document_title: Binomial nomenclature

question_text: who simplified the naming of living things by giving each species a two-part scientific name

text: Binomial nomenclature

text: Origin[edit]

text: The name split into two parts means "Bi" a Latin prefix for two, and "Nomial" which is a noun commonly used in mathematics, algebra to define a name or term.

text: General[edit]

text: Binomial nomenclature (also called binominal nomenclature or binary nomenclature) is a formal system of naming species of living things by giving each a name composed of two parts, both of which use Latin grammatical forms, although they can be based on words from other languages. Such a name is called a binomial name (which may be shortened to just "binomial"), a binomen, binominal name or a scientific name; more informally it is also called a Latin name. The first part of the name identifies the genus to which the species belongs; the second part identifies the species within the genus. For example, humans belong to the genus Homo and within this genus to the species Homo sapiens. The formal introduction of this system of naming species is credited to Carl Linnaeus, effectively beginning with his work Species Plantarum in 1753.[1] But Gaspard Bauhin, in as early as 1623, had introduced in his book Pinax theatri botanici (English, Illustrated exposition of plants) many names of genera that were later adopted by Linnaeus.[2]

text: The application of binomial nomenclature is now governed by various internationally agreed codes of rules, of which the two most important are the International Code of Zoological Nomenclature (ICZN) for animals and the International Code of Nomenclature for algae, fungi, and plants (ICN). Although the general principles underlying binomial nomenclature are common to these two codes, there are some differences, both in the terminology they use and in their precise rules.

text: In modern usage, the first letter of the first part of the name, the genus, is always capitalized in writing, while that of the second part is not, even when derived from a proper noun such as the name of a person or place. Similarly, both parts are italicized when a binomial name occurs in normal text (or underlined in handwriting). Thus the binomial name of the annual phlox (named after botanist Thomas Drummond) is now written as Phlox drummondii.

text: In scientific works, the "authority" for a binomial name is usually given, at least when it is first mentioned, and the date of publication may be specified.

text:
In zoology

"Patella vulgata Linnaeus, 1758". The name "Linnaeus" tells the reader who it was that first published a description and name for this species of limpet; 1758 is the date of the publication in which the original description can be found (in this case the 10th edition of the book Systema Naturae).
"Passer domesticus (Linnaeus, 1758)". The original name given by Linnaeus was Fringilla domestica; the parentheses indicate that the species is now considered to belong in a different genus. The ICZN does not require that the name of the person who changed the genus be given, nor the date on which the change was made, although nomenclatorial catalogs usually include such information.

In botany

"Amaranthus retroflexus L." – "L." is the standard abbreviation used in botany for "Linnaeus".
"Hyacinthoides italica (L.) Rothm. – Linnaeus first named this bluebell species Scilla italica; Rothmaler transferred it to the genus Hyacinthoides; the ICN does not require that the dates of either publication be specified.

text: History[edit]

text: Prior to the adoption of the modern binomial system of naming species, a scientific name consisted of a generic name combined with a specific name that was from one to several words long. Together they formed a system of polynomial nomenclature.[3] These names had two separate functions. First, to designate or label the species, and second, to be a diagnosis or description; however these two goals were eventually found to be incompatible.[4] In a simple genus, containing only two species, it was easy to tell them apart with a one-word genus and a one-word specific name; but as more species were discovered the names necessarily became longer and unwieldy, for instance Plantago foliis ovato-lanceolatus pubescentibus, spica cylindrica, scapo tereti ("Plantain with pubescent ovate-lanceolate leaves, a cylindric spike and a terete scape"), which we know today as Plantago media.

text: Such "polynomial names" may sometimes look like binomials, but are significantly different. For example, Gerard's herbal (as amended by Johnson) describes various kinds of spiderwort: "The first is called Phalangium ramosum, Branched Spiderwort; the second, Phalangium non ramosum, Unbranched Spiderwort. The other ... is aptly termed Phalangium Ephemerum Virginianum, Soon-Fading Spiderwort of Virginia".[5] The Latin phrases are short descriptions, rather than identifying labels.

text: The Bauhins, in particular Caspar Bauhin (1560–1624), took some important steps towards the binomial system, by pruning the Latin descriptions, in many cases to two words.[6] The adoption by biologists of a system of strictly binomial nomenclature is due to Swedish botanist and physician Carl von Linné, more commonly known by his Latinized name Carl Linnaeus (1707–1778). It was in his 1753 Species Plantarum that he first began consistently using a one-word "trivial name" together with a generic name in a system of binomial nomenclature.[7] This trivial name is what is now known as a specific epithet (ICN) or specific name (ICZN).[7] The Bauhins' genus names were retained in many of these, but the descriptive part was reduced to a single word.

text: Linnaeus's trivial names introduced an important new idea, namely that the function of a name could simply be to give a species a unique label. This meant that the name no longer need be descriptive; for example both parts could be derived from the names of people. Thus Gerard's phalangium ephemerum virginianum became Tradescantia virginiana, where the genus name honoured John Tradescant the younger,[note 1] an English botanist and gardener.[8] A bird in the parrot family was named Psittacus alexandri, meaning "Alexander's parrot", after Alexander the Great whose armies introduced eastern parakeets to Greece.[9] Linnaeus' trivial names were much easier to remember and use than the parallel polynomial names and eventually replaced them.[1]

text: Value[edit]

text: The value of the binomial nomenclature system derives primarily from its economy, its widespread use, and the uniqueness and stability of names it generally favors:

