane, because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System.

Angular momentum and torque
Newton's second law of motion can be expressed mathematically, $$\mathbf{F} = m\mathbf{a},$$ or force = mass × acceleration. The rotational equivalent for point particles may be derived as follows: $$\mathbf{L} = I\boldsymbol{\omega}$$ which means that the torque (i.e. the time derivative of the angular momentum) is

Because the moment of inertia is $$mr^2$$, it follows that $$\frac{dI}{dt} = 2mr\frac{dr}{dt} = 2rp_{||}$$, and $$\frac{d\mathbf{L}}{dt} = I\frac{d\boldsymbol{\omega}}{dt} + 2rp_{||}\boldsymbol{\omega},$$ which, reduces to

$$\boldsymbol{\tau} = I\boldsymbol{\alpha} + 2rp_{||}\boldsymbol{\omega}.$$ This is the rotational analog of Newton's second law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass.

General considerations
A rotational analog of Newton's third law of motion might be written, "In a closed system, no torque can be exerted on any matter without the exertion on some other matter of an equal and opposite torque about the same axis." Hence, angular momentum can be exchanged between objects in a closed system, but total angular momentum before and after an exchange remains constant (is conserved).

Seen another way, a rotational analogue of Newton's first law of motion might be written, "A rigid body continues in a state of uniform rotation unless acted by an external influence." Thus with no external influence to act upon it, the original angular momentum of the system remains constant.

The conservation of angular momentum is used in analyzing central force motion. If the net force on some body is directed always toward some point, the center, then there is no torque on the body with respect to the center, as all of the force is directed along the radius vector, and none is perpendicular to the radius. Mathematically, torque $$\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} = \mathbf{0}, $$ because in this case $$\mathbf{r}$$ and $$\mathbf{F}$$ are parallel vectors. Therefore, the angular momentum of the body about the center is constant. This is the case with gravitational attraction in the orbits of planets and satellites, where the gravitational force is always directed toward the primary body and orbiting bodies conserve angular momentum by exchanging distance and velocity as they move about the primary. Central force motion is also used in the analysis of the Bohr model of the atom.

For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the Earth–Moon system results in the transfer of angular momentum from Earth to Moon, due to tidal torque the Moon exerts on the Earth. This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day, and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year.

The conservation of angular momentum explains the angular acceleration of an ice skater as they bring their arms and legs close to the vertical axis of rotation. By bringing part of the mass of their body closer to the axis, they decrease their body's moment of inertia. Because angular momentum is the product of moment of inertia and angular velocity, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must increase.

The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars.

Conservation is not always a full explanation for the dynamics of a system but is a key constraint. For example, a spinning top is subject to gravitational torque making it lean over and change the angular momentum about the nutation axis, but neglecting friction at the point of spinning contact, it has a conserved angular momentum about its spinning axis, and another about its precession axis. Also, in any planetary system, the planets, star(s), comets, and asteroids can all move in numerous complicated ways, but only so that the angular momentum of the system is conserved.

Noether's theorem states that every conservation law is associated with a symmetry (invariant) of the underlying physics. The symmetry associated with conservation of angular momentum is rotational invariance. 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.

Relation to Newton's second law of motion
While angular momentum total conservation can be understood separately from Newton's laws of motion as stemming from Noether's theorem in systems symmetric under rotations, it can also be understood simply as an efficient method of calculation of results that can also be otherwise arrived at directly from Newton's second law, together with laws governing the forces of nature (such as Newton's third law, Maxwell's equations and Lorentz force). Indeed, given initial conditions of position and velocity for every point, and the forces at such a condition, one may use Newton's second law to calculate the second derivative of position, and solving for this gives full information on the development of the physical system with time. 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 terms of angular momentum conservation, we have, for angular momentum L, moment of inertia I and angular velocity ω: $$ 0 = dL = d (I\cdot \omega) = dI \cdot \omega + I \cdot d\omega $$

Using this, we see that the change requires an energy of:

$$ dE = d \left(\tfrac{1}{2} I\cdot \omega^2\right) = \tfrac{1}{2} dI \cdot \omega^2 + I \cdot \omega \cdot d\omega = -\tfrac{1}{2} dI \cdot \omega^2$$ so that a decrease in the moment of inertia requires investing energy.

