title: Immortality

text: It is suggested that achieving immortality through this mechanism would require specific consideration to be given to the role of consciousness in the functions of the mind. An uploaded mind would only be a copy of the original mind, and not the conscious mind of the living entity associated in such a transfer. Without a simultaneous upload of consciousness, the original living entity remains mortal, thus not achieving true immortality.


title: Immortality

text: Research on neural correlates of consciousness is yet inconclusive on this issue. Whatever the route to mind upload, persons in this state could then be considered essentially immortal, short of loss or traumatic destruction of the machines that maintained them.


title: Immortality

text: Transforming a human into a cyborg can include brain implants or extracting a human processing unit and placing it in a robotic life-support system. Even replacing biological organs with robotic ones could increase life span (e.g. pace makers) and depending on the definition, many technological upgrades to the body, like genetic modifications or the addition of nanobots would qualify an individual as a cyborg. Some people believe that such modifications would make one impervious to aging and disease and theoretically immortal unless killed or destroyed.


title: Immortality

text: As late as 1952, the editorial staff of the "" found in their compilation of the Great Books of the Western World, that "The philosophical issue concerning immortality cannot be separated from issues concerning the existence and nature of man's soul." Thus, the vast majority of speculation on immortality before the 21st century was regarding the nature of the afterlife.


title: Immortality

text: Immortality in ancient Greek religion originally always included an eternal union of body and soul as can be seen in Homer, Hesiod, and various other ancient texts. The soul was considered to have an eternal existence in Hades, but without the body the soul was considered dead. Although almost everybody had nothing to look forward to but an eternal existence as a disembodied dead soul, a number of men and women were considered to have gained physical immortality and been brought to live forever in either Elysium, the Islands of the Blessed, heaven, the ocean or literally right under the ground.


title: Immortality

text: Among those humans made immortal were Amphiaraus, Ganymede, Ino, Iphigenia, Menelaus, Peleus, and a great number of those who fought in the Trojan and Theban wars.


title: Immortality

text: Some were considered to have died and been resurrected before they achieved physical immortality. Asclepius was killed by Zeus only to be resurrected and transformed into a major deity. In some versions of the Trojan War myth, Achilles, after being killed, was snatched from his funeral pyre by his divine mother Thetis, resurrected, and brought to an immortal existence in either Leuce, the Elysian plains, or the Islands of the Blessed. Memnon, who was killed by Achilles, seems to have received a similar fate. Alcmene, Castor, Heracles, and Melicertes were also among the figures sometimes considered to have been resurrected to physical immortality. According to Herodotus' Histories, the 7th century BCE sage Aristeas of Proconnesus was first found dead, after which his body disappeared from a locked room. Later he was found not only to have been resurrected but to have gained immortality.


title: Immortality

text: The parallel between these traditional beliefs and the later resurrection of Jesus was not lost on early Christians, as Justin Martyr argued:


title: Immortality

text: The philosophical idea of an immortal soul was a belief first appearing with either Pherecydes or the Orphics, and most importantly advocated by Plato and his followers. This, however, never became the general norm in Hellenistic thought. As may be witnessed even into the Christian era, not least by the complaints of various philosophers over popular beliefs, many or perhaps most traditional Greeks maintained the conviction that certain individuals were resurrected from the dead and made physically immortal and that others could only look forward to an existence as disembodied and dead, though everlasting, souls.


title: Immortality

text: One of the three marks of existence in Buddhism is anattā, "non-self". This teaching states that the body does not have an eternal soul but is composed of five skandhas or aggregates. Additionally, another mark of existence is impermanence, also called anicca, which runs directly counter to concepts of immortality or permanence. According to one Tibetan Buddhist teaching, Dzogchen, individuals can transform the physical body into an immortal body of light called the rainbow body.


title: Immortality

text: Christian theology holds that Adam and Eve lost physical immortality for themselves and all their descendants in the Fall of man, although this initial "imperishability of the bodily frame of man" was "a preternatural condition".


title: Immortality

text: Christians who profess the Nicene Creed believe that every dead person (whether they believed in Christ or not) will be resurrected from the dead at the Second Coming, and this belief is known as Universal resurrection. While Paul the Apostle insisted that the resurrected body was only "spiritual" and that "flesh and blood cannot inherit the kingdom of God"., the Gospels increasingly emphasized the physical nature of the resurrection body – as the resurrected Jesus in the Gospel of Luke insisting on his still consisting of "flesh and bones". This shift may have been in response to traditional Greek expectations that immortality always included both body and soul.


title: Immortality

text: N.T. Wright, a theologian and former Bishop of Durham, has said many people forget the physical aspect of what Jesus promised. He told Time: "Jesus' resurrection marks the beginning of a restoration that he will complete upon his return. Part of this will be the resurrection of all the dead, who will 'awake', be embodied and participate in the renewal. Wright says John Polkinghorne, a physicist and a priest, has put it this way: 'God will download our software onto his hardware until the time he gives us new hardware to run the software again for ourselves.' That gets to two things nicely: that the period after death (the Intermediate state) is a period when we are in God's presence but not active in our own bodies, and also that the more important transformation will be when we are again embodied and administering Christ's kingdom." This kingdom will consist of Heaven and Earth "joined together in a new creation", he said.


title: Immortality

text: Christian apocrypha include immortal human figures such as Cartaphilus and Longinus who were cursed with physical immortality for various transgressions against Christ during the Passion.


title: Immortality

text: Hindus believe in an immortal soul which is reincarnated after death. According to Hinduism, people repeat a process of life, death, and rebirth in a cycle called "samsara". If they live their life well, their "karma" improves and their station in the next life will be higher, and conversely lower if they live their life poorly. After many life times of perfecting its karma, the soul is freed from the cycle and lives in perpetual bliss. There is no place of eternal torment in Hinduism, although if a soul consistently lives very evil lives, it could work its way down to the very bottom of the cycle.


title: Immortality

text: There are explicit renderings in the Upanishads alluding to a physically immortal state brought about by purification, and sublimation of the 5 elements that make up the body. For example, in the Shvetashvatara Upanishad (Chapter 2, Verse 12), it is stated "When earth, water, fire, air and sky arise, that is to say, when the five attributes of the elements, mentioned in the books on yoga, become manifest then the yogi's body becomes purified by the fire of yoga and he is free from illness, old age and death."


title: Immortality

text: Another view of immortality is traced to the Vedic tradition by the interpretation of Maharishi Mahesh Yogi:


title: Immortality

text:
That man indeed whom these (contacts)
do not disturb, who is even-minded in
pleasure and pain, steadfast, he is fit
for immortality, O best of men.


title: Immortality

text: To Maharishi Mahesh Yogi, the verse means, "Once a man has become established in the understanding of the permanent reality of life, his mind rises above the influence of pleasure and pain. Such an unshakable man passes beyond the influence of death and in the permanent phase of life: he attains eternal life ... A man established in the understanding of the unlimited abundance of absolute existence is naturally free from existence of the relative order. This is what gives him the status of immortal life."


title: Immortality

text: An Indian Tamil saint known as Vallalar claimed to have achieved immortality before disappearing forever from a locked room in 1874.


title: Immortality

text: The traditional concept of an immaterial and immortal soul distinct from the body was not found in Judaism before the Babylonian exile, but developed as a result of interaction with Persian and Hellenistic philosophies. Accordingly, the Hebrew word "nephesh", although translated as ""soul"" in some older English-language Bibles, actually has a meaning closer to "living being". "Nephesh" was rendered in the Septuagint as ("psūchê"), the Greek word for ‘soul’.


title: Immortality

text: The only Hebrew word traditionally translated "soul" ("nephesh") in English language Bibles refers to a living, breathing conscious body, rather than to an immortal soul.


title: Immortality

text: In the New Testament, the Greek word traditionally translated "soul" () has substantially the same meaning as the Hebrew, without reference to an immortal soul.


title: Immortality

text: ""Soul"" may refer either to the whole person, the self, as in "three thousand "souls"" were converted in (see ).


title: Immortality

text: The Hebrew Bible speaks about "Sheol" (שאול), originally a synonym of the grave – the repository of the dead or the cessation of existence, until the resurrection of the dead. This doctrine of resurrection is mentioned explicitly only in although it may be implied in several other texts. New theories arose concerning Sheol during the intertestamental period.


title: Immortality

text: The views about immortality in Judaism is perhaps best exemplified by the various references to this in Second Temple period. The concept of resurrection of the physical body is found in , according to which it will happen through recreation of the flesh. Resurrection of the dead is specified in detail in the extra-canonical books of Enoch, and in Apocalypse of Baruch. According to the British scholar in ancient Judaism P.R. Davies, there is "little or no clear reference ... either to immortality or to resurrection from the dead" in the Dead Sea scrolls texts.


title: Immortality

text: but the sources vary on the beliefs of the Pharisees. The New Testament claims that the Pharisees believed in the resurrection, but does not specify whether this included the flesh or not. According to Josephus, who himself was a Pharisee, the Pharisees held that only the soul was immortal and the souls of good people will be reincarnated and "pass into other bodies," while "the souls of the wicked will suffer eternal punishment."


title: Immortality

text: The Book of Jubilees seems to refer to the resurrection of the soul only, or to a more general idea of an immortal soul.


title: Immortality

text: Rabbinic Judaism claims that the righteous dead will be resurrected in the Messianic Age, with the coming of the messiah. They will then be granted immortality in a perfect world. The wicked dead, on the other hand, will not be resurrected at all. This is not the only Jewish belief about the afterlife. The Tanakh is not specific about the afterlife, so there are wide differences in views and explanations among believers.