text:
Economy. Compared to the polynomial system which it replaced, a binomial name is shorter and easier to remember.[1] It corresponds to the widespread system of family name plus given name(s) used to name people in many cultures.[6]
Widespread use. The binomial system of nomenclature is governed by international codes and is used by biologists worldwide.[10] A few binomials have also entered common speech, such as Homo sapiens, E. coli, and Tyrannosaurus rex.
Uniqueness. Provided that taxonomists agree as to the limits of a species, it can have only one name that is correct under the appropriate nomenclature code, generally the earliest published if two or more names are accidentally assigned to a species.[11] However, establishing that two names actually refer to the same species and then determining which has priority can be difficult, particularly if the species was named by biologists from different countries. Therefore, a species may have more than one regularly used name; all but one of these names are "synonyms".[12]
Stability. Although stability is far from absolute, the procedures associated with establishing binomial names, such as the principle of priority, tend to favor stability.[13] For example, when species are transferred between genera (as not uncommonly happens as a result of new knowledge), if possible the second part of the binomial is kept the same. Thus there is disagreement among botanists as to whether the genera Chionodoxa and Scilla are sufficiently different for them to be kept separate. Those who keep them separate give the plant commonly grown in gardens in Europe the name Chionodoxa siehei; those who do not give it the name Scilla siehei.[14] The siehei element is constant. Similarly if what were previously thought to be two distinct species are demoted to a lower rank, such as subspecies, where possible the second part of the binomial name is retained as the third part of the new name. Thus the Tenerife robin may be treated as a different species from the European robin, in which case its name is Erithacus superbus, or as only a subspecies, in which case its name is Erithacus rubecula superbus.[15] The superbus element of the name is constant.

text: Problems[edit]

text: Binomial nomenclature for species has the effect that when a species is moved from one genus to another, sometimes the specific name or epithet must be changed as well. This may happen because the specific name is already used in the new genus, or to agree in gender with the new genus. Some biologists have argued for the combination of the genus name and specific epithet into a single unambiguous name, or for the use of uninomials (as used in nomenclature of ranks above species).[16]

text: Because binomials are unique only within a kingdom, it is possible for two or more species to share the same binomial if they occur in different kingdoms. At least five instances of such binomial duplication occur.[17]

text: Relationship to classification and taxonomy[edit]

text: Nomenclature (including binomial nomenclature) is not the same as classification, although the two are related. Classification is the ordering of items into groups based on similarities or differences; in biological classification, species are one of the kinds of item to be classified.[18] In principle, the names given to species could be completely independent of their classification. This is not the case for binomial names, since the first part of a binomial is the name of the genus into which the species is placed. Above the rank of genus, binomial nomenclature and classification are partly independent; for example, a species retains its binomial name if it is moved from one family to another or from one order to another, unless it better fits a different genus in the same or different family, or it is split from its old genus and placed in a newly created genus. The independence is only partial since the names of families and other higher taxa are usually based on genera.[citation needed]

text: Taxonomy includes both nomenclature and classification. Its first stages (sometimes called "alpha taxonomy") are concerned with finding, describing and naming species of living or fossil organisms.[19] Binomial nomenclature is thus an important part of taxonomy as it is the system by which species are named. Taxonomists are also concerned with classification, including its principles, procedures and rules.[20]

text: Derivation of binomial names[edit]

text: A complete binomial name is always treated grammatically as if it were a phrase in the Latin language (hence the common use of the term "Latin name" for a binomial name). However, the two parts of a binomial name can each be derived from a number of sources, of which Latin is only one. These include:

text:
Latin, either classical or medieval. Thus, both parts of the binomial name Homo sapiens are Latin words, meaning "wise" (sapiens) "human/man" (Homo).
Classical Greek. The genus Rhododendron was named by Linnaeus from the Greek word ῥοδόδενδρον, itself derived from rhodon, "rose", and dendron, "tree".[21][22] Greek words are often converted to a Latinized form. Thus coca (the plant from which cocaine is obtained) has the name Erythroxylum coca. Erythroxylum is derived from the Greek words erythros, red, and xylon, wood.[23] The Greek neuter ending -ον (-on) is often converted to the Latin neuter ending -um.[note 2]
Other languages. The second part of the name Erythroxylum coca is derived from kuka, the name of the plant in Aymara and Quechua.[24][25] Since many dinosaur fossils were found in Mongolia, their names often use Mongolian words, e.g. Tarchia from tarkhi, meaning "brain", or Saichania meaning "beautiful one".
Names of people (often naturalists or biologists). The name Magnolia campbellii commemorates two people: Pierre Magnol, a French botanist, and Archibald Campbell, a doctor in British India.[26]
Names of places. The lone star tick, Amblyomma americanum, is widespread in the United States.[27]
Other sources. Some binominal names have been constructed from anagrams or other re-orderings of existing names. Thus the name of the genus Muilla is derived by reversing the name Allium.[28] Names may also be derived from jokes or puns. For example, Ratcliffe described a number of species of Rhinoceros beetle, including Cyclocephala nodanotherwon.[29]

text: The first part of the name, which identifies the genus, must be a word which can be treated as a Latin singular noun in the nominative case. It must be unique within each kingdom, but can be repeated between kingdoms. Thus Huia recurvata is an extinct species of plant, found as fossils in Yunnan, China,[30] whereas Huia masonii is a species of frog found in Java, Indonesia.[31]

text: The second part of the name, which identifies the species within the genus, is also treated grammatically as a Latin word. It can have one of a number of forms.