This can be compared to the work done as calculated using Newton's laws. Each point in the rotating body is accelerating, at each point of time, with radial acceleration of: $$-r\cdot \omega^2$$

Let us observe a point of mass m, whose position vector relative to the center of motion is perpendicular to the z-axis at a given point of time, and is at a distance z. The centripetal force on this point, keeping the circular motion, is: $$-m\cdot z\cdot \omega^2$$

Thus the work required for moving this point to a distance dz farther from the center of motion is: $$dW = -m\cdot z\cdot \omega^2\cdot dz = -m\cdot \omega^2\cdot d\left(\tfrac{1}{2} z^2\right)$$

For a non-pointlike body one must integrate over this, with m replaced by the mass density per unit z. This gives: $$dW = - \tfrac{1}{2}dI \cdot \omega^2$$

which is exactly the energy required for keeping the angular momentum conserved.

Note, that the above calculation can also be performed per mass, using kinematics only. Thus the phenomena of figure skater accelerating tangential velocity while pulling their hands in, can be understood as follows in layman's language: The skater's palms are not moving in a straight line, so they are constantly accelerating inwards, but do not gain additional speed because the accelerating is always done when their motion inwards is zero. However, this is different when pulling the palms closer to the body: The acceleration due to rotation now increases the speed; but because of the rotation, the increase in speed does not translate to a significant speed inwards, but to an increase of the rotation speed.

Stationary-action principle
In classical mechanics it can be shown that the rotational invariance of action functional implies conservation of angular momentum. The action is defined in classical physics as a functional of positions, $$x_i (t)$$ often represented by the use of square brackets, and the final and initial times. It assumes the following form in cartesian coordinates:$$S\left([x_{i}];t_{1},t_{2}\right)\equiv\int_{t_{1}}^{t_{2}}d t\left(\frac{1}{2}m\frac{d x_{i}}{d t}\ \frac{d x_{i}}{d t}-V(x_{i})\right)$$where the repeated indices indicate summation over the index. If the action is invariant of an infinitesimal transformation, it can be mathematically stated as: $\delta S = S\left([x_{i}+\delta x_i];t_{1},t_{2}\right)-S\left([x_{i}];t_{1},t_{2}\right) =0$.

Under the transformation, $$x_i \rightarrow x_i + \delta x_i $$, the action becomes:

$$S\left([x_{i}+\delta x_i];t_{1},t_{2}\right)=\!\int_{t_{1}}^{t_{2}}d t\left(\frac{1}{2}m\frac{d(x_{i}+\delta x_{i})}{d t}\frac{d(x_{i}+\delta x_{i})}{d t}-V(x_{i}+\delta x_{i})\right)$$

where we can employ the expansion of the terms up-to first order in $\delta x_i$ :

$$\begin{align} \frac{d(x_i+\delta x_i)}{d t} \frac{d( x_{i}+\delta x_{i})}{ d t } &\simeq\frac{d x_{i}}{d t} \frac{d x_{i}}{d t}-2\frac{d^{2}x_{i}}{d t^{2}}\delta x_{i}+2\frac{d}{d t}\left(\delta x_{i}\frac{d x_{i}}{d t}\right)\\ V(x_{i}+\delta x_{i}) & \simeq V(x_{i})+\delta x_{i}\frac{\partial V}{\partial x_i}\\ \end{align}$$giving the following change in action:

$$S[x_{i}+\delta x_{i}]\simeq S[x_{i}]+\int_{t_{1}}^{t_{2}}d t\,\delta x_{i}\left(- \frac{\partial V}{\partial x_i}-m{\frac{d^{2}x_{i}}{d t^{2}}}\right)+m\int_{t_{1}}^{t_{2}}d t{\frac{d}{d t}}\left(\delta x_{i}{\frac{d x_{i}}{d t}}\right).$$