title: Immortality

text: It is repeatedly stated in the "Lüshi Chunqiu" that death is unavoidable. Henri Maspero noted that many scholarly works frame Taoism as a school of thought focused on the quest for immortality. Isabelle Robinet asserts that Taoism is better understood as a "way of life" than as a religion, and that its adherents do not approach or view Taoism the way non-Taoist historians have done. In the Tractate of Actions and their Retributions, a traditional teaching, spiritual immortality can be rewarded to people who do a certain amount of good deeds and live a simple, pure life. A list of good deeds and sins are tallied to determine whether or not a mortal is worthy. Spiritual immortality in this definition allows the soul to leave the earthly realms of afterlife and go to pure realms in the Taoist cosmology.


title: Immortality

text: Zoroastrians believe that on the fourth day after death, the human soul leaves the body and the body remains as an empty shell. Souls would go to either heaven or hell; these concepts of the afterlife in Zoroastrianism may have influenced Abrahamic religions. The Persian word for "immortal" is associated with the month "Amurdad", meaning "deathless" in Persian, in the Iranian calendar (near the end of July). The month of Amurdad or Ameretat is celebrated in Persian culture as ancient Persians believed the "Angel of Immortality" won over the "Angel of Death" in this month.


title: Immortality

text: Alcmaeon of Croton argued that the soul is continuously and ceaselessly in motion. The exact form of his argument is unclear, but it appears to have influenced Plato, Aristotle, and other later writers.


title: Immortality

text: Plotinus offers a version of the argument that Kant calls "The Achilles of Rationalist Psychology". Plotinus first argues that the soul is simple, then notes that a simple being cannot decompose. Many subsequent philosophers have argued both that the soul is simple and that it must be immortal. The tradition arguably culminates with Moses Mendelssohn's Phaedon.


title: Immortality

text: Theodore Metochites argues that part of the soul's nature is to move itself, but that a given movement will cease only if what causes the movement is separated from the thing moved – an impossibility if they are one and the same.


title: Immortality

text: Avicenna argued for the distinctness of the soul and the body, and the incorruptibility of the former.


title: Immortality

text: The full argument for the immortality of the soul and Thomas Aquinas' elaboration of Aristotelian theory is found in Question 75 of the First Part of the "Summa Theologica".


title: Immortality

text: René Descartes endorses the claim that the soul is simple, and also that this entails that it cannot decompose. Descartes does not address the possibility that the soul might suddenly disappear.


title: Immortality

text: In early work, Gottfried Wilhelm Leibniz endorses a version of the argument from the simplicity of the soul to its immortality, but like his predecessors, he does not address the possibility that the soul might suddenly disappear. In his monadology he advances a sophisticated novel argument for the immortality of monads.


title: Immortality

text: Moses Mendelssohn's "Phaedon" is a defense of the simplicity and immortality of the soul. It is a series of three dialogues, revisiting the Platonic dialogue "Phaedo", in which Socrates argues for the immortality of the soul, in preparation for his own death. Many philosophers, including Plotinus, Descartes, and Leibniz, argue that the soul is simple, and that because simples cannot decompose they must be immortal. In the Phaedon, Mendelssohn addresses gaps in earlier versions of this argument (an argument that Kant calls the Achilles of Rationalist Psychology). The Phaedon contains an original argument for the simplicity of the soul, and also an original argument that simples cannot suddenly disappear. It contains further original arguments that the soul must retain its rational capacities as long as it exists.


title: Immortality

text: The possibility of clinical immortality raises a host of medical, philosophical, and religious issues and ethical questions. These include persistent vegetative states, the nature of personality over time, technology to mimic or copy the mind or its processes, social and economic disparities created by longevity, and survival of the heat death of the universe.


title: Immortality

text: Physical immortality has also been imagined as a form of eternal torment, as in the myth of Tithonus, or in Mary Shelley's short story "The Mortal Immortal", where the protagonist lives to witness everyone he cares about die around him. For additional examples in fiction, see Immortality in fiction.


title: Immortality

text: Kagan (2012) argues that any form of human immortality would be undesirable. Kagan's argument takes the form of a dilemma. Either our characters remain essentially the same in an immortal afterlife, or they do not:


title: Immortality

text: Either way, Kagan argues, immortality is unattractive. The best outcome, Kagan argues, would be for humans to live as long as they desired and then to accept death gratefully as rescuing us from the unbearable tedium of immortality.


title: Immortality

text: If human beings were to achieve immortality, there would most likely be a change in the world's social structures. Sociologists argue that human beings' awareness of their own mortality shapes their behavior. With the advancements in medical technology in extending human life, there may need to be serious considerations made about future social structures. The world is already experiencing a global demographic shift of increasingly ageing populations with lower replacement rates. The social changes that are made to accommodate this new population shift may be able to offer insight on the possibility of an immortal society.


title: Immortality

text: Sociology has a growing body of literature on the sociology of immortality, which details the different attempts at reaching immortality (whether actual or symbolic) and their prominence in the 21st century. These attempts include renewed attention to the dead in the West, practices of online memorialization, and biomedical attempts to increase longevity. These attempts at reaching immortality and their effects in societal structures have led some to argue that we are becoming a "Postmortal Society". Foreseen changes to societies derived from the pursuit of immortality would encompass societal paradigms and worldviews, as well as the institutional landscape. Similarly, different forms of reaching immortality might entail a significant reconfiguration of societies, from becoming more technologically-oriented to becoming more aligned with nature


title: Immortality

text: Immortality would increase population growth, bringing with it many consequences as for example the impact of population growth on the environment and planetary boundaries.


title: Immortality

text: Although some scientists state that radical life extension, delaying and stopping aging are achievable, there are no international or national programs focused on stopping aging or on radical life extension. In 2012 in Russia, and then in the United States, Israel and the Netherlands, pro-immortality political parties were launched. They aimed to provide political support to anti-aging and radical life extension research and technologies and at the same time transition to the next step, radical life extension, life without aging, and finally, immortality and aim to make possible access to such technologies to most currently living people.


title: Immortality

text: There are numerous symbols representing immortality. The ankh is an Egyptian symbol of life that holds connotations of immortality when depicted in the hands of the and pharaohs, who were seen as having control over the journey of life. The Möbius strip in the shape of a trefoil knot is another symbol of immortality. Most symbolic representations of infinity or the life cycle are often used to represent immortality depending on the context they are placed in. Other examples include the Ouroboros, the Chinese fungus of longevity, the "ten" kanji, the phoenix, the peacock in Christianity, and the colors amaranth (in Western culture) and peach (in Chinese culture).


title: Alsace–Lorraine

text: Alsace–Lorraine, now called Alsace–Moselle, is a historical region located in France. It was created in 1871 by the German Empire after it had seized the region from the Second French Empire in the Franco-Prussian War with the Treaty of Frankfurt. Alsace–Lorraine reverted to French ownership in 1918 as part of the Treaty of Versailles and Germany's defeat in World War I.


title: Alsace–Lorraine

text: When created in 1871, the region was named the Imperial Territory of Alsace–Lorraine ( or ; ; Moselle Franconian/) and as a new territory of the German Empire. The Empire annexed most of Alsace and the Moselle department of Lorraine, following its victory in the Franco-Prussian War. The Alsatian part lay in the Rhine Valley on the west bank of the Rhine River, east of the Vosges Mountains; the section originally in Lorraine was in the upper Moselle valley to the north of the Vosges.


title: Alsace–Lorraine

text: The territory encompassed almost all of Alsace (93%) and over a quarter of Lorraine (26%), while the rest of these regions remained parts of France. For historical reasons, specific legal dispositions are still applied in the territory in the form of a "local law in Alsace–Moselle". In relation to its special legal status, since reversion to France, the territory has been referred to administratively as Alsace–Moselle ().


title: Alsace–Lorraine

text: Alsace–Lorraine had a land area of . Its capital was Straßburg. It was divided in three districts ("Bezirke" in German):


title: Alsace–Lorraine

text: The modern history of Alsace–Lorraine was largely influenced by the rivalry between French and German nationalism.


title: Alsace–Lorraine

text: France long sought to attain and then preserve what it considered to be its "natural boundaries", which it considered the Pyrenees to the southwest, the Alps to the southeast, and the Rhine River to the northeast. These strategic claims led to the annexation of territories located west of the Rhine river in the Holy Roman Empire. What is now known as Alsace was progressively conquered by France under and in the 17th century, while Lorraine was incorporated from the 16th century under Henry II to the 18th century under (in the case of the Three Bishoprics, as early as 1552). These border-changes, at the time, meant more or less that one ruler (the local princes and city-governments, with some remaining power of the Holy Roman Emperor) was exchanged for another (the King of France).


title: Alsace–Lorraine

text: German nationalism on the other hand, which in its 19th century form originated as a reaction against the French occupation of large areas of Germany under Napoleon, sought to unify all the German-speaking populations of the former Holy Roman Empire into a single nation-state. As various German dialects were spoken by most of the population of Alsace and Moselle (northern Lorraine), these regions were viewed by German nationalists to be rightfully part of hoped-for united Germany in the future.


title: Alsace–Lorraine

text: In 1871, the newly created German Empire's demand for Alsace from France after its victory in the Franco-Prussian War was not simply a punitive measure. The transfer was controversial even among the Germans: The German Chancellor, Otto von Bismarck, was initially opposed to it, as he thought (correctly) it would engender permanent French enmity toward Germany. Some German industrialists did not want the competition from Alsatian industries, such as the cloth makers who would be exposed to competition from the sizeable industry in Mulhouse. Karl Marx also warned his fellow Germans:


title: Alsace–Lorraine

text: Bismarck and the South German industrialists proposed to have Alsace ceded to Switzerland, while Switzerland would compensate Germany with another territory. The Swiss rejected the proposal, preferring to remain neutral between the French and Germans.