text:
The second part of a binomial may be an adjective. The adjective must agree with the genus name in gender. Latin has three genders, masculine, feminine and neuter, shown by varying endings to nouns and adjectives. The house sparrow has the binomial name Passer domesticus. Here domesticus ("domestic") simply means "associated with the house". The sacred bamboo is Nandina domestica[32] rather than Nandina domesticus, since Nandina is feminine whereas Passer is masculine. The tropical fruit langsat is a product of the plant Lansium parasiticum, since Lansium is neuter. Some common endings for Latin adjectives in the three genders (masculine, feminine, neuter) are -us, -a, -um (as in the previous example of domesticus); -is, -is, -e (e.g. tristis, meaning "sad"); and -or, -or, -us (e.g. minor, meaning "smaller"). For further information, see Latin declension: Adjectives.
The second part of a binomial may be a noun in the nominative case. An example is the binomial name of the lion, which is Panthera leo. Grammatically the noun is said to be in apposition to the genus name and the two nouns do not have to agree in gender; in this case, Panthera is feminine and leo is masculine.
The second part of a binomial may be a noun in the genitive (possessive) case. The genitive case is constructed in a number of ways in Latin, depending on the declension of the noun. Common endings for masculine and neuter nouns are -ii or -i in the singular and -orum in the plural, and for feminine nouns -ae in the singular and -arum in the plural. The noun may be part of a person's name, often the surname, as in the Tibetan antelope Pantholops hodgsonii, the shrub Magnolia hodgsonii, or the olive-backed pipit Anthus hodgsoni. The meaning is "of the person named", so that Magnolia hodgsonii means "Hodgson's magnolia". The -ii or -i endings show that in each case Hodgson was a man (not the same one); had Hodgson been a woman, hodgsonae would have been used. The person commemorated in the binomial name is not usually (if ever) the person who created the name; for example Anthus hodgsoni was named by Charles Wallace Richmond, in honour of Hodgson. Rather than a person, the noun may be related to a place, as with Latimeria chalumnae, meaning "of the Chalumna River". Another use of genitive nouns is in, for example, the name of the bacterium Escherichia coli, where coli means "of the colon". This formation is common in parasites, as in Xenos vesparum, where vesparum means "of the wasps", since Xenos vesparum is a parasite of wasps.

text: Whereas the first part of a binomial name must be unique within a kingdom, the second part is quite commonly used in two or more genera (as is shown by examples of hodgsonii above). The full binomial name must be unique within a kingdom.

text: Codes[edit]

text: From the early 19th century onwards it became ever more apparent that a body of rules was necessary to govern scientific names. In the course of time these became nomenclature codes. The International Code of Zoological Nomenclature (ICZN) governs the naming of animals,[33] the International Code of Nomenclature for algae, fungi, and plants (ICN) that of plants (including cyanobacteria), and the International Code of Nomenclature of Bacteria (ICNB) that of bacteria (including Archaea). Virus names are governed by the International Committee on Taxonomy of Viruses (ICTV), a taxonomic code, which determines taxa as well as names. These codes differ in certain ways, e.g.:

text:
"Binomial nomenclature" is the correct term for botany,[34] although it is also used by zoologists.[35] Since 1953, "binominal nomenclature" is the technically correct term in zoology. A binominal name is also called a binomen (plural binomina).[36]
Both codes consider the first part of the two-part name for a species to be the "generic name". In the zoological code (ICZN), the second part of the name is a "specific name". In the botanical code (ICN), it is a "specific epithet". Together, these two parts are referred to as a "species name" or "binomen" in the zoological code; or "species name", "binomial", or "binary combination" in the botanical code. "Species name" is the only term common to the two codes.
The ICN, the plant Code, does not allow the two parts of a binomial name to be the same (such a name is called a tautonym), whereas the ICZN, the animal Code, does. Thus the American bison has the binomial Bison bison; a name of this kind would not be allowed for a plant.
The starting points, the time from which these codes are in effect (retroactively), vary from group to group. In botany the starting point will often be in 1753 (the year Carl Linnaeus first published Species Plantarum). In zoology the starting point is 1758 (1 January 1758 is considered the date of the publication of Linnaeus's Systema Naturae, 10th Edition, and also Clerck's Aranei Svecici). Bacteriology started anew, with a starting point on 1 January 1980.[37]

text:
Summary of terminology for the names of species in the ICZN and ICN

Code
Full name
First part
Second part

ICZN
species name, binomen, binominal name
generic name, genus name
specific name

ICN
species name, binary combination, binomial (name)
generic name
specific epithet

text: Unifying the different codes into a single code, the "BioCode", has been suggested, although implementation is not in sight. (There is also a code in development for a different system of classification which does not use ranks, but instead names clades. This is called the PhyloCode.)

text: Writing binomial names[edit]

text: By tradition, the binomial names of species are usually typeset in italics; for example, Homo sapiens.[38] Generally, the binomial should be printed in a font style different from that used in the normal text; for example, "Several more Homo sapiens fossils were discovered." When handwritten, each part of a binomial name should be underlined; for example, Homo sapiens.[39]

text: The first part of the binomial, the genus name, is always written with an initial capital letter. In current usage, the second part is never written with an initial capital.[40][41] Older sources, particularly botanical works published before the 1950s, use a different convention. If the second part of the name is derived from a proper noun, e.g. the name of a person or place, a capital letter was used. Thus the modern form Berberis darwinii was written as Berberis Darwinii. A capital was also used when the name is formed by two nouns in apposition, e.g. Panthera Leo or Centaurea Cyanus.[42][note 3]

text: When used with a common name, the scientific name often follows in parentheses, although this varies with publication.[43] For example, "The house sparrow (Passer domesticus) is decreasing in Europe."

