text:
Hal Clement

Harry Clement Stubbs (May 30, 1922 – October 29, 2003), better known by the pen name Hal Clement, was an American science fiction writer and a leader of the hard science fiction subgenre. He also painted astronomically oriented artworks under the name George Richard.

In 1998, Clement was inducted into the Science Fiction and Fantasy Hall of Fame. He was named the 17th SFWA Grand Master by the Science Fiction and Fantasy Writers of America in 1999.

Biography
Harry Clement Stubbs was born in Somerville, Massachusetts, on May 30, 1922.

He went to Harvard, graduating with a B.S. in astronomy in 1943. While there he wrote his first published story, "Proof", which appeared in the June 1942 issue of Astounding Science Fiction, edited by John W. Campbell; three more appeared in later 1942 numbers. His further educational background includes an M.Ed. (Boston University 1946) and M.S. in chemistry (Simmons College 1963).

During World War II Clement was a pilot and copilot of a B-24 Liberator and flew 35 combat missions over Europe with the 68th Bomb Squadron, 44th Bomb Group, based in England with 8th Air Force. After the war, he served in the United States Air Force Reserve, and retired with the rank of colonel. He taught chemistry and astronomy for many years at Milton Academy in Milton, Massachusetts.

From 1949 to 1953, Clement's first three novels were two-, three-, and four-part Astounding serials under Campbell: Needle (Doubleday, 1950), Iceworld (Gnome Press, 1953), and Mission of Gravity (1954), his best-known novel, published by Doubleday's Science Fiction Book Club (established 1953). The latter novel features a land and sea expedition across the superjovian planet Mesklin to recover a stranded scientific probe. The natives of Mesklin are centipede-like intelligent beings about 50 centimeters long. Various episodes hinge on the fact that Mesklin's fast rotational speed causes it to be considerably deformed from the spherical, with effective surface gravity that varies from approximately 3 'g'n at the equator to approximately 700 gn at the poles.

Clement's article "Whirligig World" describes his approach to writing a science fiction story:

"Writing a science fiction story is fun, not work. ... the fun ... lies in treating the whole thing as a game.... [T]he rules must be quite simple. They are; for the reader of a science-fiction story, they consist of finding as many as possible of the author's statements or implications which conflict with the facts as science currently understands them. For the author, the rule is to make as few such slips as he possibly can... Certain exceptions are made [e.g., to allow travel faster than the speed of light], but fair play demands that all such matters be mentioned as early as possible in the story..."

Clement was a frequent guest at science fiction conventions, especially in the eastern United States, where he usually presented talks and slide shows about writing and astronomy.

Clement died in his sleep in Massachusetts at the Milton Hospital on October 29, 2003, at age 81.

Awards and honors
Clement has been honored several times for his cumulative contributions including 1998 Hall of Fame induction, when Clement and Frederik Pohl were the fifth and sixth living persons honored, and the 1999 SFWA Grand Master Award.

For the 1945 short story "Uncommon Sense" he received a 50-year Retro Hugo Award at the 1996 World Science Fiction Convention. Mission of Gravity, first published as a serial during 1953, was named best foreign novel by the Spanish Science Fiction Association in 1994 and it was a finalist for a 50-year Retro Hugo Award in 2004.

The Hal Clement Award for Young Adults for Excellence in Children's Science Fiction Literature was presented in Clement's name from 1992 to 2016.

Wayne Barlowe illustrated two of Clement's fictional species, the Abyormenites and the Mesklinites, in his Barlowe's Guide to Extraterrestrials.