title: Alsace–Lorraine

text: The German Emperor, Wilhelm I, eventually sided with army commander Helmuth von Moltke, other Prussian generals and other officials who argued that a westward shift in the French border was necessary for strategic military and ethnographic reasons. From a linguistic perspective, the transfer involved people who for the most part spoke Alemannic German dialects. At the time, ethnic identity was often based primarily on language, unlike the more multifaceted approach focusing on self identification in use today. From a military perspective, by early 1870s standards, shifting the frontier away from the Rhine would give the Germans a strategic buffer against feared future French attacks. Due to the annexation, the Germans gained control of the fortifications of Metz and Strasbourg (Straßburg) on the left bank of the Rhine and most of the iron resources of Lorraine.


title: Alsace–Lorraine

text: Creating a new Imperial Territory () out of formerly French territory would achieve this goal: Although a Reichsland would not technically be part of the Kingdom of Prussia, being governed directly by the Empire (headed by the King of Prussia as Emperor, and the minister-president of Prussia as Imperial Chancellor) would in practical terms amount to the same thing. Thus, by annexing Alsace–Lorraine, Berlin was able to avoid complications with Baden and Bavaria on matters such as new fortifications.


title: Alsace–Lorraine

text: Memory of the Napoleonic Wars was still fresh in the 1870s. Wilhelm I himself had had to flee with the Prussian royal family to East Prussia as a nine year old in 1806 and had served in the Battle of Waterloo. Until the Franco-Prussian War, the French had maintained a long-standing desire to establish their entire eastern frontier on the Rhine, and thus they were viewed by most 19th century Germans as an aggressive and acquisitive people. In the years before 1870 the Germans feared the French more than the French feared the Germans. Many Germans at the time thought that the creation of the new Empire in itself would be enough to earn permanent French enmity, and thus desired a defensible border with their long-standing enemy. Any additional enmity that would be earned from territorial concessions was downplayed as marginal and insignificant in the overall scheme of things.


title: Alsace–Lorraine

text: This area corresponded to the present French "départements" of Bas-Rhin (in its entirety), Haut-Rhin (except the area of Belfort and Montbéliard), and a small northeast section of the Vosges département, all of which made up Alsace, and most of the "départements" of Moselle (four-fifths of Moselle) and the northeast of Meurthe (one-third of Meurthe), which were the eastern part of Lorraine.


title: Alsace–Lorraine

text: The remaining two-thirds of the département of Meurthe and the westernmost one-fifth of Moselle, which had escaped German annexation were joined to form the new French département of Meurthe-et-Moselle.


title: Alsace–Lorraine

text: The new border between France and Germany mainly followed the geolinguistic divide between French and German dialects, except in a few valleys of the Alsatian side of the Vosges mountains, the city of Metz and its region and in the area of Château-Salins (formerly in the Meurthe "département"), which were annexed by Germany although most people there spoke French. In 1900, 11.6% of the population of Alsace–Lorraine spoke French as their first language (11.0% in 1905, 10.9% in 1910).


title: Alsace–Lorraine

text: That small francophone areas were affected was used in France to denounce the new border as hypocrisy, since Germany had justified the annexation on linguistic grounds. The German administration was tolerant of the use of the French language (in sharp contrast to the use of the Polish language in the Province of Posen), and French was permitted as an official language and school language in those areas where it was spoken by a majority. This changed in 1914 with the First World War.


title: Alsace–Lorraine

text: The Treaty of Frankfurt gave the residents of the region until 1 October 1872, to choose between emigrating to France or remaining in the region and having their nationality legally changed to German. About 161,000 people, or around 10.4% of the Alsace–Lorraine population, opted for French citizenship (the so-called "Optanden"); but, only about 50,000 actually emigrated, while the rest acquired German citizenship.


title: Alsace–Lorraine

text: The sentiment of attachment to France stayed strong at least during the first 16 years of the annexation. During the Reichstag elections, the 15 deputies of 1874, 1881, 1884 (but one) and 1887 were called "protester deputies" (fr: ) because they expressed to the Reichstag their opposition to the annexation by means of the "1874 motion" in the French language:


title: Alsace–Lorraine

text: The abusive and oppressive behaviour by the German military towards the population of the town of Saverne (the Saverne Affair, usually known in English-language accounts as the Zabern Affair) led to protests not just in Alsace, but in other regions, which put a severe strain on the relationship between the people of Alsace–Lorraine and the rest of the German Empire.


title: Alsace–Lorraine

text: Under the German Empire of 1871–1918, the annexed territory constituted the "Reichsland" or Imperial Territory of (German for Alsace–Lorraine). The area was administered directly from Berlin, but was granted limited autonomy in 1911. This included its constitution and state assembly, its own flag, and the ("Alsatian Flag Song") as its anthem.


title: Alsace–Lorraine

text: In French foreign policy, the demand for the return of Alsace and Lorraine faded in importance after 1880 with the decline of the monarchist element. When the World War broke out in 1914, recovery of the two lost provinces became the top French war goal.


title: Alsace–Lorraine

text: In the early 20th century, the increased militarization of Europe, and the lack of negotiation between major powers, led to harsh and rash actions taken by both sides in respect to Alsace–Lorraine during World War I. As soon as war was declared, both the French and German authorities used the inhabitants of Alsace–Lorraine as propaganda pawns.


title: Alsace–Lorraine

text: Germans living in France were arrested and placed into camps by the French authorities. Upon occupying certain villages, veterans of the 1870 conflict were sought out and arrested by the French army.


title: Alsace–Lorraine

text: The Germans responded to the outbreak of war with harsh measures against the Alsace–Lorraine populace: the Saverne Affair had convinced the high command that the population was hostile to the German Empire and that it should be forced into submission. German troops occupied some homes. The German military feared French partisans – or "francs-tireurs", as they had been called during the Franco-Prussian War – would reappear.


title: Alsace–Lorraine

text: German authorities developed policies aimed at reducing the influence of French. In Metz, French street names, which had been displayed in French and German, were suppressed in January 1915. Six months later, on 15 July 1915, German became the only official language in the region, leading to the Germanization of the towns’ names effective 2 September 1915.


title: Alsace–Lorraine

text: Prohibiting the speaking of French in public further increased the exasperation of some of the natives, who were long accustomed to mixing their conversation with French language (see code-switching); still, the use even of one word, as innocent as ""bonjour"", could incur a fine. Some ethnic Germans in the region cooperated in the persecution as a way to demonstrate German patriotism.


title: Alsace–Lorraine

text: German authorities became increasingly worried about renewed French nationalism. The Reichsland governor stated in February 1918: "Sympathies towards France and repulsion for Germans have penetrated to a frightening depth the petty bourgeoisie and the peasantry". But in order to spare them possible confrontations with relatives in France but also to avoid any desertion from the Alsatian soldiers to the French army, German Army draftees from Alsace–Lorraine were sent mainly to the Eastern front, or the Navy (). About 15,000 Alsatians and Lorrainers served in the German Navy.


title: Alsace–Lorraine

text: In the general revolutionary atmosphere of the expiring German Empire, Marxist councils of workers and soldiers () formed in Mulhouse, Colmar, and Strasbourg in November 1918, in imitation of the "soviets" of revolutionary Russia, and in parallel to other such bodies set up in Germany.


title: Alsace–Lorraine

text: In this chaotic situation, Alsace–Lorraine's Landtag proclaimed itself the supreme authority of the land with the name of , the Strasbourg Soviet proclaimed the foundation of a "Republic of Alsace–Lorraine", and Jacques Peirotes, the SPD Reichstag representative for Colmar, announced the establishment of French rule, urging Paris to send troops quickly.


title: Alsace–Lorraine

text: The soviet councils disbanded themselves with the departure of the German troops between 11 and 17 November. The arrival of the French Army stabilized the situation: French troops put the region under military occupation and entered Strasbourg on 5 November. The ""Nationalrat"" proclaimed the annexation of Alsace to France on 5 December, but this action was not internationally recognized until the Treaty of Versailles was concluded in 1919.


title: Alsace–Lorraine

text: France divided Alsace–Lorraine into the "départements" of Haut-Rhin, Bas-Rhin, and Moselle (the same political structure as before the annexation and as created by the French Revolution, with slightly different limits). Even today, laws in these three regions are somewhat different from the rest of France – these specific provisions are known as the "local law in Alsace–Moselle".


title: Alsace–Lorraine

text: The "département" Meurthe-et-Moselle was maintained even after France recovered Alsace–Lorraine in 1919. The area of Belfort became a special-status area and was not reintegrated into Haut-Rhin in 1919 but instead was made a full-status "département" in 1922 under the name Territoire-de-Belfort.


title: Alsace–Lorraine

text: The French Government immediately started a Francization campaign that included the forced deportation of all Germans who had settled in the area after 1870. For that purpose, the population was divided in four categories:  (French citizens before 1870),  (descendants of such French citizens),  (citizens of Allied or neutral states), and  (enemy aliens – Germans). By July 1921, 111,915 people categorized as "" were expelled to Germany. All place names were francizised (e.g., Straßburg → Strasbourg, Mülhausen → Mulhouse, Schlettstadt → Sélestat, etc.).