Since all rotations can be expressed as matrix exponential of skew-symmetric matrices, i.e. as $$R(\hat n,\theta) = e^{M \theta}$$ where $$M $$ is a skew-symmetric matrix and $$\theta $$ is angle of rotation, we can express the change of coordinates due to the rotation $$R(\hat n,\delta \theta )$$, up-to first order of infinitesimal angle of rotation, $$\delta \theta $$ as:

$$\delta x_i  =  M_{ij} x_j \delta \theta. $$

Combining the equation of motion and rotational invariance of action, we get from the above equations that:$$0=\delta S=\int_{t_{1}}^{t_{2}}d t\frac{d}{d t}\left(m\frac{d x_{i}}{d t}\delta x_{i}\right)= M_{i j}\,\delta \theta \, m \,x_{j}\frac{d x_{i}}{d t}\Bigg\vert_{t_{1}}^{t_{2}}$$Since this is true for any matrix $$M_{ij} $$ that satisfies $$M_{ij} = - M_{ji}   , $$ it results in the conservation of the following quantity:

$$\ell_{ij}(t) := m\left(x_i \frac{dx_j}{dt}-x_j \frac{dx_i}{dt}\right),$$

as $$\ell_{ij}(t_1)=\ell_{ij}(t_2)$$. This corresponds to the conservation of angular momentum throughout the motion.

Lagrangian formalism
In Lagrangian mechanics, angular momentum for rotation around a given axis, is the conjugate momentum of the generalized coordinate of the angle around the same axis. For example, $$L_z$$, the angular momentum around the z axis, is: $$L_z = \frac{\partial \cal{L}}{\partial \dot\theta_z}$$ where $$\cal{L}$$ is the Lagrangian and $$\theta_z$$ is the angle around the z axis.

Note that $$\dot\theta_z$$, the time derivative of the angle, is the angular velocity $$\omega_z$$. Ordinarily, the Lagrangian depends on the angular velocity through the kinetic energy: The latter can be written by separating the velocity to its radial and tangential part, with the tangential part at the x-y plane, around the z-axis, being equal to: $$\sum_i \tfrac{1}{2}m_i {v_T}_i^2 = \sum_i \tfrac{1}{2} m_i \left(x_i^2 + y_i^2\right) { {\omega_z}_i}^2$$ where the subscript i stands for the i-th body, and m, vT and ωz stand for mass, tangential velocity around the z-axis and angular velocity around that axis, respectively.

For a body that is not point-like, with density ρ, we have instead: $$\frac{1}{2}\int \rho(x,y,z) \left(x_i^2 + y_i^2\right) { {\omega_z}_i}^2\,dx\,dy = \frac{1}{2} {I_z}_i { {\omega_z}_i}^2$$ where integration runs over the area of the body, and Iz is the moment of inertia around the z-axis.

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: $$\begin{align} {L_z}_i &= \frac{\partial \cal{L} }{\partial { {\omega_z}_i} } = \frac{\partial E_k}{\partial { {\omega_z}_i} } \\ &= {I_z}_i \cdot {\omega_z}_i \end{align}$$

We have thus far rotated each object by a separate angle; we may also define an overall angle θz by which we rotate the whole system, thus rotating also each object around the z-axis, and have the overall angular momentum: $$L_z = \sum_i {I_z}_i \cdot {\omega_z}_i$$

From Euler–Lagrange equations it then follows that: $$0 = \frac{\partial \cal{L} }{\partial { {\theta_z}_i} } - \frac{d}{dt}\left(\frac{\partial \cal{L} }{\partial { {\dot\theta_z}_i}}\right) = \frac{\partial \cal{L} }{\partial { {\theta_z}_i} } - \frac{d{L_z}_i}{dt}$$

Since the lagrangian is dependent upon the angles of the object only through the potential, we have: $$\frac{d{L_z}_i}{dt} = \frac{\partial \cal{L}}{\partial { {\theta_z}_i} } = -\frac{\partial V}{\partial { {\theta_z}_i} }$$ which is the torque on the ith object.

Suppose the system is invariant to rotations, so that the potential is independent of an overall rotation by the angle θz (thus it may depend on the angles of objects only through their differences, in the form $$V({\theta_z}_i, {\theta_z}_j) = V({\theta_z}_i - {\theta_z}_j)$$). We therefore get for the total angular momentum:

$$\frac{d L_z}{dt} = -\frac{\partial V}{\partial {\theta_z} } = 0 $$ And thus the angular momentum around the z-axis is conserved.

This analysis can be repeated separately for each axis, giving conversation of the angular momentum vector. However, the angles around the three axes cannot be treated simultaneously as generalized coordinates, since they are not independent; in particular, two angles per point suffice to determine its position. While it is true that in the case of a rigid body, fully describing it requires, in addition to three translational degrees of freedom, also specification of three rotational degrees of freedom; however these cannot be defined as rotations around the Cartesian axes (see Euler angles). This caveat is reflected in quantum mechanics in the non-trivial commutation relations of the different components of the angular momentum operator.