Short stories, novelettes and novellas

 * "Proof" (June 1942). Short story. Published in Astounding. Collected in The Essential Hal Clement Volume 2, Possible Worlds of Science Fiction (1951), SF: Author's Choice 2 (1970), Where Do We Go From Here? (1971), The Great SF Stories 4 (1942) (1980), First Voyages (1981), The Golden Years of Science Fiction (Second Series) (1983), Encounters (1988), Ascent of Wonder (1994) and Wondrous Beginnings (2003).
 * Impediment (August 1942). Novelette. Published in Astounding. Collected in Natives of Space, The Best of Hal Clement and The Essential Hal Clement Volume 2.
 * Avenue of Escape (November 1942). Published in Astounding's series Probability Zero. Collected in The Essential Hal Clement Volume 2.
 * "Attitude" (September 1943). Novella. Published in Astounding. Collected in The Essential Hal Clement Volume 2 and Travellers of Space (1951).
 * Technical Error" (January 1944). Novelette. Published in Astounding. Collected in Natives of Space, The Best of Hal Clement and The Essential Hal Clement Volume 2.
 * "Trojan Fall" (June 1944). Short story. Published in Astounding. Collected in Small Changes.
 * "Uncommon Sense" (September 1945). His most famous short story. Part of the Laird Cunningham Series. Hugo Award for Best Short Story of 1945. Published in Astounding. Collected in Small Changes, The Best of Hal Clement, Intuit, The Essential Hal Clement Volume 2, The Old Masters  (1970), Out of This World 10  (1973) and Nebula Awards Showcase 2000 (2000).
 * "Cold Front" (July 1946). Novelette. Published in Astounding. Collected in The Essential Hal Clement Volume 2, Men Against the Stars (1950, 1956) and Astounding Stories: The 60th Anniversary Collection, Vol. 2 (1990).
 * Assumption Unjustified (October 1946). Novelette. Published in Astounding. Collected in Natives of Space, The Best of Hal Clement and Crossroads in Time  (1953).
 * "Answer" (April 1947). Short story. Published in Astounding SF. Collected in The Best of Hal Clement and Science Fiction Thinking Machines (1954).
 * "Fireproof" (March 1949). Short story. Published in Astounding. Collected in Small Changes, Decade of the 1940s (1975) and Combat SF (1981).
 * "Halo" (October 1952). Novelette. Published in Galaxy. Collected in Small Changes, The Essential Hal Clement Volume 2 and Shadow of Tomorrow  (1953).
 * "Critical Factor" (1953). Short story not included in any of the Hal Clement's compilations. Published in Star Science Fiction Stories #2  (1953). Collected in Titan 4 (1977) and The Road to Science Fiction #3: From Heinlein to here (1979).
 * "Ground" (December 1953). Short story not included in any of the Hal Clement's compilations. Published in Science Fiction Adventures.
 * "Dust Rag" (September 1956). Short story. Published in Astounding. Collected in Small Changes, The Best of Hal Clement, Where Do We Go From Here? (1971) and The Prentice Hall Anthology of Science Fiction and Fantasy (2001).
 * "Planet for Plunder" (February 1957). Published in Satellite SF jointly with Sam Merwin, Jr. A previous version of "Planetfall". Collected in Men of the Morning Star/Planet for Plunder.
 * "The Lunar Lichen" (February 1960). Novelette not included in any of the Hal Clement's compilations. Published in Future Science Fiction. Collected in The Time Trap/The Lunar Lichen.
 * "Sun Spot" (November 1960). Short story. Published in Analog. Collected in Small Changes, The Essential Hal Clement Volume 2, Isaac Asimov’s Wonderful Worlds of Science Fiction # 4: Comets (1986) and Analog’s Expanding Universe
 * "The Green World" (May 1963). Novella not included in any of the Hal Clement's compilations. Published in If. Collected in The Moon is Hell!/The Green World.
 * "Hot Planet" (August 1963). Novelette not included in any of the Hal Clement's compilations to date. Published in Galaxy. Collected in The 9th Annual of the Year's Best SF (1964), Spectrum 4 (1965), The Eighth Galaxy Reader  (1965), Isaac Asimov Presents the Great SF Stories #25 (1963) (1992) and Science Fiction Century (1997).