title: Alsace–Lorraine

text: On 1 September 1939, residents of Alsace and Moselle living in the Franco-German border region were evacuated. This comprised about one third of the population of Alsace and Moselle, or about 600,000 residents. The evacuation was aimed at providing space for military operations and for protecting citizens from attack. The evacuees were allowed to return in July 1940, after France surrendered to Germany.


title: Alsace–Lorraine

text: The area then came under German occupation. Nazi laws against homosexuality were applied to Alsace–Moselle, and homosexuals were deported. The Nazis also deported refugee and resident Jews.


title: Alsace–Lorraine

text: After the defeat of France in the spring of 1940, Alsace and Moselle were not formally annexed by Nazi Germany. Although the terms of the armistice specified that the integrity of the whole French territory could not be modified in any way, Adolf Hitler, the German Führer, drafted an annexation law in 1940 that he kept secret, expecting to announce it in the event of a German victory. Through a series of laws which individually seemed minor, Berlin took de facto control of Alsace–Lorraine, and Alsatians–Lorrainians could be drafted into the German Army. During the occupation, Moselle was integrated into a Reichsgau named Westmark and Alsace was amalgamated with Baden. Beginning in 1942, people from Alsace and Moselle were made German citizens by decree of the Nazi government.


title: Alsace–Lorraine

text: Beginning in October 1942, young Alsatian and Lorrainian men were inducted into the German armed forces. Sometimes they were known as the , which could be translated into English as "against our will". A small minority volunteered, notably the author of "The Forgotten Soldier", known by the pseudonym Guy Sajer. Ultimately, 100,000 Alsatians and 30,000 Mosellans were enrolled, many of them to fight against the Soviet Red Army, on Germany's Eastern Front. Most of those who survived the war were interned in Tambov in Russia in 1945. Many others fought in Normandy against the Allies as the of the 2nd SS Panzer Division Das Reich, some of whom were involved in the Oradour sur Glane and Tulle war crimes.


title: Alsace–Lorraine

text: When Alsace and the Lorraine department became part of Germany, the French laws regarding religious bodies were preserved, with special privileges to the then recognised religions of Calvinism, Judaism, Lutheranism and Roman Catholicism, under a system known as the Concordat. However, the Roman Catholic dioceses of Metz and of Strasbourg became exempt jurisdictions. The Church of Augsburg Confession of France, with its directory, supreme consistory and the bulk of its parishioners residing in Alsace, was reorganised as the Protestant Church of Augsburg Confession of Alsace and Lorraine (EPCAAL) in 1872, but territorially reconfined to Alsace–Lorraine only. The five local Calvinist consistories, originally part of the Reformed Church of France, formed a statewide synod in 1895, the Protestant Reformed Church of Alsace and Lorraine (EPRAL). The three Israelite consistories in , and were disentangled from supervision by the Israelite Central Consistory of France and continued as separate statutory corporations which never formed a joint body, but cooperated. All the mentioned religious bodies retained the status as "établissements publics de culte" (public bodies of Religion). When the new Alsace–Lorraine constitution of 1911 provided for a bicameral state parliament () each recognised religion was entitled to send a representative into the first chamber of the Landtag as ex officio members (the bishops of Strasbourg and of Metz, the presidents of EPCAAL and EPRAL, and a delegate of the three Israelite consistories).


title: Alsace–Lorraine

text: Both Germanic and Romance dialects were traditionally spoken in Alsace–Lorraine before the 20th century.


title: TT pistol

text: The TT-30, commonly known simply as the Tokarev, is an out-of-production Soviet semi-automatic pistol. It was developed in 1930 by Fedor Tokarev as a service pistol for the Soviet military to replace the Nagant M1895 revolver that had been in use since the Russian Empire, though it ended up being used in conjunction with, rather than replacing, the M1895. It served until 1952, when it was replaced by the Makarov pistol.


title: TT pistol

text: In 1930, the Revolutionary Military Council approved a resolution to test new small arms to replace its aging Nagant M1895 revolvers. During these tests, on 7 January 1931, the potential of a pistol designed by Fedor Tokarev was noted. A few weeks later, 1,000 TT-30s were ordered for troop trials, and the pistol was adopted for service in the Red Army. The TT-30 was manufactured between 1930 and 1936, with about 93,000 being produced.


title: TT pistol

text: Even as the TT-30 was being put into production, design changes were made to simplify manufacturing. Minor changes to the barrel, disconnector, trigger and frame were implemented, the most notable ones being the omission of the removable hammer assembly and changes to the full-circumference locking lugs. This redesigned pistol was the TT-33. Most TT-33s were issued to commanding officers. The TT-33 was widely used by Soviet troops during World War II, but did not completely replace the Nagant. From 1931-1945, a total of 1,330,000 Tokarevs were produced in the Soviet Union.


title: TT pistol

text: Externally, the TT-33 is very similar to John Browning's blowback operated FN Model 1903 semiautomatic pistol, and internally it uses Browning's short recoil tilting-barrel system from the M1911 pistol. In other areas the TT-33 differs more from Browning's designs—it employs a much simpler hammer/sear assembly than the M1911. This assembly is removable from the pistol as a modular unit and includes machined magazine feed lips, preventing misfeeds when a damaged magazine is loaded into the magazine well. Soviet engineers made several alterations to make the mechanism easier to produce and maintain, most notably the simplifications of the barrel's locking lugs, allowing fewer machining steps. Some models use a captive recoil spring secured to the guide rod, which does depend on the barrel bushing to hold it under tension. The TT-33 is chambered for the 7.62×25mm Tokarev cartridge, which was itself based on the similar 7.63×25mm Mauser cartridge used in the Mauser C96 pistol. The 7.62×25mm cartridge is powerful, has an extremely flat trajectory, and is capable of penetrating thick clothing and soft body armor. Because of their reliability, large numbers of the TT-33 were produced during World War II and well into the 1950s. In modern times, the robust TT-33 has been converted to many powerful cartridges including .38 Super and 9×23mm Winchester. The TT-33 omitted a safety catch other than the half cock notch, which rendered the trigger inoperable until the hammer was pulled back to full cock and then lowered manually to the half cock position. Many variants imported into the US have manual safeties added, which vary greatly in placement and function.


title: TT pistol

text: The Wehrmacht captured TT-33s and issued them to units under the "Pistole 615(r)" designation. This was made possible by the fact that Russian 7.62 mm Model 1930 Type P cartridges were nearly identical to the German 7.63×25mm Mauser cartridge, although in German service the 9×19mm Parabellum round was more common. Due to much higher pressures, using Russian cartridges in German Mauser pistols can induce damages, and is advised against.


title: TT pistol

text: Interarms marketed World War II–surplus Russian-made Tokarevs in Europe and the United States as the "Phoenix". They had new wooden grips with a phoenix design on them and were overstamped INTERARMS on the barrel. Later gun laws banned their sale due to their lack of a safety.


title: TT pistol

text: In 1949, a silenced variant was produced. Uniquely, the silencer is attached to the barrel bushing rather than the barrel itself. The combined weight of the suppressor with the slide prevents semi-auto cycling of the action, forcing the user to manually cycle it in the same manner as pump action firearms. It would later be replaced by the PB pistol in 1967.


title: TT pistol

text: The TT-33 was eventually replaced by the 8-round, 9×18mm Makarov PM pistol in 1952. Production of the TT-33 in Russia ended in 1954, but copies (licensed or otherwise) were also made by other countries. At one time or another, most communist or Soviet bloc countries made a variation of the TT-33 pistol.


title: TT pistol

text: Norinco, the People's Liberation Army's state armaments manufacturer in China, manufactured a commercial variant of the Tokarev pistol chambered in the more common 9×19mm Parabellum round, known as the "Tokarev Model 213", as well as in the original 7.62×25mm caliber.


title: TT pistol

text: The 9mm model features a safety catch, which was absent on Russian-produced TT-33 handguns. Furthermore, the Model 213 features the thin slide grip grooves, as opposed to the original Russian wide-types. The 9mm model is featured with a magazine well block mounted in the rear of the magazine well to accept 9mm type magazines without frame modification.


title: TT pistol

text: The Norinco model in current production is not available for sale in the United States due to import prohibitions on Chinese firearms, although older handguns of the Model 213 type imported in the 1980s and 1990s are common.


title: TT pistol

text: 7.62×25mm ammo is also rather inexpensive and locally produced or imported from China, also made by Norinco. The Type-54 was replaced in the mid to late 1990s by the QSZ-92 in PLA service.


title: TT pistol

text: Hungary rebarreled the TT to fire 9×19mm Parabellum as the "M48", as well as an export version for Egypt known as the "Tokagypt 58" which was widely used by police forces there. Tokagypts differ from the original Tokarevs by an external safety lever that can be engaged in safety decocking as well as cocked hammer position. By changing the barrel and magazine into original TT parts, a calibre change system can be made easily (after proof-shooting in countries affiliated with the CIP).


title: TT pistol

text: Egypt, however, cancelled much of the order of the Tokagypt and PP-like pistols manufactured in Hungary; those were marketed then in the West, like the then Federal Republic of Germany, where it was imported by Hege.


title: TT pistol

text: In mid-50s a training version of PW wz. 33 was created, chambered in .22lr called "TT Sportowy". All of those pistols were converted between 1954 and 1958 from the 7.62mm variant by changing the barrel and removing the locking lugs from slide.


title: TT pistol

text: Additionally, the Radom M48 was created in Radom, Poland as another slightly modified copy of the TT33.