text: The binomial name should generally be written in full. The exception to this is when several species from the same genus are being listed or discussed in the same paper or report, or the same species is mentioned repeatedly; in which case the genus is written in full when it is first used, but may then be abbreviated to an initial (and a period/full stop).[44] For example, a list of members of the genus Canis might be written as "Canis lupus, C. aureus, C. simensis". In rare cases, this abbreviated form has spread to more general use; for example, the bacterium Escherichia coli is often referred to as just E. coli, and Tyrannosaurus rex is perhaps even better known simply as T. rex, these two both often appearing in this form in popular writing even where the full genus name has not already been given.

text: The abbreviation "sp." is used when the actual specific name cannot or need not be specified. The abbreviation "spp." (plural) indicates "several species". These abbreviations are not italicised (or underlined).[45] For example: "Canis sp." means "an unspecified species of the genus Canis", while "Canis spp." means "two or more species of the genus Canis". (The abbreviations "sp." and "spp." can easily be confused with the abbreviations "ssp." (zoology) or "subsp." (botany), plurals "sspp." or "subspp.", referring to one or more subspecies. See trinomen (zoology) and infraspecific name.)

text: The abbreviation "cf." (i.e. confer in Latin) is used to compare individuals/taxa with known/described species. Conventions for use of the "cf." qualifier vary.[46] In paleontology, it is typically used when the identification is not confirmed.[47] For example, "Corvus cf. nasicus" was used to indicate "a fossil bird similar to the Cuban crow but not certainly identified as this species".[48] In molecular systematics papers, "cf." may be used to indicate one or more undescribed species assumed related to a described species. For example, in a paper describing the phylogeny of small benthic freshwater fish called darters, five undescribed putative species (Ozark, Sheltowee, Wildcat, Ihiyo, and Mamequit darters), notable for brightly colored nuptial males with distinctive color patterns,[49] were referred to as "Etheostoma cf. spectabile" because they had been viewed as related to, but distinct from, Etheostoma spectabile (orangethroat darter).[50] This view was supported in varying degrees by DNA analysis. The somewhat informal use of taxa names with qualifying abbreviations is referred to as open nomenclature and it is not subject to strict usage codes.

text: In some contexts the dagger symbol ("†") may be used before or after the binomial name to indicate that the species is extinct.

text: Authority[edit]

text: In scholarly texts, at least the first or main use of the binomial name is usually followed by the "authority" – a way of designating the scientist(s) who first published the name. The authority is written in slightly different ways in zoology and botany. For names governed by the ICZN the surname is usually written in full together with the date (normally only the year) of publication. The ICZN recommends that the "original author and date of a name should be cited at least once in each work dealing with the taxon denoted by that name."[51] For names governed by the ICN the name is generally reduced to a standard abbreviation and the date omitted. The International Plant Names Index maintains an approved list of botanical author abbreviations. Historically, abbreviations were used in zoology too.

text: When the original name is changed, e.g. the species is moved to a different genus, both Codes use parentheses around the original authority; the ICN also requires the person who made the change to be given. Some examples:

text:
(Plant) Amaranthus retroflexus L. – "L." is the standard abbreviation for "Linnaeus"; the absence of parentheses shows that this is his original name.
(Plant) Hyacinthoides italica (L.) Rothm. – Linnaeus first named the Italian bluebell Scilla italica; Rothmaler transferred it to the genus Hyacinthoides.
(Animal) Passer domesticus (Linnaeus, 1758) – the original name given by Linnaeus was Fringilla domestica; unlike the ICN, the ICZN does not require the name of the person who changed the genus to be given.

text: Other ranks[edit]

text: This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2011) (Learn how and when to remove this template message)

text: Binomial nomenclature, as described here, is a system for naming species. Implicitly it includes a system for naming genera, since the first part of the name of the species is a genus name. In a classification system based on ranks there are also ways of naming ranks above the level of genus and below the level of species. Ranks above genus (e.g., family, order, class) receive one-part names, which are conventionally not written in italics. Thus the house sparrow, Passer domesticus, belongs to the family Passeridae. Family names are normally based on genus names, although the endings used differ between zoology and botany.

text: Ranks below species receive three-part names, conventionally written in italics like the names of species. There are significant differences between the ICZN and the ICN. In zoology, the only rank below species is subspecies and the name is written simply as three parts (a trinomen). Thus one of the subspecies of the olive-backed pipit is Anthus hodgsoni berezowskii. In botany, there are many ranks below species and although the name itself is written in three parts, a "connecting term" (not part of the name) is needed to show the rank. Thus the American black elder is Sambucus nigra subsp. canadensis; the white-flowered form of the ivy-leaved cyclamen is Cyclamen hederifolium f. albiflorum.


document_title: Guillén

question_text: where does the last name guillen come from

root

text: Guillén

text: Guillén or Guillen is a Spanish or French surname. Notable people with the surname include:

tag: p

text:
Ambrosio Guillen (1929–1953), United States Marine who was posthumously awarded the Medal of Honor
Carlos Guillén, retired Venezuelan baseball player
Fernando Guillén (disambiguation), multiple people
Jorge Guillén, Spanish poet
José Guillén (born 1976), Dominican baseball player
Laurice Guillén (born 1947), Filipina actress
Rafael Guillen ("Subcomandante Marcos"), spokesman for the Zapatista Army of National Liberation
Nancy Guillén (born 1976), retired Salvadoran hammer thrower
Néstor Guillén Olmos (1890–1966), President of Bolivia for 27 days
Nicolás Guillén, Afro-Cuban poet
Ozzie Guillén (born 1964), Venezuelan baseball coach and former player
Ricardo Guillén (born 1967), Spanish basketball player
Sammy Guillen (1924–2013), West Indian and New Zealand cricket player
Sara Sotillo (maternal name Guillén, 1900–1961), Panamanian academic and activist

tag: ul

text: This page lists people with the surname Guillén. If an internal link intending to refer to a specific person led you to this page, you may wish to change that link by adding the person's given name(s) to the link.