Hamiltonian formalism
Equivalently, in Hamiltonian mechanics the Hamiltonian can be described as a function of the angular momentum. As before, the part of the kinetic energy related to rotation around the z-axis for the ith object is: $$\frac{1}{2} {I_z}_i { {\omega_z}_i}^2 = \frac{ { {L_z}_i}^2}{2{I_z}_i}$$

which is analogous to the energy dependence upon momentum along the z-axis, $$\frac{ { {p_z}_i}^2}{ {2m}_i}$$.

Hamilton's equations relate the angle around the z-axis to its conjugate momentum, the angular momentum around the same axis: $$\begin{align} \frac{d{\theta_z}_i}{dt} &= \frac{\partial \cal{H} }{\partial {L_z}_i} = \frac{ {L_z}_i}{ {I_z}_i} \\ \frac{d{L_z}_i}{dt} &= -\frac{\partial \cal{H} }{\partial {\theta_z}_i} = -\frac{\partial V}{\partial {\theta_z}_i} \end{align}$$

The first equation gives $${L_z}_i = {I_z}_i \cdot { {\dot\theta_z}_i} = {I_z}_i \cdot {\omega_z}_i$$

And so we get the same results as in the Lagrangian formalism.

Note, that for combining all axes together, we write the kinetic energy as: $$ E_k = \frac{1}{2}\sum_i \frac{|\bf{p}_i|^2}{2m_i} = \sum_i \left(\frac{ { {p_r}_i}^2}{2m_i} + \frac{1}{2}{\bf{L}_i}^\textsf{T}{I_i}^{-1} \bf{L}_i\right) $$

where pr is the momentum in the radial direction, and the moment of inertia is a 3-dimensional matrix; bold letters stand for 3-dimensional vectors.

For point-like bodies we have: $$E_k = \sum_i \left(\frac{ { {p_r}_i}^2}{2m_i} + \frac{|{\bf{L}_i}|^2}{2m_i {r_i}^2}\right)$$

This form of the kinetic energy part of the Hamiltonian is useful in analyzing central potential problems, and is easily transformed to a quantum mechanical work frame (e.g. in the hydrogen atom problem).

Angular momentum in orbital mechanics
While in classical mechanics the language of angular momentum can be replaced by Newton's laws of motion, it is particularly useful for motion in central potential such as planetary motion in the solar system. Thus, the orbit of a planet in the solar system is defined by its energy, angular momentum and angles of the orbit major axis relative to a coordinate frame.

In astrodynamics and celestial mechanics, a quantity closely related to angular momentum is defined as $$\mathbf{h} = \mathbf{r} \times \mathbf{v}, $$ called specific angular momentum. Note that $$\mathbf{L} = m\mathbf{h}.$$ Mass is often unimportant in orbital mechanics calculations, because motion of a body is determined by gravity. The primary body of the system is often so much larger than any bodies in motion about it that the gravitational effect of the smaller bodies on it can be neglected; it maintains, in effect, constant velocity. The motion of all bodies is affected by its gravity in the same way, regardless of mass, and therefore all move approximately the same way under the same conditions.

Solid bodies
Angular momentum is also an extremely useful concept for describing rotating rigid bodies such as a gyroscope or a rocky planet. For a continuous mass distribution with density function ρ(r), a differential volume element dV with position vector r within the mass has a mass element dm = ρ(r)dV. Therefore, the infinitesimal angular momentum of this element is:

$$d\mathbf{L} = \mathbf{r}\times dm \mathbf{v} = \mathbf{r}\times \rho(\mathbf{r}) dV \mathbf{v} = dV \mathbf{r}\times \rho(\mathbf{r}) \mathbf{v}$$

and integrating this differential over the volume of the entire mass gives its total angular momentum:

$$\mathbf{L}=\int_V dV \mathbf{r}\times \rho(\mathbf{r}) \mathbf{v}$$

In the derivation which follows, integrals similar to this can replace the sums for the case of continuous mass.