 * "Raindrop" (May 1965). Novelette. Published in If. Collected in Small Changes, The Essential Hal Clement Volume 2 and Isaac Asimov’s Wonderful Worlds of Science Fiction # 4: Comets.
 * "The Foundling Stars" (August 1966). Short story. Published in If. Collected in Small Changes and The Second If Reader of Science Fiction (1968).
 * "The Mechanic" (September 1966). Novelette. Published in Analog. Collected in Small Changes, The Essential Hal Clement Volume 2 and Analog: Writers’ Choice, Volume II (1984).
 * "Bulge" (September 1968). Novelette. Published in If. Collected in Small Changes and The Essential Hal Clement Volume 2.
 * '"Planetfall" (1972). Original version of "Planet for Plunder" (1957). Published in Strange Tomorrows (1972). Collected in The Essential Hal Clement Volume 2.
 * "Lecture Demonstration" (1973). Short story from the Mesklin Series (of Mission of Gravity fame). Published in the book Astounding (1973). Collected in The Essential Hal Clement Volume 3, Heavy Planet and Mission of Gravity  (1978).
 * "Mistaken for Granted" (January/February 1974). Novella. Published in Worlds of If. Collected in The Best of Hal Clement.
 * "The Logical Life" (1974). Second short story in the Laird Cunningham Series. Published in Stellar #1 (1974). Collected in Intuit and The Essential Hal Clement Volume 2.
 * "Question of Guilt" (1976). Novelette. Published in The Year's Best Horror Stories: Series IV (1976). Collected in The Best of Hal Clement.
 * "Stuck with It" (1976). Novelette innthe Laird Cunningham Series. Published in Stellar #2  (1976). Collected in The Best of Hal Clement, Intuit and The Essential Hal Clement Volume 2].
 * "Longline" (1976). Novelette. Published in Faster than Light (1976). Collected in The Essential Hal Clement Volume 2.
 * "Seasoning" (September/October 1978). Novelette set in Harlan Ellison's Medea world. Not included in any of Hal Clement's compilations. Published in IASFM. Collected in Medea: Harlan's World (1985) and Aliens and UFO's (1993).
 * "Status Symbol" (1987). Novelette, the last story in the Laird Cunningham Series. Published in Intuit. Collected in The Essential Hal Clement Volume 2.
 * "Blot" (1989). Novelette about Asimov's positronic robots. Not included in any of Hal Clement's compilations to date. Published in Foundation's Friends (1989).
 * "Phases in Chaos'" (1991). Novella not included in any of Hal Clement's compilations. Published in Isaac’s Universe Volume Two: Phases in Chaos.
 * "Eyeball Vectors" (1992). Novella not included in any of Clement's compilations to date. Published in Isaac's Universe Volume 3: Unnatural Diplomacy.
 * "Sortie" (spring/summer 1994). First part of the Sortie series. Novella not included in any of Hal Clement's compilations to date. Published in Harsh Mistress.
 * "Settlement" (fall/winter 1994). Second part of the Sortie series. Novella not included in any of Clement's compilations. Published in Absolute Magnitude.
 * "Seismic Sidetrack" (spring 1995). Third part of the Sortie series. Novella not included in any of Hal Clement's compilations to date. Published in Absolute Magnitude.
 * "Simile" (summer 1995). Fourth and last part of the Sortie series. Novella not included in any of Hal Clement's compilations. Published in Absolute Magnitude.
 * "Oh, Natural" (spring 1998). Novelette not included in any of Hal Clement's compilations. Published in Absolute Magnitude. Collected in Hal's Worlds: Stories and Essays in Memory of Hal Clement.
 * Options" (1998). Short story not included in any of Clement's compilations to date. Published as Harry C. Stubbs in Lamps on the Brow.
 * "Exchange Rate" (winter 1999). Novella not included in any of Hal Clement's compilations to date. Published in Absolute Magnitude. Collected in The Year’s Best Science Fiction: Seventeenth Annual Collection and The Hard SF Renaissance (2002).
 * "Under" (January 2000). Short story, last story in the 'Mesklin series. Published in Analog. Collected in The Essential Hal Clement Volume 3 and Heavy Planet.
 * "Office politics" (2003). Short story not included in any of Clement's compilations. Published in Readercon 15 Souvenir Book (This may be an article and not a fiction story)