tag: table

text: Surnames

text: All set index articles


document_title: Momentum

question_text: a quantity 15 m/s to the north is a measure of

question_tokens: a quantity 15 m / s to the north is a measure of

text: Momentum

text: Newtonian[edit]

text: Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations (see multiple dimensions).

text: Single particle[edit]

text: The momentum of a particle is conventionally represented by the letter p. It is the product of two quantities, the particle's mass (represented by the letter m) and its velocity (v):[1]

text: The unit of momentum is the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second then the momentum is in kilogram meters per second (kg⋅m/s). In cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters per second (g⋅cm/s).

text: Being a vector, momentum has magnitude and direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground.

text: Many particles[edit]

text: The momentum of a system of particles is the sum of their momenta. If two particles have respective masses m1 and m2, and velocities v1 and v2, the total momentum is

text: The momenta of more than two particles can be added more generally with the following:

text: A system of particles has a center of mass, a point determined by the weighted sum of their positions:

text: If all the particles are moving, the center of mass will generally be moving as well (unless the system is in pure rotation around it). If the center of mass is moving at velocity vcm, the momentum is:

text: This is known as Euler's first law.[2][3]

text: Relation to force[edit]

text: If the net force applied to a particle is a constant F, and is applied for a time interval Δt, the momentum of the particle changes by an amount

text: In differential form, this is Newton's second law; the rate of change of the momentum of a particle is equal to the instantaneous force F acting on it,[1]

text: If the net force experienced by a particle changes as a function of time, F(t), the change in momentum (or impulse J ) between times t1 and t2  is

text: Impulse is measured in the derived units of the newton second (1 N⋅s = 1 kg⋅m/s) or dyne second (1 dyne⋅s = 1 g⋅m/s)

text: Under the assumption of constant mass m, it is equivalent to write

text: hence the net force is equal to the mass of the particle, times its acceleration.[1]

text: Example: A model airplane of mass 1 kg accelerates from rest to a velocity of 6 m/s due north in 2 s. The net force required to produce this acceleration is 3 newtons due north. The change in momentum is 6 kg⋅m/s. The rate of change of momentum is 3 (kg⋅m/s)/s = 3 N.

text: Conservation[edit]

text: In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum is constant. This fact, known as the law of conservation of momentum, is implied by Newton's laws of motion.[4][5] Suppose, for example, that two particles interact. Because of the third law, the forces between them are equal and opposite. If the particles are numbered 1 and 2, the second law states that F1 = dp1/dt and F2 = dp2/dt. Therefore,

text: with the negative sign indicating that the forces oppose. Equivalently,

text: If the velocities of the particles are u1 and u2 before the interaction, and afterwards they are v1 and v2, then

text: This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero. This conservation law applies to all interactions, including collisions and separations caused by explosive forces.[4] It can also be generalized to situations where Newton's laws do not hold, for example in the theory of relativity and in electrodynamics.[6]

text: Dependence on reference frame[edit]

text: Momentum is a measurable quantity, and the measurement depends on the motion of the observer. For example: if an apple is sitting in a glass elevator that is descending, an outside observer, looking into the elevator, sees the apple moving, so, to that observer, the apple has a non-zero momentum. To someone inside the elevator, the apple does not move, so, it has zero momentum. The two observers each have a frame of reference, in which, they observe motions, and, if the elevator is descending steadily, they will see behavior that is consistent with those same physical laws.

text: Suppose a particle has position x in a stationary frame of reference. From the point of view of another frame of reference, moving at a uniform speed u, the position (represented by a primed coordinate) changes with time as

text: This is called a Galilean transformation. If the particle is moving at speed dx/dt = v in the first frame of reference, in the second, it is moving at speed

text: Since u does not change, the accelerations are the same:

text: Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form, in both frames, Newton's second law is unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of reference frame is called Newtonian relativity or Galilean invariance.[7]

text:
A change of reference frame, can, often, simplify calculations of motion. For example, in a collision of two particles, a reference frame can be chosen, where, one particle begins at rest. Another, commonly used reference frame, is the center of mass frame – one that is moving with the center of mass. In this frame,
the total momentum is zero.

text: Application to collisions[edit]

text: By itself, the law of conservation of momentum is not enough to determine the motion of particles after a collision. Another property of the motion, kinetic energy, must be known. This is not necessarily conserved. If it is conserved, the collision is called an elastic collision; if not, it is an inelastic collision.

text: Elastic collisions[edit]

text: An elastic collision is one in which no kinetic energy is absorbed in the collision. Perfectly elastic "collisions" can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps them apart. A slingshot maneuver of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two pool balls is a good example of an almost totally elastic collision, due to their high rigidity, but when bodies come in contact there is always some dissipation.[8]

text: A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are u1 and u2 before the collision and v1 and v2 after, the equations expressing conservation of momentum and kinetic energy are:

text: A change of reference frame can simplify analysis of a collision. For example, suppose there are two bodies of equal mass m, one stationary and one approaching the other at a speed v (as in the figure). The center of mass is moving at speed v/2 and both bodies are moving towards it at speed v/2. Because of the symmetry, after the collision both must be moving away from the center of mass at the same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed v. The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion. Therefore, the final velocities are given by[4]

text: In general, when the initial velocities are known, the final velocities are given by[9]

text: If one body has much greater mass than the other, its velocity will be little affected by a collision while the other body will experience a large change.