Collection of particles

For a collection of particles in motion about an arbitrary origin, it is informative to develop the equation of angular momentum by resolving their motion into components about their own center of mass and about the origin. Given,
 * $$m_i$$ is the mass of particle $$i$$,
 * $$\mathbf{R}_i$$ is the position vector of particle $$i$$ w.r.t. the origin,
 * $$\mathbf{V}_i$$ is the velocity of particle $$i$$ w.r.t. the origin,
 * $$\mathbf{R}$$ is the position vector of the center of mass w.r.t. the origin,
 * $$\mathbf{V}$$ is the velocity of the center of mass w.r.t. the origin,
 * $$\mathbf{r}_i$$ is the position vector of particle $$i$$ w.r.t. the center of mass,
 * $$\mathbf{v}_i$$ is the velocity of particle $$i$$ w.r.t. the center of mass,

The total mass of the particles is simply their sum, $$M=\sum_i m_i.$$

The position vector of the center of mass is defined by, $$M\mathbf{R}=\sum_i m_i \mathbf{R}_i.$$

By inspection,
 * $$\mathbf{R}_i = \mathbf{R} + \mathbf{r}_i$$ and $$\mathbf{V}_i = \mathbf{V} + \mathbf{v}_i.$$

The total angular momentum of the collection of particles is the sum of the angular momentum of each particle, $$

Expanding $$\mathbf{R}_i$$,
 * $$\begin{align}

\mathbf{L} &= \sum_i \left[\left(\mathbf{R} + \mathbf{r}_i\right) \times m_i\mathbf{V}_i \right] \\ &= \sum_i \left[ \mathbf{R} \times m_i\mathbf{V}_i + \mathbf{r}_i \times m_i\mathbf{V}_i \right] \end{align}$$

Expanding $$\mathbf{V}_i$$,
 * $$\begin{align}

\mathbf{L} &= \sum_i \left[ \mathbf{R} \times m_i\left(\mathbf{V} + \mathbf{v}_i\right) + \mathbf{r}_i \times m_i(\mathbf{V} + \mathbf{v}_i) \right ]\\ &= \sum_i \left[ \mathbf{R} \times m_i\mathbf{V} + \mathbf{R} \times m_i\mathbf{v}_i + \mathbf{r}_i \times m_i\mathbf{V} + \mathbf{r}_i \times m_i\mathbf{v}_i \right]\\ &= \sum_i \mathbf{R} \times m_i\mathbf{V} + \sum_i \mathbf{R} \times m_i\mathbf{v}_i + \sum_i \mathbf{r}_i \times m_i\mathbf{V} + \sum_i \mathbf{r}_i \times m_i\mathbf{v}_i \end{align}$$

It can be shown that (see sidebar),

 * $$\sum_i m_i\mathbf{r}_i = \mathbf{0}$$ and $$\sum_i m_i\mathbf{v}_i = \mathbf{0},$$

therefore the second and third terms vanish,
 * $$\mathbf{L} = \sum_i \mathbf{R} \times m_i\mathbf{V} + \sum_i \mathbf{r}_i \times m_i\mathbf{v}_i .$$

The first term can be rearranged,
 * $$\sum_i \mathbf{R} \times m_i\mathbf{V} = \mathbf{R} \times \sum_i m_i\mathbf{V} = \mathbf{R} \times M\mathbf{V},$$

and total angular momentum for the collection of particles is finally, $$

The first term is the angular momentum of the center of mass relative to the origin. Similar to ', below, it is the angular momentum of one particle of mass M at the center of mass moving with velocity V'''. The second term is the angular momentum of the particles moving relative to the center of mass, similar to , below. The result is general—the motion of the particles is not restricted to rotation or revolution about the origin or center of mass. The particles need not be individual masses, but can be elements of a continuous distribution, such as a solid body.

Rearranging equation ($$) by vector identities, multiplying both terms by "one", and grouping appropriately,

$$\begin{align} \mathbf{L} &= M(\mathbf{R} \times \mathbf{V}) + \sum_i \left[m_i\left(\mathbf{r}_i \times \mathbf{v}_i\right)\right], \\ &= \frac{R^2}{R^2}M\left(\mathbf{R} \times \mathbf{V}\right) + \sum_i \left[ \frac{r_i^2}{r_i^2}m_i\left(\mathbf{r}_i \times \mathbf{v}_i\right)\right], \\ &= R^2M \left( \frac{\mathbf{R} \times \mathbf{V} }{R^2} \right) + \sum_i \left[ r_i^2 m_i \left( \frac{\mathbf{r}_i \times \mathbf{v}_i}{r_i^2} \right) \right], \\ \end{align}$$