Books

 * Needle (1950), ISBN 0-380-00635-9 (The first novel in the Needle series. Also published as From Outer Space. Published as young adult fiction although it includes abstract hard science fiction.)
 * Iceworld (1953), ISBN 0-345-25805-3
 * Mission of Gravity (1954), ISBN 0-345-31622-3 (first book in the Mesklin series).
 * The Ranger Boys in Space (1956) (for children)
 * Cycle of Fire (1957), ISBN 0-345-24368-4
 * Close to Critical (1958), ISBN 0-345-24508-3 (part of the Mesklin series. Magazine publication in 1958, book in 1964)
 * Natives of Space (1965), ISBN 0345219503 (three novelettes)
 * Small Changes (1969), ISBN 0709110006 (collection of 9 short stories)
 * Space Lash (1969), ISBN (reprint in paperback of Small Changes)
 * First Flights to the Moon (1970), ASIN B000BCHC4Y (anthology of short stories from others, edited by Hal Clement)
 * Star Light (1971), ISBN 0-345-02361-7 (part of the Mesklin series, sequel to Mission of Gravity. It also shares some characters with Close to Critical)
 * Ocean on Top (1973), ISBN 1-4510-1057-5 (magazine publications in 1967)
 * Left of Africa (1976), ISBN 0936414014 (historical novel for young adults, apparently limited to 750 copies )
 * Through the Eye of a Needle (1978), ISBN 0-345-25850-9 (the second and last novel in the Needle series)
 * The Best of Hal Clement (1979), ISBN 0345276892 (collection of 10 short stories, including all of Natives of Space and two from Small Changes: "Uncommon Sense" and "Dust Rag")
 * The Nitrogen Fix (1980), ISBN 0-441-58116-1
 * Intuit (1987), ISBN 0-915368-35-8 (complete collection of the 4 Laird Cunningham stories, edition limited to 820 copies)
 * Still River (1987), ISBN 0-345-32916-3
 * Fossil (1993), ISBN 0-88677-573-6 (set in Isaac Asimov's Universe)
 * Half Life (1999), ISBN 0-312-86920-7 (Humanity is going extinct due to disease, scientists are sent to Titan in the faint hope of finding biochemical clues to a cure)
 * The Essential Hal Clement, Volume 1: Trio for Slide Rule and Typewriter  (1999), ISBN 1-886778-06-X (collection of the novels Needle, Iceworld and Close to Critical)
 * The Essential Hal Clement, Volume 2: Music of Many Spheres  (2000), ISBN 1-886778-07-8 (collection of 17 short stories, including most from Small Changes and from The Best of Hal Clement)
 * The Essential Hal Clement, Volume 3: Variations on a Theme by Sir Isaac Newton  (2000), ISBN 1-886778-08-6 (collection of all Mesklin stories except Close to Critical: Mission of Gravity, Star Light, "Lecture Demonstration" and "Under"; also the how-to-write-science-fiction article "Whirligig World")
 * Heavy Planet (2002), ISBN 0-7653-0368-X (reprint of The Essential Hal Clement, Volume 3)
 * Noise (2003), ISBN 0-7653-0857-6 (set on an ocean planet)
 * Men of the Morning Star/Planet for Plunder (2011), ISBN 978-1-61287-018-2 (two novellas, the first by Edmond Hamilton and the second by Hal Clement and Sam Merwin Jr.)
 * The Moon is Hell!/The Green World (2012), ISBN 978-1-61287-087-8 (two novellas, the first by John W. Campbell Jr. and the second by Hal Clement)
 * The Time Trap/The Lunar Lichen (2013), ISBN 978-1-61287-142-4 (two novellas, the first by Henry Kuttner and the second by Hal Clement)
 * Hal Clement SF Gateway Omnibus (2014), ISBN 978-0575110151 (collection of the novels Iceworld, Cycle of Fire and Close to Critical)

About Hal Clement

 * Starmont Readers Guide 11: Hal Clement (1982), ISBN 978-0893700423. Donald M. Hassler.
 * Hal Clement, Scientist with a Mission: a Working Bibliography (1989), ASIN B0006OUUAU. Gordon Benson Jr.
 * Hal's Worlds: Stories and Essays in Memory of Hal Clement (2005), ISBN 978-0-8095-5073-9. Several authors.

Articles and introductions
"In the early 1940s, in Astounding, there was a small department called Probability Zero! that ran short-short stories. Or items. Or lies. Things. These things were usually funny and always impossible - echoing the description of the title."
 * Probability Zero! (nov 1942). Published jointly with Malcolm Jameson, Harry Warner Jr., Dennis Tucker and P. Schuyler Miller in Astounding. About Probability Zero, Harry Harrison said in the John Campbell Memorial Anthology:
 * Whirligig World (jun 1953). About how to write science fiction, and specifically, about how he wrote Mission of Gravity. Published in Astounding. Collected in The Essential Hal Clement Volume 3, Heavy Planet and Mission of Gravity (1978).
 * Some Notes on Xi Bootis. Published by Advent Publishers.
 * Gravity insufficient (nov 1961). Published in Analog Science Fact.
 * Chips on Distant Shoulders (1980). Published in The Future at War Vol. 3.
 * Basic Concepts: Astrophysics, Geology (1985). About Harlan Ellison's world Medea. Published in Medea: Harlan's World.
 * Second Thoughts (1985). About Harlan Ellison's world Medea, jointly written with Poul Anderson, Thomas M. Disch, Larry Niven & Frederik Pohl. Published in Medea: Harlan's World.
 * The Home System (oct 1986). Published in Aboriginal.
 * Intuition: The Guide Who Needs Steering (1987). Published in Intuit.
 * The Magic Picture (1989). Published in L. Ron Hubbard Presents Writers of the Future v5.
 * Whatever Happened to the Science in Science Fiction? (sep 1993). Published in Science Fiction Age.
 * Ardent Thuria, Chilly Cluros: Seeing, and Seeing From, Low Orbiting Satellites (1994). Published in Mindsparks.
 * Only Once (1994). Published in Fractal.
 * Will (1998). Introduction to First Contacts: The Essential Murray Leinster.
 * Jack Williamson, especulator (1999). Introduction to The Collected Stories of Jack Williamson, Volume One: The Metal Man and Others.
 * Alfred E. van Vogt (2003). Introduction to Transfinite: The Essential A.E. van Vogt.
 * About Proof, of Course (2003). Introduction to Proof in Wondrous Beginnings.