title: TT pistol

text: Romania produced a TT-33 copy as the "TTC", or "Tokarev Cugir" well into the 1950s. These have been made available for commercial sale in great numbers in recent years. However, to be importable into the United States, a trigger blocking safety was added.


title: TT pistol

text: The K54 is a copy of the TT-33. An updated version known as the K14-VN is made by Factory Z111, and has an increased capacity of 13 rounds, with a wider grip to incorporate a double stack magazine. Research and development started in 2001. The K14-VN began to see service with PAVN forces on May 10, 2014.


title: TT pistol

text: The M57 has a longer grip and longer 9-round magazine (versus 8 rounds in TT). A 9×19mm version is also made by Zastava designated "M70A" as well as a compact version "M88".


title: TT pistol

text: Zastava manufactures a sub compact pistol M70 (a.k.a."Pčelica" ("little bee")) roughly based on TT design in 7,65mm Browning (.32 ACP) or 9mm Kratak (.380 ACP).


title: TT pistol

text: Since 2012, the M57A, M70A and M88A was formerly imported into the U.S. by Century International Arms, but has since been replaced by Zastava USA.


title: TT pistol

text: The TT-33 is still in service in the Bangladeshi and North Korean armed forces today, while police in Pakistan still commonly use the TT pistol as a sidearm, though unofficially, as it is being replaced by modern 9 mm Beretta and SIG Sauer pistols. In China, the TT-33 pistol is also occasionally supplied to the People's Armed Police and People's Liberation Army under the name Type 54.


title: TT pistol

text: The Tokarev, as well as its variants in 9mm, have been renowned for their simplicity, power and accuracy.


title: Lagrangian mechanics

text: In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, "Mécanique analytique".


title: Lagrangian mechanics

text: Lagrangian mechanics describes a mechanical system as a pair formula_1 consisting of a configuration space formula_2 and a smooth function formula_3 within that space called a "Lagrangian". By convention, formula_4 where formula_5 and formula_6 are the kinetic and potential energy of the system, respectively.


title: Lagrangian mechanics

text: The stationary action principle requires that the action functional of the system derived from formula_3 must remain at a stationary point (a maximum, minimum, or saddle) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.


title: Lagrangian mechanics

text: Suppose there exists a bead sliding around on a wire, or a swinging simple pendulum, etc. If one tracks each of the massive objects (bead, pendulum bob, etc.) as a particle, calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion (reaction force exerted by the wire on the bead, or tension in the pendulum rod). For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of "independent" generalized coordinates that completely characterize the possible motion of the particle. This choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.


title: Lagrangian mechanics

text: For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a point particle. For a system of "N" point particles with masses "m", "m", ..., "m", each particle has a position vector, denoted r, r, ..., r. Cartesian coordinates are often sufficient, so r = ("x", "y", "z"), r = ("x", "y", "z") and so on. In three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3"N" coordinates to uniquely define the configuration of the system. These are all specific points in space to locate the particles; a general point in space is written r = ("x", "y", "z"). The velocity of each particle is how fast the particle moves along its path of motion, and is the time derivative of its position, thus


title: Lagrangian mechanics

text: In Newtonian mechanics, the equations of motion are given by Newton's laws. The second law "net force equals mass times acceleration",formula_9applies to each particle. For an "N" particle system in 3 dimensions, there are 3"N" second order ordinary differential equations in the positions of the particles to solve for.


title: Lagrangian mechanics

text: Instead of forces, Lagrangian mechanics uses the energies in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The "non-relativistic" Lagrangian for a system of particles can be defined by


title: Lagrangian mechanics

text: is the total kinetic energy of the system, equalling the sum Σ of the kinetic energies of the particles, and "V" is the potential energy of the system.


title: Lagrangian mechanics

text: Kinetic energy is the energy of the system's motion, and "v" = v · v is the magnitude squared of velocity, equivalent to the dot product of the velocity with itself. The kinetic energy is a function only of the velocities v, not the positions r nor time "t", so "T" = "T"(v, v, ...).


title: Lagrangian mechanics

text: The potential energy of the system reflects the energy of interaction between the particles, i.e. how much energy any one particle will have due to all the others and other external influences. For conservative forces (e.g. Newtonian gravity), it is a function of the position vectors of the particles only, so "V" = "V"(r, r, ...). For those non-conservative forces which can be derived from an appropriate potential (e.g. electromagnetic potential), the velocities will appear also, "V" = "V"(r, r, ..., v, v, ...). If there is some external field or external driving force changing with time, the potential will change with time, so most generally "V" = "V"(r, r, ..., v, v, ..., "t").


title: Lagrangian mechanics

text: The above form of "L" does not hold in relativistic Lagrangian mechanics, and must be replaced by a function consistent with special or general relativity. Also, for dissipative forces another function must be introduced alongside "L".


title: Lagrangian mechanics

text: One or more of the particles may each be subject to one or more holonomic constraints; such a constraint is described by an equation of the form "f"(r, "t") = 0. If the number of constraints in the system is "C", then each constraint has an equation, "f"(r, "t") = 0, "f"(r, "t") = 0, ..., "f"(r, "t") = 0, each of which could apply to any of the particles. If particle "k" is subject to constraint "i", then "f"(r, "t") = 0. At any instant of time, the coordinates of a constrained particle are linked together and not independent. The constraint equations determine the allowed paths the particles can move along, but not where they are or how fast they go at every instant of time. Nonholonomic constraints depend on the particle velocities, accelerations, or higher derivatives of position. Lagrangian mechanics "can only be applied to systems whose constraints, if any, are all holonomic". Three examples of nonholonomic constraints are: when the constraint equations are nonintegrable, when the constraints have inequalities, or with complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may have to revert to Newtonian mechanics, or use other methods.


title: Lagrangian mechanics

text: If "T" or "V" or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian "L"(r, r, ... v, v, ... "t") is "explicitly time-dependent". If neither the potential nor the kinetic energy depend on time, then the Lagrangian "L"(r, r, ... v, v, ...) is "explicitly independent of time". In either case, the Lagrangian will always have implicit time-dependence through the generalized coordinates.


title: Lagrangian mechanics

text: where "k" = 1, 2, ..., "N" labels the particles, there is a Lagrange multiplier "λ" for each constraint equation "f", and


title: Lagrangian mechanics

text: are each shorthands for a vector of partial derivatives with respect to the indicated variables (not a derivative with respect to the entire vector). Each overdot is a shorthand for a time derivative. This procedure does increase the number of equations to solve compared to Newton's laws, from 3"N" to 3"N" + "C", because there are 3"N" coupled second order differential equations in the position coordinates and multipliers, plus "C" constraint equations. However, when solved alongside the position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations.


title: Lagrangian mechanics

text: In the Lagrangian, the position coordinates and velocity components are all independent variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usual differentiation rules (e.g. the partial derivative of "L" with respect to the "z"-velocity component of particle 2, defined by formula_13, is just formula_14; no awkward chain rules or total derivatives need to be used to relate the velocity component to the corresponding coordinate "z").


title: Lagrangian mechanics

text: In each constraint equation, one coordinate is redundant because it is determined from the other coordinates. The number of "independent" coordinates is therefore "n" = 3"N" − "C". We can transform each position vector to a common set of "n" generalized coordinates, conveniently written as an "n"-tuple q = ("q", "q", ... "q"), by expressing each position vector, and hence the position coordinates, as functions of the generalized coordinates and time,


title: Lagrangian mechanics

text: The vector q is a point in the configuration space of the system. The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the total derivative of its position with respect to time, is


title: Lagrangian mechanics

text: Given this v, the kinetic energy "in generalized coordinates" depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so formula_17.


title: Lagrangian mechanics

text: are mathematical results from the calculus of variations, which can also be used in mechanics. Substituting in the Lagrangian "L"(q, dq/d"t", "t"), gives the equations of motion of the system. The number of equations has decreased compared to Newtonian mechanics, from 3"N" to "n" = 3"N" − "C" coupled second order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for.


title: Lagrangian mechanics

text: Although the equations of motion include partial derivatives, the results of the partial derivatives are still ordinary differential equations in the position coordinates of the particles. The total time derivative denoted d/d"t" often involves implicit differentiation. Both equations are linear in the Lagrangian, but will generally be nonlinear coupled equations in the coordinates.


title: Lagrangian mechanics

text: For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for a system of "N" particles, all of these equations apply to each particle in the system). The equation of motion for a particle of mass "m" is Newton's second law of 1687, in modern vector notation


title: Lagrangian mechanics

text: where a is its acceleration and F the resultant force acting "on" it. In three spatial dimensions, this is a system of three coupled second order ordinary differential equations to solve, since there are three components in this vector equation. The solution is the position vector r of the particle at time "t", subject to the initial conditions of r and v when "t" = 0.


title: Lagrangian mechanics

text: Newton's laws are easy to use in Cartesian coordinates, but Cartesian coordinates are not always convenient, and for other coordinate systems the equations of motion can become complicated. In a set of curvilinear coordinates ξ = ("ξ", "ξ", "ξ"), the law in tensor index notation is the ""Lagrangian form""


title: Lagrangian mechanics

text: where "F" is the "a"th contravariant component of the resultant force acting on the particle, Γ"" are the Christoffel symbols of the second kind,


title: Lagrangian mechanics

text: is the kinetic energy of the particle, and "g" the covariant components of the "metric tensor" of the curvilinear coordinate system. All the indices "a", "b", "c", each take the values 1, 2, 3. Curvilinear coordinates are not the same as generalized coordinates.