gives the total angular momentum of the system of particles in terms of moment of inertia $$I$$ and angular velocity $$\boldsymbol{\omega}$$,

$$

Single particle case
In the case of a single particle moving about the arbitrary origin, $$\begin{align} \mathbf{r}_i &= \mathbf{v}_i = \mathbf{0}, \\ \mathbf{r} &= \mathbf{R}, \\ \mathbf{v} &= \mathbf{V}, \\ m &= M, \end{align}$$ $$\sum_i \mathbf{r}_i \times m_i\mathbf{v}_i = \mathbf{0},$$ $$\sum_i I_i\boldsymbol{\omega}_i = \mathbf{0},$$ and equations ($$) and ($$) for total angular momentum reduce to, $$\mathbf{L} = \mathbf{R} \times m\mathbf{V} = I_R\boldsymbol{\omega}_R.$$

Case of a fixed center of mass
For the case of the center of mass fixed in space with respect to the origin, $$\mathbf{V} = \mathbf{0},$$ $$\mathbf{R} \times M\mathbf{V} = \mathbf{0},$$ $$I_R\boldsymbol{\omega}_R = \mathbf{0},$$ and equations ($$) and ($$) for total angular momentum reduce to, $$\mathbf{L} = \sum_i \mathbf{r}_i \times m_i\mathbf{v}_i = \sum_i I_i\boldsymbol{\omega}_i.$$

Angular momentum in general relativity
In modern (20th century) theoretical physics, angular momentum (not including any intrinsic angular momentum – see below) is described using a different formalism, instead of a classical pseudovector. In this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance. As a result, angular momentum is generally not conserved locally for general curved spacetimes, unless they have rotational symmetry; whereas globally the notion of angular momentum itself only makes sense if the spacetime is asymptotically flat. If the spacetime is only axially symmetric like for the Kerr metric, the total angular momentum is not conserved but $$p_{\phi}$$ is conserved which is related to the invariance of rotating around the symmetry-axis, where note that $$p_{\phi}=g_{\mu \phi}p^{\phi}=mg_{\mu \phi} dX^{\mu}/d\tau$$ where $$g_{\mu\nu}$$ is the metric, $$m=\sqrt{|p_\mu p^{\mu}|}$$ is the rest mass, $$dX^{\mu}/d\tau$$ is the four-velocity, and $$X^{\mu}=(t,r,\theta,\phi)$$ is the four-position in spherical coordinates.

In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element: $$\mathbf{L} = \mathbf{r} \wedge \mathbf{p} \,,$$ in which the exterior product (∧) replaces the cross product (×) (these products have similar characteristics but are nonequivalent). 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 Cartesian coordinates: $$\begin{align} \mathbf{L} &= \left(xp_y - yp_x\right)\mathbf{e}_x \wedge \mathbf{e}_y + \left(yp_z - zp_y\right)\mathbf{e}_y \wedge \mathbf{e}_z + \left(zp_x - xp_z\right)\mathbf{e}_z \wedge \mathbf{e}_x\\ &= L_{xy}\mathbf{e}_x \wedge \mathbf{e}_y + L_{yz}\mathbf{e}_y \wedge \mathbf{e}_z + L_{zx}\mathbf{e}_z \wedge \mathbf{e}_x \,, \end{align}$$ or more compactly in index notation: $$L_{ij} = x_i p_j - x_j p_i\,.$$

The angular velocity can also be defined as an anti-symmetric second order tensor, with components ωij. The relation between the two anti-symmetric tensors is given by the moment of inertia which must now be a fourth order tensor: $$L_{ij} = I_{ijk\ell} \omega_{k\ell} \,. $$

Again, this equation in L and ω as tensors is true in any number of dimensions. This equation also appears in the geometric algebra formalism, in which L and ω are bivectors, and the moment of inertia is a mapping between them.

In relativistic mechanics, the relativistic angular momentum of a particle is expressed as an anti-symmetric tensor of second order: $$M_{\alpha\beta} = X_\alpha P_\beta - X_\beta P_\alpha$$ in terms of four-vectors, namely the four-position X and the four-momentum P, and absorbs the above L together with the moment of mass, i.e., the product of the relativistic mass of the particle and its centre of mass, which can be thought of as describing the motion of its centre of mass, since mass–energy is conserved.