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title: Hal Clement


text:
Halldór Laxness

Halldór Kiljan Laxness (born Halldór Guðjónsson; 23 April 1902 – 8 February 1998) was an Icelandic writer and winner of the 1955 Nobel Prize in Literature. He wrote novels, poetry, newspaper articles, essays, plays, travelogues and short stories. Writers who influenced Laxness include August Strindberg, Sigmund Freud, Knut Hamsun, Sinclair Lewis, Upton Sinclair, Bertolt Brecht, and Ernest Hemingway.

Early life
Halldór Guðjónsson was born in Reykjavík in 1902. When he was three, his family moved to the Laxnes farm in Mosfellssveit parish. He was brought up and enormously influenced by his grandmother, who "sang me ancient songs before I could talk, told me stories from heathen times and sang me cradle songs from the Catholic era". He started to read books and write stories at an early age and attended the technical school in Reykjavík from 1915 to 1916. His earliest published writings appeared in 1916 in Morgunblaðið and in the children's periodical Æskan. The same year, two letters-to-the-editor Halldór wrote also appeared in the North American-Icelandic children's newspapers Sólskin, which was published in Winnipeg, Manitoba. Laxness then attended and in 1918 graduated from the Reykjavík Lyceum. By the time his first novel, Barn náttúrunnar (Child of Nature, 1919), was published he had already begun his travels on the European continent.

1920s
In 1922, Halldór moved into and considered joining the Abbaye Saint-Maurice et Saint-Maur in Clervaux, Luxembourg, where the monks followed the rules of Saint Benedict of Nursia. In 1923 he was baptized and confirmed in the Catholic Church, adopting the surname Laxness after the homestead on which he was raised and adding the name Kiljan (the Icelandic name of Irish martyr Saint Killian); Laxness practiced self-study, read books, and studied French, Latin, theology and philosophy. He became a member of a group that prayed for reversion of the Nordic countries to Catholicism. Laxness wrote of his experiences in the essay Kaþólsk viðhorf (1925) and in the novels Undir Helgahnúk (1924) and Vefarinn mikli frá Kasmír (1927), the latter hailed by Icelandic critic Kristján Albertsson: "Finally, finally, a grand novel which towers like a cliff above the flatland of contemporary Icelandic poetry and fiction! Iceland has gained a new literary giant - it is our duty to celebrate the fact with joy!" Laxness's religious period did not last long. He lived in the United States from 1927 to 1929, giving lectures on Iceland and attempting to write screenplays for Hollywood films. During this time he became attracted to socialism:

[Laxness] did not become a socialist in America from studying manuals of socialism but from watching the starving unemployed in the parks.

Laxness joined the socialist bandwagon… with a book Alþýðubókin (The Book of the People, 1929) of brilliant burlesque and satirical essays

Beside the fundamental idea of socialism, the strong sense of Icelandic individuality is also the sustaining element in Alþýðubókin. The two elements are entwined together in characteristic fashion and in their very union give the work its individual character.

In 1929 Laxness published an article critical of the U.S. in Heimskringla, a Canadian newspaper. This resulted in charges against him, his detention, and the forfeiture of his passport. With the aid of Upton Sinclair and the ACLU, the charges were dropped and Laxness returned to Iceland.

1930s
By the 1930s Laxness "had become the apostle of the younger generation" of Icelandic writers. "Salka Valka (1931–32) began the great series of sociological novels, often coloured with socialist ideas, continuing almost without a break for nearly twenty years. This was probably the most brilliant period of his career, and it is the one which produced those of his works that have become most famous. But Laxness never attached himself permanently to a particular dogma." In addition to the two parts of Salka Valka, Laxness published Fótatak manna (Steps of Men) in 1933, a collection of short stories, as well as other essays, notably Dagleið á fjöllum (A Day's Journey in the Mountains) in 1937.

Laxness's next novel was Sjálfstætt fólk (Independent People (1934 and 1935), which has been called "one of the best books of the twentieth century."

When Salka Valka was published in English in 1936 a reviewer at the Evening Standard wrote: "No beauty is allowed to exist as ornamentation in its own right in these pages; but the work is replete from cover to cover with the beauty of its perfection."

In 1937 Laxness wrote the poem Maístjarnan (The May Star), which was set to music by Jón Ásgeirsson and became a socialist anthem.

This was followed by the four-part novel Heimsljós (World Light, 1937, 1938, 1939, and 1940), which is loosely based on the life of Magnús Hjaltason Magnusson, a minor Icelandic poet of the late 19th century. It has been "consistently regarded by many critics as his most important work."

Laxness also traveled to the Soviet Union in 1938 and wrote approvingly of the Soviet system and culture. He was present at the "Trial of the Twenty-one" and wrote about it in detail in his book Gerska æfintýrið (The Russian Adventure).