title: Lagrangian mechanics

text: It may seem like an overcomplication to cast Newton's law in this form, but there are advantages. The acceleration components in terms of the Christoffel symbols can be avoided by evaluating derivatives of the kinetic energy instead. If there is no resultant force acting on the particle, F = 0, it does not accelerate, but moves with constant velocity in a straight line. Mathematically, the solutions of the differential equation are "geodesics", the curves of extremal length between two points in space (these may end up being minimal so the shortest paths, but that is not necessary). In flat 3D real space the geodesics are simply straight lines. So for a free particle, Newton's second law coincides with the geodesic equation, and states that free particles follow geodesics, the extremal trajectories it can move along. If the particle is subject to forces, F ≠ 0, the particle accelerates due to forces acting on it, and deviates away from the geodesics it would follow if free. With appropriate extensions of the quantities given here in flat 3D space to 4D curved spacetime, the above form of Newton's law also carries over to Einstein's general relativity, in which case free particles follow geodesics in curved spacetime that are no longer "straight lines" in the ordinary sense.


title: Lagrangian mechanics

text: However, we still need to know the total resultant force F acting on the particle, which in turn requires the resultant non-constraint force N plus the resultant constraint force C,


title: Lagrangian mechanics

text: The constraint forces can be complicated, since they will generally depend on time. Also, if there are constraints, the curvilinear coordinates are not independent but related by one or more constraint equations.


title: Lagrangian mechanics

text: The constraint forces can either be eliminated from the equations of motion so only the non-constraint forces remain, or included by including the constraint equations in the equations of motion.


title: Lagrangian mechanics

text: A fundamental result in analytical mechanics is D'Alembert's principle, introduced in 1708 by Jacques Bernoulli to understand static equilibrium, and developed by D'Alembert in 1743 to solve dynamical problems. The principle asserts for "N" particles the virtual work, i.e. the work along a virtual displacement, δr, is zero:


title: Lagrangian mechanics

text: The "virtual displacements", δr, are by definition infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the system "at an instant of time", i.e. in such a way that the constraint forces maintain the constrained motion. They are not the same as the actual displacements in the system, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate and move it. Virtual work is the work done along a virtual displacement for any force (constraint or non-constraint).


title: Lagrangian mechanics

text: Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero:


title: Lagrangian mechanics

text: Thus D'Alembert's principle allows us to concentrate on only the applied non-constraint forces, and exclude the constraint forces in the equations of motion. The form shown is also independent of the choice of coordinates. However, it cannot be readily used to set up the equations of motion in an arbitrary coordinate system since the displacements δr might be connected by a constraint equation, which prevents us from setting the "N" individual summands to 0. We will therefore seek a system of mutually independent coordinates for which the total sum will be 0 if and only if the individual summands are 0. Setting each of the summands to 0 will eventually give us our separated equations of motion.


title: Lagrangian mechanics

text: If there are constraints on particle "k", then since the coordinates of the position r = ("x", "y", "z") are linked together by a constraint equation, so are those of the virtual displacements "δr = ("δx", "δy", "δz"). Since the generalized coordinates are independent, we can avoid the complications with the "δr by converting to virtual displacements in the generalized coordinates. These are related in the same form as a total differential,


title: Lagrangian mechanics

text: There is no partial time derivative with respect to time multiplied by a time increment, since this is a virtual displacement, one along the constraints in an "instant" of time.


title: Lagrangian mechanics

text: The first term in D'Alembert's principle above is the virtual work done by the non-constraint forces N along the virtual displacements "δ"r, and can without loss of generality be converted into the generalized analogues by the definition of generalized forces


title: Lagrangian mechanics

text: This is half of the conversion to generalized coordinates. It remains to convert the acceleration term into generalized coordinates, which is not immediately obvious. Recalling the Lagrange form of Newton's second law, the partial derivatives of the kinetic energy with respect to the generalized coordinates and velocities can be found to give the desired result:


title: Lagrangian mechanics

text: and since these virtual displacements "δq" are independent and nonzero, the coefficients can be equated to zero, resulting in Lagrange's equations or the generalized equations of motion,


title: Lagrangian mechanics

text: These equations are equivalent to Newton's laws "for the non-constraint forces". The generalized forces in this equation are derived from the non-constraint forces only – the constraint forces have been excluded from D'Alembert's principle and do not need to be found. The generalized forces may be non-conservative, provided they satisfy D'Alembert's principle.


title: Lagrangian mechanics

text: For a non-conservative force which depends on velocity, it "may" be possible to find a potential energy function "V" that depends on positions and velocities. If the generalized forces "Q" can be derived from a potential "V" such that


title: Lagrangian mechanics

text: equating to Lagrange's equations and defining the Lagrangian as "L" = "T" − "V" obtains Lagrange's equations of the second kind or the Euler–Lagrange equations of motion


title: Lagrangian mechanics

text: However, the Euler–Lagrange equations can only account for non-conservative forces "if" a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations.


title: Lagrangian mechanics

text: The Euler–Lagrange equations also follow from the calculus of variations. The "variation" of the Lagrangian is


title: Lagrangian mechanics

text: which has a form similar to the total differential of "L", but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. An integration by parts with respect to time can transfer the time derivative of "δq" to the ∂"L"/∂(d"q"/d"t"), in the process exchanging d("δq")/d"t" for "δq", allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian,


title: Lagrangian mechanics

text: Now, if the condition "δq"("t") = "δq"("t") = 0 holds for all "j", the terms not integrated are zero. If in addition the entire time integral of "δL" is zero, then because the "δq" are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of "δq" must also be zero. Then we obtain the equations of motion. This can be summarized by Hamilton's principle:


title: Lagrangian mechanics

text: which is a "functional"; it takes in the Lagrangian function for all times between "t" and "t" and returns a scalar value. Its dimensions are the same as [ angular momentum ], [energy]·[time], or [length]·[momentum]. With this definition Hamilton's principle is


title: Lagrangian mechanics

text: Thus, instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton's principle is sometimes referred to as the "principle of least action", however the action functional need only be "stationary", not necessarily a maximum or a minimum value. Any variation of the functional gives an increase in the functional integral of the action.


title: Lagrangian mechanics

text: Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the calculus of variations to mechanical problems, such as the Brachistochrone problem solved by Jean Bernoulli in 1696, as well as Leibniz, Daniel Bernoulli, L'Hôpital around the same time, and Newton the following year. Newton himself was thinking along the lines of the variational calculus, but did not publish. These ideas in turn lead to the variational principles of mechanics, of Fermat, Maupertuis, Euler, Hamilton, and others.


title: Lagrangian mechanics

text: Hamilton's principle can be applied to nonholonomic constraints if the constraint equations can be put into a certain form, a linear combination of first order differentials in the coordinates. The resulting constraint equation can be rearranged into first order differential equation. This will not be given here.


title: Lagrangian mechanics

text: Hamilton's principle is still valid even if the coordinates "L" is expressed in are not independent, here r, but the constraints are still assumed to be holonomic. As always the end points are fixed "δr("t") = "δr("t") = 0 for all "k". What cannot be done is to simply equate the coefficients of δr to zero because the δr are not independent. Instead, the method of Lagrange multipliers can be used to include the constraints. Multiplying each constraint equation "f"(r, "t") = 0 by a Lagrange multiplier "λ" for "i" = 1, 2, ..., "C", and adding the results to the original Lagrangian, gives the new Lagrangian


title: Lagrangian mechanics

text: The Lagrange multipliers are arbitrary functions of time "t", but not functions of the coordinates r, so the multipliers are on equal footing with the position coordinates. Varying this new Lagrangian and integrating with respect to time gives


title: Lagrangian mechanics

text: The introduced multipliers can be found so that the coefficients of "δ"r are zero, even though the r are not independent. The equations of motion follow. From the preceding analysis, obtaining the solution to this integral is equivalent to the statement


title: Lagrangian mechanics

text: which are Lagrange's equations of the first kind. Also, the "λ" Euler-Lagrange equations for the new Lagrangian return the constraint equations


title: Lagrangian mechanics

text: For the case of a conservative force given by the gradient of some potential energy "V", a function of the r coordinates only, substituting the Lagrangian "L" = "T" − "V" gives


title: Lagrangian mechanics

text: and identifying the derivatives of kinetic energy as the (negative of the) resultant force, and the derivatives of the potential equaling the non-constraint force, it follows the constraint forces are


title: Lagrangian mechanics

text: thus giving the constraint forces explicitly in terms of the constraint equations and the Lagrange multipliers.


title: Lagrangian mechanics

text: The Lagrangian of a given system is not unique. A Lagrangian "L" can be multiplied by a nonzero constant "a" and shifted by an arbitrary constant "b", and the new Lagrangian "L' = aL" + "b" will describe the same motion as "L". If one restricts as above to trajectories formula_45 over a given time interval formula_46 and fixed end points formula_47 and formula_48, then two Lagrangians describing the same system can differ by the "total time derivative" of a function formula_49:


title: Lagrangian mechanics

text: Both Lagrangians formula_53 and formula_54 produce the same equations of motion since the corresponding actions formula_55 and formula_56 are related via


title: Lagrangian mechanics

text: Given a set of generalized coordinates q, if we change these variables to a new set of generalized coordinates s according to a point transformation q = q(s, "t"), the new Lagrangian "L"′ is a function of the new coordinates


title: Lagrangian mechanics

text: and by the chain rule for partial differentiation, Lagrange's equations are invariant under this transformation;


title: Lagrangian mechanics

text: An important property of the Lagrangian is that conserved quantities can easily be read off from it. The "generalized momentum" "canonically conjugate to" the coordinate "q" is defined by


title: Lagrangian mechanics

text: If the Lagrangian "L" does "not" depend on some coordinate "q", it follows immediately from the Euler–Lagrange equations that


title: Lagrangian mechanics

text: and integrating shows the corresponding generalized momentum equals a constant, a conserved quantity. This is a special case of Noether's theorem. Such coordinates are called "cyclic" or "ignorable".