In each of the above cases, for a system of particles the total angular momentum is just the sum of the individual particle angular momenta, and the centre of mass is for the system.

Angular momentum in quantum mechanics
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. Because of this, the axis of rotation of a quantum particle is undefined. Quantum particles do possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to a spinning motion. In relativistic quantum mechanics the above relativistic definition becomes a tensorial operator.

Spin, orbital, and total angular momentum
[[File:Classical angular momentum.svg|upright=1.25|thumb|Angular momenta of a classical object.Left: "spin" angular momentum S is really orbital angular momentum of the object at every point.

Right: extrinsic orbital angular momentum L about an axis.

Top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω).

Bottom: momentum p and its radial position r from the axis. The total angular momentum (spin plus orbital) is J. For a quantum particle the interpretations are different; particle spin does not have the above interpretation. ]]

The classical definition of angular momentum as $$\mathbf{L} = \mathbf{r}\times\mathbf{p}$$ can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator. L is then an operator, specifically called the orbital angular momentum operator. The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum Hilbert space. (See also the discussion below of the angular momentum operators as the generators of rotations.)

However, in quantum physics, there is another type of angular momentum, called spin angular momentum, represented by the spin operator S. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. All elementary particles have a characteristic spin (possibly zero), and almost all elementary particles have nonzero spin. For example electrons have "spin 1/2" (this actually means "spin ħ/2"), photons have "spin 1" (this actually means "spin ħ"), and pi-mesons have spin 0.

Finally, there is total angular momentum J, which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, .) Conservation of angular momentum applies to J, but not to L or S; for example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have half-integer values.

In molecules the total angular momentum F is the sum of the rovibronic (orbital) angular momentum N, the electron spin angular momentum S, and the nuclear spin angular momentum I. For electronic singlet states the rovibronic angular momentum is denoted J rather than N. As explained by Van Vleck, the components of the molecular rovibronic angular momentum referred to molecule-fixed axes have different commutation relations from those for the components about space-fixed axes.

Quantization
In quantum mechanics, angular momentum is quantized – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where $$\hbar$$ is the reduced Planck constant and $$\hat n$$ is any Euclidean vector such as x, y, or z: The reduced Planck constant $$\hbar$$ is tiny by everyday standards, about 10−34 J s, and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of electron shells and subshells in chemistry is significantly affected by the quantization of angular momentum.

Quantization of angular momentum was first postulated by Niels Bohr in his model of the atom and was later predicted by Erwin Schrödinger in his Schrödinger equation.

Uncertainty
In the definition $$\mathbf{L}=\mathbf{r}\times\mathbf{p}$$, six operators are involved: The position operators $$r_x$$, $$r_y$$, $$r_z$$, and the momentum operators $$p_x$$, $$p_y$$, $$p_z$$. However, the Heisenberg uncertainty principle tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.

The uncertainty is closely related to the fact that different components of an angular momentum operator do not commute, for example $$L_xL_y \neq L_yL_x$$. (For the precise commutation relations, see angular momentum operator.)

Total angular momentum as generator of rotations
As mentioned above, orbital angular momentum L is defined as in classical mechanics: $$\mathbf{L}=\mathbf{r}\times\mathbf{p}$$, but total angular momentum J is defined in a different, more basic way: J is defined as the "generator of rotations". More specifically, J is defined so that the operator $$R(\hat{n},\phi) \equiv \exp\left(-\frac{i}{\hbar}\phi\, \mathbf{J}\cdot \hat{\mathbf{n} }\right)$$ is the rotation operator that takes any system and rotates it by angle $$\phi$$ about the axis $$\hat{\mathbf{n}}$$. (The "exp" in the formula refers to operator exponential.) To put this the other way around, whatever our quantum Hilbert space is, we expect that the rotation group SO(3) will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators.

The relationship between the angular momentum operator and the rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics. The close relationship between angular momentum and rotations is reflected in Noether's theorem that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.