text: Inelastic collisions[edit]

text: In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as heat or sound). Examples include traffic collisions,[10] in which the effect of lost kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms (as in the Franck–Hertz experiment);[11] and particle accelerators in which the kinetic energy is converted into mass in the form of new particles.

text: In a perfectly inelastic collision (such as a bug hitting a windshield), both bodies have the same motion afterwards. If one body is motionless to begin with, the equation for conservation of momentum is

text: so

text: In a frame of reference moving at the speed v), the objects are brought to rest by the collision and 100% of the kinetic energy is converted to other forms of energy.

text: One measure of the inelasticity of the collision is the coefficient of restitution CR, defined as the ratio of relative velocity of separation to relative velocity of approach. In applying this measure to a ball bouncing from a solid surface, this can be easily measured using the following formula:[12]

text: The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an explosion is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation. Rockets also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket.[13]

text: Multiple dimensions[edit]

text: Real motion has both direction and velocity and must be represented by a vector. In a coordinate system with x, y, z axes, velocity has components vx in the x-direction, vy in the y-direction, vz in the z-direction. The vector is represented by a boldface symbol:[14]

text: Similarly, the momentum is a vector quantity and is represented by a boldface symbol:

text: The equations in the previous sections, work in vector form if the scalars p and v are replaced by vectors p and v. Each vector equation represents three scalar equations. For example,

text: represents three equations:[14]

text: The kinetic energy equations are exceptions to the above replacement rule. The equations are still one-dimensional, but each scalar represents the magnitude of the vector, for example,

text: Each vector equation represents three scalar equations. Often coordinates can be chosen so that only two components are needed, as in the figure. Each component can be obtained separately and the results combined to produce a vector result.[14]

text: A simple construction involving the center of mass frame can be used to show that if a stationary elastic sphere is struck by a moving sphere, the two will head off at right angles after the collision (as in the figure).[15]

text: Objects of variable mass[edit]

text: The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a rocket ejecting fuel or a star accreting gas. In analyzing such an object, one treats the object's mass as a function that varies with time: m(t). The momentum of the object at time t is therefore p(t) = m(t)v(t). One might then try to invoke Newton's second law of motion by saying that the external force F on the object is related to its momentum p(t) by F = dp/dt, but this is incorrect, as is the related expression found by applying the product rule to d(mv)/dt:[16]

text: This equation does not correctly describe the motion of variable-mass objects. The correct equation is

text: where u is the velocity of the ejected/accreted mass as seen in the object's rest frame.[16] This is distinct from v, which is the velocity of the object itself as seen in an inertial frame.

text: This equation is derived by keeping track of both the momentum of the object as well as the momentum of the ejected/accreted mass (dm). When considered together, the object and the mass (dm) constitute a closed system in which total momentum is conserved.

text: Relativistic[edit]

text: Lorentz invariance[edit]

text: Newtonian physics assumes that absolute time and space exist outside of any observer; this gives rise to Galilean invariance. It also results in a prediction that the speed of light can vary from one reference frame to another. This is contrary to observation. In the special theory of relativity, Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light c is invariant. As a result, position and time in two reference frames are related by the Lorentz transformation instead of the Galilean transformation.[17]

text: Consider, for example, a reference frame moving relative to another at velocity v in the x direction. The Galilean transformation gives the coordinates of the moving frame as

text: while the Lorentz transformation gives[18]

text: where γ is the Lorentz factor:

text: Newton's second law, with mass fixed, is not invariant under a Lorentz transformation. However, it can be made invariant by making the inertial mass m of an object a function of velocity:

text: m0 is the object's invariant mass.[19]

text: The modified momentum,

text: obeys Newton's second law:

text: Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, γm0v is approximately equal to m0v, the Newtonian expression for momentum.

text: Four-vector formulation[edit]

text: In the theory of special relativity, physical quantities are expressed in terms of four-vectors that include time as a fourth coordinate along with the three space coordinates. These vectors are generally represented by capital letters, for example R for position. The expression for the four-momentum depends on how the coordinates are expressed. Time may be given in its normal units or multiplied by the speed of light so that all the components of the four-vector have dimensions of length. If the latter scaling is used, an interval of proper time, τ, defined by[20]

text: is invariant under Lorentz transformations (in this expression and in what follows the (+ − − −) metric signature has been used, different authors use different conventions). Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as Euclidean vectors and multiplying time by √−1; or by keeping time a real quantity and embedding the vectors in a Minkowski space.[21] In a Minkowski space, the scalar product of two four-vectors U = (U0,U1,U2,U3) and V = (V0,V1,V2,V3) is defined as

text: In all the coordinate systems, the (contravariant) relativistic four-velocity is defined by

text: and the (contravariant) four-momentum is

text: where m0 is the invariant mass. If R = (ct,x,y,z) (in Minkowski space), then

text: Using Einstein's mass-energy equivalence, E = mc2, this can be rewritten as

text: Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.

text: The magnitude of the momentum four-vector is equal to m0c:

text: and is invariant across all reference frames.

text: The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting m0 = 0 it follows that

text: In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle. This is unlike the non-relativistic case where they travel at right angles.[22]

text: The four-momentum of a planar wave can be related to a wave four-vector[23]

text: For a particle, the relationship between temporal components,  E = ħ ω, is the Planck–Einstein relation, and the relation between spatial components, p= ħ k, describes a de Broglie matter wave.