title: Lagrangian mechanics

text: where "r" and "z" are lengths along straight lines, "s" is an arc length along some curve, and "θ" and "φ" are angles. Notice "z", "s", and "φ" are all absent in the Lagrangian even though their velocities are not. Then the momenta


title: Lagrangian mechanics

text: are all conserved quantities. The units and nature of each generalized momentum will depend on the corresponding coordinate; in this case "p" is a translational momentum in the "z" direction, "p" is also a translational momentum along the curve "s" is measured, and "p" is an angular momentum in the plane the angle "φ" is measured in. However complicated the motion of the system is, all the coordinates and velocities will vary in such a way that these momenta are conserved.


title: Lagrangian mechanics

text: At every time instant formula_69 the energy is invariant under configuration space coordinate changes formula_70, i.e.


title: Lagrangian mechanics

text: Besides this result, the proof below shows that, under such change of coordinates, the derivatives formula_72 change as coefficients of a linear form.


title: Lagrangian mechanics

text: In Lagrangian mechanics, the system is closed if and only if its Lagrangian formula_53 does not explicitly depend on time. The energy conservation law states that the energy formula_74 of a closed system is an integral of motion.


title: Lagrangian mechanics

text: More precisely, let formula_75 be an "extremal". (In other words, formula_45 satisfies the Euler-Lagrange equations). Taking the total time-derivative of formula_53 along this extremal and using the EL equations leads to


title: Lagrangian mechanics

text: If the Lagrangian formula_53 does not explicitly depend on time, then formula_80 so formula_74 is, indeed, an integral of motion, meaning that


title: Lagrangian mechanics

text: It also follows that the kinetic energy is a homogenous function of degree 2 in the generalized velocities. If in addition the potential "V" is only a function of coordinates and independent of velocities, it follows by direct calculation, or use of Euler's theorem for homogenous functions, that


title: Lagrangian mechanics

text: is the total energy of the system. The kinetic and potential energies still change as the system evolves, but the motion of the system will be such that their sum, the total energy, is constant. This is a valuable simplification, since the energy "E" is a constant of integration that counts as an arbitrary constant for the problem, and it may be possible to integrate the velocities from this energy relation to solve for the coordinates. In the case the velocity or kinetic energy or both depends on time, then the energy is "not" conserved.


title: Lagrangian mechanics

text: If the potential energy is a homogeneous function of the coordinates and independent of time, and all position vectors are scaled by the same nonzero constant "α", r′ = "α"r, so that


title: Lagrangian mechanics

text: and time is scaled by a factor "β", "t"′ = "βt", then the velocities v are scaled by a factor of "α"/"β" and the kinetic energy "T" by ("α"/"β"). The entire Lagrangian has been scaled by the same factor if


title: Lagrangian mechanics

text: Since the lengths and times have been scaled, the trajectories of the particles in the system follow geometrically similar paths differing in size. The length "l" traversed in time "t" in the original trajectory corresponds to a new length "l′" traversed in time "t′" in the new trajectory, given by the ratios


title: Lagrangian mechanics

text: For a given system, if two subsystems "A" and "B" are non-interacting, the Lagrangian "L" of the overall system is the sum of the Lagrangians "L" and "L" for the subsystems:


title: Lagrangian mechanics

text: If they do interact this is not possible. In some situations, it may be possible to separate the Lagrangian of the system "L" into the sum of non-interacting Lagrangians, plus another Lagrangian "L" containing information about the interaction,


title: Lagrangian mechanics

text: This may be physically motivated by taking the non-interacting Lagrangians to be kinetic energies only, while the interaction Lagrangian is the system's total potential energy. Also, in the limiting case of negligible interaction, "L" tends to zero reducing to the non-interacting case above.


title: Lagrangian mechanics

text: The extension to more than two non-interacting subsystems is straightforward – the overall Lagrangian is the sum of the separate Lagrangians for each subsystem. If there are interactions, then interaction Lagrangians may be added.


title: Lagrangian mechanics

text: A particle of mass "m" moves under the influence of a conservative force derived from the gradient ∇ of a scalar potential,


title: Lagrangian mechanics

text: If there are more particles, in accordance with the above results, the total kinetic energy is a sum over all the particle kinetic energies, and the potential is a function of all the coordinates.


title: Lagrangian mechanics

text: The equations of motion for the particle are found by applying the Euler–Lagrange equation, for the "x" coordinate


title: Lagrangian mechanics

text: Using the spherical coordinates (r, "θ", "φ") as commonly used in physics (ISO 80000-2:2019 convention), where "r" is the radial distance to origin, "θ" is polar angle (also known as colatitude, zenith angle, normal angle, or inclination angle), and "φ" is the azimuthal angle, the Lagrangian for a central potential is


title: Lagrangian mechanics

text: The "φ" coordinate is cyclic since it does not appear in the Lagrangian, so the conserved momentum in the system is the angular momentum


title: Lagrangian mechanics

text: The Lagrangian in two-dimensional polar coordinates is recovered by fixing "θ" to the constant value "π"/2.


title: Lagrangian mechanics

text: Consider a pendulum of mass "m" and length "ℓ", which is attached to a support with mass "M", which can move along a line in the formula_101-direction. Let formula_101 be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle formula_103 from the vertical. The coordinates and velocity components of the pendulum bob are


title: Lagrangian mechanics

text: The generalized coordinates can be taken to be formula_101 and formula_103. The kinetic energy of the system is then


title: Lagrangian mechanics

text: Since formula_101 is absent from the Lagrangian, it is a cyclic coordinate. The conserved momentum is


title: Lagrangian mechanics

text: These equations may look quite complicated, but finding them with Newton's laws would have required carefully identifying all forces, which would have been much more laborious and prone to errors. By considering limit cases, the correctness of this system can be verified: For example, formula_117 should give the equations of motion for a simple pendulum that is at rest in some inertial frame, while formula_118 should give the equations for a pendulum in a constantly accelerating system, etc. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by stepping through the results iteratively.


title: Lagrangian mechanics

text: Two bodies of masses and with position vectors and are in orbit about each other due to an attractive central potential . We may write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce the Jacobi coordinates; the separation of the bodies and the location of the center of mass . The Lagrangian is then


title: Lagrangian mechanics

text: where is the total mass, is the reduced mass, and the potential of the radial force, which depends only on the magnitude of the separation . The Lagrangian splits into a "center-of-mass" term and a "relative motion" term .


title: Lagrangian mechanics

text: Since the relative motion only depends on the magnitude of the separation, it is ideal to use polar coordinates and take ,


title: Lagrangian mechanics

text: The radial coordinate and angular velocity can vary with time, but only in such a way that is constant. The Lagrange equation for is


title: Lagrangian mechanics

text: This equation is identical to the radial equation obtained using Newton's laws in a "co-rotating" reference frame, that is, a frame rotating with the reduced mass so it appears stationary. Eliminating the angular velocity from this radial equation,


title: Lagrangian mechanics

text: which is the equation of motion for a one-dimensional problem in which a particle of mass is subjected to the inward central force and a second outward force, called in this context the centrifugal force


title: Lagrangian mechanics

text: Of course, if one remains entirely within the one-dimensional formulation, enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general two-dimensional problem from which the one-dimensional problem originated.


title: Lagrangian mechanics

text: If one arrives at this equation using Newtonian mechanics in a co-rotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation of the frame itself. If one arrives at this equation directly by using the generalized coordinates and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth of "using polar coordinates". As Hildebrand says:


title: Lagrangian mechanics

text: "Since such quantities are not true physical forces, they are often called "inertia forces". Their presence or absence depends, not upon the particular problem at hand, but "upon the coordinate system chosen"." In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides the centripetal force for a curved motion.


title: Lagrangian mechanics

text: This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates. For example, see for a comparison of Lagrangians in an inertial and in a noninertial frame of reference. See also the discussion of "total" and "updated" Lagrangian formulations in. Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not an inertial frame of reference), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces as "generalized" inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we deal "always" with "generalized" forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" will be omitted frequently."


title: Lagrangian mechanics

text: It is known that the Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the symmetry of the system.


title: Lagrangian mechanics

text: A test particle is a particle whose mass and charge are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles like electrons and up quarks are more complex and have additional terms in their Lagrangians.


title: Lagrangian mechanics

text: The Lagrangian for a charged particle with electrical charge , interacting with an electromagnetic field, is the prototypical example of a velocity-dependent potential. The electric scalar potential and magnetic vector potential are defined from the electric field and magnetic field as follows:


title: Lagrangian mechanics

text: Note that the canonical momentum (conjugate to position ) is the kinetic momentum plus a contribution from the field (known as the potential momentum):


title: Lagrangian mechanics

text: This relation is also used in the minimal coupling prescription in quantum mechanics and quantum field theory. From this expression, we can see that the canonical momentum is not gauge invariant, and therefore not a measurable physical quantity; However, if is cyclic (i.e. Lagrangian is independent of position ), which happens if the and fields are uniform, then this canonical momentum given here is the conserved momentum, while the measurable physical kinetic momentum is not.


title: Lagrangian mechanics

text: Dissipation (i.e. non-conservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom.