In the late 1930s Laxness developed a unique spelling system that was closer to pronunciation than standard Icelandic. This characteristic of his writing is lost in translation.

1940s
In 1941 Laxness translated Hemingway's A Farewell to Arms into Icelandic, which caused controversy because of his use of neologisms. He continued to court controversy over the next few years through the publication of new editions of several Icelandic sagas using modern Icelandic rather than the Old Norse orthography that had become customary. Laxness and his publishing partners were taken to court after the publication of his edition of Hrafnkels saga in 1942. They were found guilty of violating a recent copyright law, but eventually acquitted when the copyright law was deemed a violation of the freedom of the press.

Laxness's "epic" three-part work of historical fiction, Íslandsklukkan (Iceland's Bell), was published between 1943 and 1946. It has been described as a novel of broad "geographical and political scope… expressly concerned with national identity and the role literature plays in forming it… a tale of colonial exploitation and the obdurate will of a suffering people." "Laxness’s three-volume Íslandsklukkan … is probably the most significant [Icelandic] novel of the 1940s."

In 1946 the English translation of Independent People was published as a Book of the Month Club selection in the U.S. and sold over 450,000 copies.

In 1945 Halldór and his second wife, Auður Sveinsdóttir, moved into Gljúfrasteinn, a new house built in the countryside near Mosfellsbær, where they started a family. In addition to her domestic duties, Auður assumed the roles of personal secretary and business manager.

In response to the establishment of a permanent U.S. military base in Keflavík, Halldór wrote the satire Atómstöðin (The Atom Station), which may have contributed to a blacklisting of his novels in the U.S. "The demoralization of the occupation period is described ... nowhere as dramatically as in Halldór Kiljan Laxness' Atómstöðin (1948)... [where he portrays] postwar society in Reykjavík, completely torn from its moorings by the avalanche of foreign gold." For its examination of modern Reykjavík, many critics and readers consider Atómstöðin the exemplary "Reykjavík Novel."

1950s

In 1952 Halldór was awarded the Soviet-sponsored World Peace Council literary prize.

A Swedish film adaptation of his novel Salka Valka, directed by Arne Mattsson and filmed by Sven Nykvist, was released in 1954.

In 1955 Laxness was awarded the Nobel Prize in Literature "for his vivid epic power, which has renewed the great narrative art of Iceland".

"His chief literary works belong to the genre… [of] narrative prose fiction. In the history of our literature Laxness is mentioned beside Snorri Sturluson, the author of 'Njals saga', and his place in world literature is among writers such as Cervantes, Zola, Tolstoy, and Hamsun… He is the most prolific and skillful essayist in Icelandic literature both old and new…"

In the presentation address for the Nobel, Elias Wessén said:"He is an excellent painter of Icelandic scenery and settings. Yet this is not what he has conceived of as his chief mission. 'Compassion is the source of the highest poetry. Compassion with Asta Sollilja on earth,' he says in one of his best books… And a social passion underlies everything Halldór Laxness has written. His personal championship of contemporary social and political questions is always very strong, sometimes so strong that it threatens to hamper the artistic side of his work. His safeguard then is the astringent humour which enables him to see even people he dislikes in a redeeming light, and which also permits him to gaze far down into the labyrinths of the human soul."

In his acceptance speech, Laxness spoke of:"… the moral principles [my grandmother] instilled in me: never to harm a living creature; throughout my life, to place the poor, the humble, the meek of this world above all others; never to forget those who were slighted or neglected or who had suffered injustice, because it was they who, above all others, deserved our love and respect…" Laxness grew increasingly disenchanted with the Soviet bloc after the suppression of the Hungarian Revolution of 1956.

In 1957 Halldór and his wife went on a world tour, stopping in New York City, Washington, DC, Chicago, Madison, Salt Lake City, San Francisco, Peking (Beijing), Bombay (Mumbai), Cairo, and Rome.

Major works in this decade were Gerpla, (The Happy Warriors/Wayward Heroes, 1952), Brekkukotsannáll, (The Fish Can Sing, 1957), and Paradísarheimt, (Paradise Reclaimed, 1960).

Later years
In the 1960s Laxness was very active in Icelandic theater. He wrote and produced plays, the most successful of which was The Pigeon Banquet (Dúfnaveislan, 1966).

In 1968 Laxness published the "visionary novel" Kristnihald undir Jökli (Under the Glacier / Christianity at the Glacier). In the 1970s he published what he called "essay novels": Innansveitarkronika (A Parish Chronicle, 1970) and Guðsgjafaþula (A Narration of God's Gifts, 1972). Neither has been translated into English.