text: Generalized[edit]

text: Newton's laws can be difficult to apply to many kinds of motion because the motion is limited by constraints. For example, a bead on an abacus is constrained to move along its wire and a pendulum bob is constrained to swing at a fixed distance from the pivot. Many such constraints can be incorporated by changing the normal Cartesian coordinates to a set of generalized coordinates that may be fewer in number.[24] Refined mathematical methods have been developed for solving mechanics problems in generalized coordinates. They introduce a generalized momentum, also known as the canonical or conjugate momentum, that extends the concepts of both linear momentum and angular momentum. To distinguish it from generalized momentum, the product of mass and velocity is also referred to as mechanical, kinetic or kinematic momentum.[6][25][26] The two main methods are described below.

text: Lagrangian mechanics[edit]

text: In Lagrangian mechanics, a Lagrangian is defined as the difference between the kinetic energy T and the potential energy V:

text: If the generalized coordinates are represented as a vector q = (q1, q2, ... , qN)  and time differentiation is represented by a dot over the variable, then the equations of motion (known as the Lagrange or Euler–Lagrange equations) are a set of N equations:[27]

text: If a coordinate qi is not a Cartesian coordinate, the associated generalized momentum component pi does not necessarily have the dimensions of linear momentum. Even if qi is a Cartesian coordinate, pi will not be the same as the mechanical momentum if the potential depends on velocity.[6]  Some sources represent the kinematic momentum by the symbol Π.[28]

text: In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as

text: Each component pj is said to be the conjugate momentum for the coordinate qj.

text: Now if a given coordinate qi does not appear in the Lagrangian (although its time derivative might appear), then

text: This is the generalization of the conservation of momentum.[6]

text: Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates. An example is found in the section on electromagnetism.

text: Hamiltonian mechanics[edit]

text: In Hamiltonian mechanics, the Lagrangian (a function of generalized coordinates and their derivatives) is replaced by a Hamiltonian that is a function of generalized coordinates and momentum. The Hamiltonian is defined as

text: where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian equations of motion are[29]

text: As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its conjugate momentum component is conserved.[30]

text: Symmetry and conservation[edit]

text: Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem.[31]

text: Electromagnetic[edit]

text: Particle in a field[edit]

text: In Maxwell's equations, the forces between particles are mediated by electric and magnetic fields. The electromagnetic force (Lorentz force) on a particle with charge q due to a combination of electric field E and magnetic field B is

text:
(in SI units).[32]:2
It has an electric potential φ(r, t) and magnetic vector potential A(r, t).[28]
In the non-relativistic regime, its generalized momentum is

text: while in relativistic mechanics this becomes

text: Conservation[edit]

text: In Newtonian mechanics, the law of conservation of momentum can be derived from the law of action and reaction, which states that every force has a reciprocating equal and opposite force. Under some circumstances, moving charged particles can exert forces on each other in non-opposite directions.[33] Nevertheless, the combined momentum of the particles and the electromagnetic field is conserved.

text: Vacuum[edit]

text: The Lorentz force imparts a momentum to the particle, so by Newton's second law the particle must impart a momentum to the electromagnetic fields.[34]

text: In a vacuum, the momentum per unit volume is

text: where μ0 is the vacuum permeability and c is the speed of light. The momentum density is proportional to the Poynting vector S which gives the directional rate of energy transfer per unit area:[34][35]

text: If momentum is to be conserved over the volume V over a region Q, changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If Pmech is the momentum of all the particles in Q, and the particles are treated as a continuum, then Newton's second law gives

text: The electromagnetic momentum is

text: and the equation for conservation of each component i of the momentum is

text: The term on the right is an integral over the surface area Σ of the surface σ representing momentum flow into and out of the volume, and nj is a component of the surface normal of S. The quantity Tij is called the Maxwell stress tensor, defined as

text: Media[edit]

text: The above results are for the microscopic Maxwell equations, applicable to electromagnetic forces in a vacuum (or on a very small scale in media). It is more difficult to define momentum density in media because the division into electromagnetic and mechanical is arbitrary. The definition of electromagnetic momentum density is modified to

text: where the H-field H is related to the B-field and the magnetization M by

text: The electromagnetic stress tensor depends on the properties of the media.[34]

text: Quantum mechanical[edit]

text: In quantum mechanics, momentum is defined as a self-adjoint operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables.

text: For a single particle described in the position basis the momentum operator can be written as

text: where ∇ is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit. This is a commonly encountered form of the momentum operator, though the momentum operator in other bases can take other forms. For example, in momentum space the momentum operator is represented as

text: where the operator p acting on a wave function ψ(p) yields that wave function multiplied by the value p, in an analogous fashion to the way that the position operator acting on a wave function ψ(x) yields that wave function multiplied by the value x.

text:
For both massive and massless objects, relativistic momentum is related to the phase constant

β

{\displaystyle \beta }

  by[36]

text: Electromagnetic radiation (including visible light, ultraviolet light, and radio waves) is carried by photons. Even though photons (the particle aspect of light) have no mass, they still carry momentum. This leads to applications such as the solar sail. The calculation of the momentum of light within dielectric media is somewhat controversial (see Abraham–Minkowski controversy).[37][38]

text: In deformable bodies and fluids[edit]

text: Conservation in a continuum[edit]

text: In fields such as fluid dynamics and solid mechanics, it is not feasible to follow the motion of individual atoms or molecules. Instead, the materials must be approximated by a continuum in which there is a particle or fluid parcel at each point that is assigned the average of the properties of atoms in a small region nearby. In particular, it has a density ρ and velocity v that depend on time t and position r. The momentum per unit volume is ρv.[39]

text: Consider a column of water in hydrostatic equilibrium. All the forces on the water are in balance and the water is motionless. On any given drop of water, two forces are balanced. The first is gravity, which acts directly on each atom and molecule inside. The gravitational force per unit volume is ρg, where g is the gravitational acceleration. The second force is the sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the pressure p. The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is[40]

text: If the forces are not balanced, the droplet accelerates. This acceleration is not simply the partial derivative ∂v/∂t because the fluid in a given volume changes with time. Instead, the material derivative is needed:[41]

text: Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes due to advection as fluid is carried past the point. Per unit volume, the rate of change in momentum is equal to ρDv/Dt. This is equal to the net force on the droplet.