Angular momentum in electrodynamics
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. As a consequence, the canonical angular momentum L = r × P is not gauge invariant either. Instead, the momentum that is physical, the so-called kinetic momentum (used throughout this article), is (in SI units)

$$ \mathbf{p} = m\mathbf{v} = \mathbf{P} - e \mathbf{A} $$

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

$$\mathbf{K}= \mathbf{r} \times ( \mathbf{P} - e\mathbf{A} )$$

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.

$$\mathbf{S}(\mathbf{r}, t) = \epsilon_0 c^2 \mathbf{E}(\mathbf{r}, t) \times \mathbf{B}(\mathbf{r}, t). $$

The angular momentum density vector $$\mathbf{L}(\mathbf{r}, t)$$ is given by a vector product as in classical mechanics:

$$\mathbf{L}(\mathbf{r}, t) = \epsilon_0 \mu_0 \mathbf{r} \times \mathbf{S}(\mathbf{r}, t). $$

The above identities are valid locally, i.e. in each space point $$\mathbf{r}$$ in a given moment $$t$$.

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 (~3.8 cm per year) and Earth to be decelerated (by −25.858 ± 0.003″/cy²) in its rotation (the length of the day increases by ~1.7 ms per century, +2.3 ms from tidal effect and −0.6 ms from post-glacial rebound). 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, but are also crucial to all forms of modern navigation.

Rifled bullets use the stability provided by conservation of angular momentum to be more true in their trajectory. The invention of rifled firearms and cannons gave their users significant strategic advantage in battle, and thus were a technological turning point in history.

History
Isaac Newton, in the Principia, hinted at angular momentum in his examples of the first law of motion, A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. 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. 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 centres. But these are cases which I do not consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject."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's derivation

As a planet orbits the Sun, the line between the Sun and the planet sweeps out equal areas in equal intervals of time. This had been known since Kepler expounded his second law of planetary motion. 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.

During the first interval of time, an object is in motion from point A to point B. Undisturbed, it would continue to point c during the second interval. When the object arrives at B, it receives an impulse directed toward point S. The impulse gives it a small added velocity toward S, such that if this were its only velocity, it would move from B to V during the second interval. By the rules of velocity composition, these two velocities add, and point C is found by construction of parallelogram BcCV. Thus the object's path is deflected by the impulse so that it arrives at point C at the end of the second interval. Because the triangles SBc and SBC have the same base SB and the same height Bc or VC, they have the same area. By symmetry, triangle SBc also has the same area as triangle SAB, therefore the object has swept out equal areas SAB and SBC in equal times.

At point C, the object receives another impulse toward S, again deflecting its path during the third interval from d to D. Thus it continues to E and beyond, the triangles SAB, SBc, SBC, SCd, SCD, SDe, SDE all having the same area. Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve.

Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero.

Conservation of angular momentum in the Law of Areas
The proportionality of angular momentum to the area swept out by a moving object can be understood by realizing that the bases of the triangles, that is, the lines from S to the object, are equivalent to the radius $p = mv$, and that the heights of the triangles are proportional to the perpendicular component of velocity $(length of moment arm) × (linear momentum)$. Hence, if the area swept per unit time is constant, then by the triangular area formula $r$, the product $v _{⊥}$ and therefore the product $1⁄2(base)(height)$ are constant: if $(base)(height)$ and the base length are decreased, $rv _{⊥}$ and height must increase proportionally. Mass is constant, therefore angular momentum $r$ is conserved by this exchange of distance and velocity.

In the case of triangle SBC, area is equal to $1⁄2$(SB)(VC). Wherever C is eventually located due to the impulse applied at B, the product (SB)(VC), and therefore $v _{⊥}$ remain constant. Similarly so for each of the triangles.

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

After Newton
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. It is unlikely that they realized the implications for ordinary rotating matter.

In 1736 Euler, like Newton, touched on some of the equations of angular momentum in his Mechanica without further developing them.

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

In 1799, Pierre-Simon Laplace first realized that a fixed plane was associated with rotation—his invariable plane.

Louis Poinsot in 1803 began representing rotations as a line segment perpendicular to the rotation, and elaborated on the "conservation of moments".

In 1852 Léon Foucault used a gyroscope in an experiment to display the Earth's rotation.

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," 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, which was introduced in 1856, and published in 1864. Rankine was mistaken, as numerous publications feature the term starting in the late 18th to early 19th centuries. However, Hayward's article apparently was the first use of the term and the concept seen by much of the English-speaking world. Before this, angular momentum was typically referred to as "momentum of rotation" in English.

title: Angular momentum