Laxness was awarded the Sonning Prize in 1969.

In 1970 Laxness published an influential ecological essay, Hernaðurinn gegn landinu (The War Against the Land). He continued to write essays and memoirs into the 1980s. As he grew older he began to suffer from Alzheimer's disease and eventually moved into a nursing home, where he died on 8 February 1998, at the age of 95.

Family and legacy
In 1922, Laxness met Málfríður Jónsdóttir (29 August 1896 - 7 November 2003), who gave birth to his first daughter, María, on 10 April 1923.

In 1930, he married Ingibjörg Einarsdóttir (3 May 1908 - 22 January 1994), who gave birth to his son Einar on 9 August 1931. In 1940 they divorced.

In 1939, he met Auður Sveinsdóttir (30 June 1918 - 29 October 2012) at Laugavatn. Auður waited for Laxness and made sacrifices so he could focus on his work. They married in 1945 and moved into their home, Gljúfrasteinn, in Mosfellsbær later that year. Auður and Halldór had two daughters: Sigríður, born 26 May 1951, and Guðný, born 23 May 1954.

His daughter Guðný Halldórsdóttir is a filmmaker whose first work was the 1989 adaptation of Kristnihald undir jōkli (Under the Glacier). In 1999 her adaptation of Laxness's story Úngfrúin góða og Húsið (The Honour of the House) was submitted for consideration for the Academy Award for Best Foreign Film. Guðný's son, Halldór Laxness Halldórsson, is a writer, actor, and poet. A grandchild, Auður Jónsdóttir, is an author and playwright. Gljúfrasteinn (Laxness's house, grounds, and personal effects) is now a museum operated by the government of Iceland.

In the 21st century, interest in Laxness in English-speaking countries increased after several of his novels were reissued and the first English-language publications of Iceland's Bell (2003) and The Great Weaver from Kashmir (2008). In 2016 a new English-language translation of Gerpla was published as Wayward Heroes. A new English-language translation of Salka Valka was released in 2022 to widespread acclaim.

Halldór Guðmundsson's book The Islander: A Biography of Halldór Laxness won the Icelandic Literary Prize for best work of nonfiction in 2004.

Numerous dramatic adaptations of Laxness's work have been staged in Iceland. In 2005 the Icelandic National Theatre premiered a play by Ólafur Haukur Símonarson, Halldór í Hollywood (Halldór in Hollywood), about Laxness's time in the United States in the 1920s.

A biennial Halldór Laxness International Literary Prize is awarded at the Reykjavík International Literary Festival.

Novels

 * 1919: Barn náttúrunnar (Child of Nature)
 * 1924: Undir Helgahnúk (Under the Holy Mountain)
 * 1927: Vefarinn mikli frá Kasmír (The Great Weaver from Kashmir)
 * 1931: Þú vínviður hreini (O Thou Pure Vine) – Part I of Salka Valka
 * 1932: Fuglinn í fjörunni (The Bird on the Beach) – Part II of Salka Valka
 * 1933: Úngfrúin góða og Húsið (The Honour of the House), as part of Fótatak manna: sjö þættir
 * 1934: Sjálfstætt fólk — Part I, Landnámsmaður Íslands (Icelandic Pioneers), Independent People
 * 1935: Sjálfstætt fólk – Part II, Erfiðir tímar (Hard Times), Independent People
 * 1937: Ljós heimsins (The Light of the World) – Part I of Heimsljós (World Light)
 * 1938: Höll sumarlandsins (The Palace of the Summerland) – Part II of Heimsljós (World Light)
 * 1939: Hús skáldsins (The Poet's House) – Part III of Heimsljós (World Light)
 * 1940: Fegurð himinsins (The Beauty of the Skies) – Part IV of Heimsljós (World Light)
 * 1943: Íslandsklukkan (Iceland's Bell) – Part I of Íslandsklukkan (Iceland's Bell)
 * 1944: Hið ljósa man (The Bright Maiden) – Part II of Íslandsklukkan (Iceland's Bell)
 * 1946: Eldur í Kaupinhafn (Fire in Copenhagen) – Part III of Íslandsklukkan (Iceland's Bell)
 * 1948: Atómstöðin (The Atom Station)
 * 1952: Gerpla (The Happy Warriors (1958) / Wayward Heroes (2016))
 * 1957: Brekkukotsannáll (The Fish Can Sing)
 * 1960: Paradísarheimt (Paradise Reclaimed)
 * 1968: Kristnihald undir Jökli (Under the Glacier / Christianity at the Glacier)
 * 1970: Innansveitarkronika (A Parish Chronicle)
 * 1972: Guðsgjafaþula (A Narration of God's Gifts)