text: Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as above. In addition, surface forces can deform the droplet. In the simplest case, a shear stress τ, exerted by a force parallel to the surface of the droplet, is proportional to the rate of deformation or strain rate. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another. If the speed in the x direction varies with z, the  tangential force in direction x per unit area normal to the z direction is

text: where μ is the viscosity. This is also a flux, or flow per unit area, of x-momentum through the surface.[42]

text: Including the effect of viscosity, the momentum balance equations for the incompressible flow of a Newtonian fluid are

text: These are known as the Navier–Stokes equations.[43]

text: The momentum balance equations can be extended to more general materials, including solids. For each surface with normal in direction i and force in direction j, there is a stress component σij. The nine components make up the Cauchy stress tensor σ, which includes both pressure and shear. The local conservation of momentum is expressed by the Cauchy momentum equation:

text: where f is the body force.[44]

text: The Cauchy momentum equation is broadly applicable to deformations of solids and liquids. The relationship between the stresses and the strain rate depends on the properties of the material (see Types of viscosity).

text: Acoustic waves[edit]

text: A disturbance in a medium gives rise to oscillations, or waves, that propagate away from their source. In a fluid, small changes in pressure p can often be described by the acoustic wave equation:

text: where c is the speed of sound. In a solid, similar equations can be obtained for propagation of pressure (P-waves) and shear (S-waves).[45]

text: The flux, or transport per unit area, of a momentum component ρvj by a velocity vi is equal to ρ vjvj. In the linear approximation that leads to the above acoustic equation, the time average of this flux is zero. However, nonlinear effects can give rise to a nonzero average.[46] It is possible for momentum flux to occur even though the wave itself does not have a mean momentum.[47]

text: History of the concept[edit]

text:
In about 530 AD, working in Alexandria, Byzantine philosopher John Philoponus developed a concept of momentum in his commentary to Aristotle's Physics. Aristotle claimed that everything that is moving must be kept moving by something. For example, a thrown ball must be kept moving by motions of the air.  Most writers continued to accept Aristotle's theory until the time of Galileo, but a few were skeptical. Philoponus pointed out the absurdity in Aristotle's claim that motion of an object is promoted by the same air that is resisting its passage. He proposed instead that an impetus was imparted to the object in the act of throwing it.[48] Ibn Sīnā (also known by his Latinized name Avicenna) read Philoponus and published his own theory of motion in The Book of Healing in 1020. He agreed that an impetus is imparted to a projectile by the thrower; but unlike Philoponus, who believed that it was a temporary virtue that would decline even in a vacuum, he viewed it as a persistent, requiring external forces such as air resistance to dissipate it.[49][50][51]
The work of Philoponus, and possibly that of Ibn Sīnā,[51] was read and refined by the European philosophers Peter Olivi and Jean Buridan. Buridan, who in about 1350 was made rector of the University of Paris, referred to impetus being proportional to the weight times the speed. Moreover, Buridan's theory was different from his predecessor's in that he did not consider impetus to be self-dissipating, asserting that a body would be arrested by the forces of air resistance and gravity which might be opposing its impetus.[52][53]

text: René Descartes believed that the total "quantity of motion" (Latin: quantitas motus) in the universe is conserved,[54] where the quantity of motion is understood as the product of size and speed. This should not be read as a statement of the modern law of momentum, since he had no concept of mass as distinct from weight and size, and more importantly he believed that it is speed rather than velocity that is conserved. So for Descartes if a moving object were to bounce off a surface, changing its direction but not its speed, there would be no change in its quantity of motion.[55][56] Galileo, in his Two New Sciences, used the Italian word impeto to similarly describe Descarte's quantity of motion.

text: Leibniz, in his "Discourse on Metaphysics", gave an argument against Descartes' construction of the conservation of the "quantity of motion" using an example of dropping blocks of different sizes different distances.  He points out that force is conserved but quantity of motion, construed as the product of size and speed of an object, is not conserved.[57]

text: The first correct statement of the law of conservation of momentum was by English mathematician John Wallis in his 1670 work, Mechanica sive De Motu, Tractatus Geometricus: "the initial state of the body, either of rest or of motion, will persist" and "If the force is greater than the resistance, motion will result".[58] Wallis uses momentum and vis for force. Newton's Philosophiæ Naturalis Principia Mathematica, when it was first published in 1687, showed a similar casting around for words to use for the mathematical momentum. His Definition II defines quantitas motus, "quantity of motion", as "arising from the velocity and quantity of matter conjointly", which identifies it as momentum.[59] Thus when in Law II he refers to mutatio motus, "change of motion", being proportional to the force impressed, he is generally taken to mean momentum and not motion.[60] It remained only to assign a standard term to the quantity of motion. The first use of "momentum" in its proper mathematical sense is not clear but by the time of Jenning's Miscellanea in 1721, five years before the final edition of Newton's Principia Mathematica, momentum M or "quantity of motion" was being defined for students as "a rectangle", the product of Q and V, where Q is "quantity of material" and V is "velocity", s/t.[61]