title: Lagrangian mechanics

text: In a more general formulation, the forces could be both conservative and viscous. If an appropriate transformation can be found from the F, Rayleigh suggests using a dissipation function, "D", of the following form:


title: Lagrangian mechanics

text: where "C" are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them. If "D" is defined this way, then


title: Lagrangian mechanics

text: The ideas in Lagrangian mechanics have numerous applications in other areas of physics, and can adopt generalized results from the calculus of variations.


title: Lagrangian mechanics

text: A closely related formulation of classical mechanics is Hamiltonian mechanics. The Hamiltonian is defined by


title: Lagrangian mechanics

text: and can be obtained by performing a Legendre transformation on the Lagrangian, which introduces new variables canonically conjugate to the original variables. For example, given a set of generalized coordinates, the variables canonically conjugate are the generalized momenta. This doubles the number of variables, but makes differential equations first order. The Hamiltonian is a particularly ubiquitous quantity in quantum mechanics (see Hamiltonian (quantum mechanics)).


title: Lagrangian mechanics

text: Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates.


title: Lagrangian mechanics

text: The Euler–Lagrange equations can also be formulated in terms of the generalized momenta rather than generalized coordinates. Performing a Legendre transformation on the generalized coordinate Lagrangian "L"(q, dq/d"t", "t") obtains the generalized momenta Lagrangian "L"′(p, dp/d"t", "t") in terms of the original Lagrangian, as well the EL equations in terms of the generalized momenta. Both Lagrangians contain the same information, and either can be used to solve for the motion of the system. In practice generalized coordinates are more convenient to use and interpret than generalized momenta.


title: Lagrangian mechanics

text: There is no mathematical reason to restrict the derivatives of generalized coordinates to first order only. It is possible to derive modified EL equations for a Lagrangian containing higher order derivatives, see Euler–Lagrange equation for details. However, from the physical point-of-view there is an obstacle to include time derivatives higher than the first order, which is implied by Ostrogradsky's construction of a canonical formalism for nondegenerate higher derivative Lagrangians, see Ostrogradsky_instability


title: Lagrangian mechanics

text: Lagrangian mechanics can be applied to geometrical optics, by applying variational principles to rays of light in a medium, and solving the EL equations gives the equations of the paths the light rays follow.


title: Lagrangian mechanics

text: Lagrangian mechanics can be formulated in special relativity and general relativity. Some features of Lagrangian mechanics are retained in the relativistic theories but difficulties quickly appear in other respects. In particular, the EL equations take the same form, and the connection between cyclic coordinates and conserved momenta still applies, however the Lagrangian must be modified and is not simply the kinetic minus the potential energy of a particle. Also, it is not straightforward to handle multiparticle systems in a manifestly covariant way, it may be possible if a particular frame of reference is singled out.


title: Lagrangian mechanics

text: In quantum mechanics, action and quantum-mechanical phase are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions.


title: Lagrangian mechanics

text: In 1948, Feynman discovered the path integral formulation extending the principle of least action to quantum mechanics for electrons and photons. In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, and Fermat's principle in optics.


title: Lagrangian mechanics

text: In Lagrangian mechanics, the generalized coordinates form a discrete set of variables that define the configuration of a system. In classical field theory, the physical system is not a set of discrete particles, but rather a continuous field "ϕ"(r, "t") defined over a region of 3D space. Associated with the field is a Lagrangian density


title: Lagrangian mechanics

text: defined in terms of the field and its space and time derivatives at a location r and time "t". Analogous to the particle case, for non-relativistic applications the Lagrangian density is also the kinetic energy density of the field, minus its potential energy density (this is not true in general, and the Lagrangian density has to be "reverse engineered"). The Lagrangian is then the volume integral of the Lagrangian density over 3D space


title: Lagrangian mechanics

text: where dr is a 3D differential volume element. The Lagrangian is a function of time since the Lagrangian density has implicit space dependence via the fields, and may have explicit spatial dependence, but these are removed in the integral, leaving only time in as the variable for the Lagrangian.


title: Lagrangian mechanics

text: The action principle, and the Lagrangian formalism, are tied closely to Noether's theorem, which connects physical conserved quantities to continuous symmetries of a physical system.


title: Lagrangian mechanics

text: If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with either special relativity or general relativity.


title: Prescription drug

text: A prescription drug (also prescription medication or prescription medicine) is a pharmaceutical drug that legally requires a medical prescription to be dispensed. In contrast, over-the-counter drugs can be obtained without a prescription. The reason for this difference in substance control is the potential scope of misuse, from drug abuse to practicing medicine without a license and without sufficient education. Different jurisdictions have different definitions of what constitutes a prescription drug.


title: Prescription drug

text: In North America, ℞, usually printed as "Rx", is used as an abbreviation of the word "prescription". It is a contraction of the Latin word ""recipe"" (an imperative form of "recipere") meaning "take". Prescription drugs are often dispensed together with a monograph (in Europe, a Patient Information Leaflet or PIL) that gives detailed information about the drug.


title: Prescription drug

text: In Australia, the Standard for the Uniform Scheduling of Medicines and Poisons (SUSMP) governs the manufacture and supply of drugs with several categories:


title: Prescription drug

text: As in other developed countries, the person requiring a prescription drug attends the clinic of a qualified health practitioner, such as a physician, who may write the prescription for the required drug.


title: Prescription drug

text: Many prescriptions issued by health practitioners in Australia are covered by the Pharmaceutical Benefits Scheme, a scheme that provides subsidised prescription drugs to residents of Australia to ensure that all Australians have affordable and reliable access to a wide range of necessary medicines. When purchasing a drug under the PBS, the consumer pays no more than the patient co-payment contribution, which, as of January 1, 2022, is A$42.50 for general patients. Those covered by government entitlements (low-income earners, welfare recipients, Health Care Card holders, etc.) and or under the Repatriation Pharmaceutical Benefits Scheme (RPBS) have a reduced co-payment, which is A$6.80 in 2022. The co-payments are compulsory and can be discounted by pharmacies up to a maximum of A$1.00 at cost to the pharmacy.


title: Prescription drug

text: In the United Kingdom, the Medicines Act 1968 and the Prescription Only Medicines (Human Use) Order 1997 contain regulations that cover the supply of sale, use, prescribing and production of medicines. There are three categories of medicine:


title: Prescription drug

text: The possession of a prescription-only medicine without a prescription is legal unless it is covered by the Misuse of Drugs Act 1971.


title: Prescription drug

text: A patient visits a medical practitioner or dentist, who may prescribe drugs and certain other medical items, such as blood glucose-testing equipment for diabetics. Also, qualified and experienced nurses and pharmacists may be independent prescribers. Both may prescribe all POMs (including controlled drugs), but may not prescribe Schedule 1 controlled drugs, and 3 listed controlled drugs for the treatment of addiction; which is similar to doctors, who require a special license from the Home Office to prescribe schedule 1 drugs. Schedule 1 drugs have little or no medical benefit, hence their limitations on prescribing. District nurses and health visitors have had limited prescribing rights since the mid-1990s; until then, prescriptions for dressings and simple medicines had to be signed by a doctor. Once issued, a prescription is taken by the patient to a pharmacy, which dispenses the medicine.


title: Prescription drug

text: Most prescriptions are NHS prescriptions, subject to a standard charge that is unrelated to what is dispensed. The NHS prescription fee was increased to £9.15 per item in England on 1 April 2020; prescriptions are free of charge if prescribed and dispensed in Scotland, Wales and Northern Ireland, and for some patients in England, such as inpatients, children, those over 60s or with certain medical conditions, and claimants of certain benefits. The pharmacy charges the NHS the actual cost of the medicine, which may vary from a few pence to hundreds of pounds. A patient can consolidate prescription charges by using a prescription payment certificate (informally a "season ticket"), effectively capping costs at £29.60 per quarter or £105.90 per year.


title: Prescription drug

text: Outside the NHS, private prescriptions are issued by private medical practitioner and sometimes under the NHS for medicines that are not covered by the NHS. A patient pays the pharmacy the normal price for medicine prescribed outside the NHS.


title: Prescription drug

text: Survey results published by Ipsos MORI in 2008 found that around 800,000 people in England were not collecting prescriptions or getting them dispensed because of the cost, the same as in 2001.


title: Prescription drug

text: In the United States, the Federal Food, Drug, and Cosmetic Act defines what substances require a prescription for them to be dispensed by a pharmacy. The federal government authorizes physicians (of any specialty), physician assistants, nurse practitioners and other advanced practice nurses, veterinarians, dentists, and optometrists to prescribe any controlled substance. They are then issued unique DEA numbers; many other mental and physical health technicians, including basic-level registered nurses, medical assistants, emergency medical technicians, most psychologists, and social workers, for example, do not have the authority to prescribe any legend drugs or controlled drugs. Legend drugs are another name for drugs requiring a prescription.


title: Prescription drug

text: The Controlled Substances Act (CSA) was enacted into law by the US Congress of the United States in 1970. It is the federal drug law that regulates manufacture, importation, possession, use, and distribution of "controlled" substances. The legislation classes these substances into five schedules, with varying qualifications for each schedule. The schedules are designated schedule I, schedule II, schedule III, schedule IV, and schedule V. Many drugs require a prescription, even though they are not a controlled substance.


title: Prescription drug

text: The safety and the effectiveness of prescription drugs in the US are regulated by the 1987 Prescription Drug Marketing Act (PDMA). The Food and Drug Administration (FDA) is charged with implementing the law.


title: Prescription drug

text: Misuse or abuse of prescription drugs can lead to adverse drug events, including those due to dangerous drug interactions.