Stories

 * 1923: Nokkrar sögur
 * 1933: Fótatak manna
 * 1935: Þórður gamli halti
 * 1942: Sjö töframenn
 * 1954: Þættir (collection)
 * 1964: Sjöstafakverið
 * 1981: Við Heygarðshornið
 * 1987: Sagan af brauðinu dýra
 * 1992: Jón í Brauðhúsum
 * 1996: Fugl á garðstaurnum og fleiri smásögur
 * 1999: Úngfrúin góða og Húsið
 * 2000: Smásögur
 * 2001: Kórvilla á Vestfjörðum og fleiri sögur

Plays

 * 1934: Straumrof
 * 1950: Snæfríður Íslandssól (from the novel Íslandsklukkan)
 * 1954: Silfurtúnglið
 * 1961: Strompleikurinn
 * 1962: Prjónastofan Sólin
 * 1966: Dúfnaveislan
 * 1970: Úa (from the novel Kristnihald undir Jökli)
 * 1972: Norðanstúlkan (from the novel Atómstöðin)

Poetry

 * 1925: Únglíngurinn í skóginum
 * 1930: Kvæðakver

Travelogues and essays

 * 1925: Kaþólsk viðhorf (Catholic View)
 * 1929: Alþýðubókin (The Book of the People)
 * 1933: Í Austurvegi (In the Baltic)
 * 1938: Gerska æfintýrið (The Russian Adventure)

Memoirs

 * 1952: Heiman eg fór (subtitle: sjálfsmynd æskumanns)
 * 1963: Skáldtími
 * 1975: Í túninu heima, part I
 * 1976: Úngur eg var, part II
 * 1978: Sjömeistarasagan, part III
 * 1980: Grikklandsárið, part IV
 * 1987: Dagar hjá múnkum

Translations

 * 1941: Vopnin kvödd (A Farewell to Arms), Ernest Hemingway
 * 1943: Kirkjan á fjallinu (Kirken på bjerget), Gunnar Gunnarsson
 * 1945: Birtingur (Candide), Voltaire
 * 1966: Veisla í Farángrinum (A Moveable Feast), Ernest Hemingway

Other

 * 1941: Laxdaela Saga, edited with preface
 * 1942: Hrafnkatla, edited with preface
 * 1945: Brennunjal's Saga, edited with afterword
 * 1945: Alexander's Saga, edited with preface
 * 1946: Grettis Saga, edited with preface
 * 1952: Kvaedi og ritgerdir by Johann Jonsson, edited with preface

title: Halldór Laxness


text:
Talk:History of the Republic of Turkey

Renaming
I think that this article should be renamed "Republican history of Turkey" in line with Economic history of Turkey or Consitutional history of Turkey. What do others think? Baristarim 14:01, 5 December 2006 (UTC)

 * Republic of Turkey is the name of the country itself. --Mugsywwiii 03:55, 28 December 2006 (UTC)

 * Agree that the article should be renamed. But I think the appropriate name should be History of the Republic of Turkey, this mirrors the name of many "History of the Republic of ..." articles in Wikipedia (see ). History of Turkey should include more ancient history, for instance, the Ottoman Empire. LK (talk) 08:29, 19 August 2010 (UTC)

 * I agree with you. Kavas (talk) 12:42, 19 August 2010 (UTC)

 * I have moved it as there's no objection. Kavas (talk) 22:31, 20 August 2010 (UTC)

Reference 9
The quote in reference 9 makes no grammatical sense. This makes me think it is either a misquote or mistranslation. 2A02:C7D:86B:4A00:1C53:B14C:9DF7:E656 (talk) 09:15, 25 February 2019 (UTC)

title: Talk:History of the Republic of Turkey


text:
Harmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

where k is a positive constant.

If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).

If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
 * Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time (underdamped oscillator).
 * Decay to the equilibrium position, without oscillations (overdamped oscillator).

The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.

If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.

Mechanical examples include pendulums (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.

Simple harmonic oscillator
A simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a mass m, which experiences a single force F, which pulls the mass in the direction of the point $x = 0$ and depends only on the position x of the mass and a constant k. Balance of forces (Newton's second law) for the system is $$F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = m\ddot{x} = -k x. $$

Solving this differential equation, we find that the motion is described by the function

where

The motion is periodic, repeating itself in a sinusoidal fashion with constant amplitude A. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period $$T = 2\pi/\omega$$, the time for a single oscillation or its frequency $$f=1/T$$, the number of cycles per unit time. The position at a given time t also depends on the phase φ, which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass m and the force constant k, while the amplitude and phase are determined by the starting position and velocity.

The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. The velocity is maximal for zero displacement, while the acceleration is in the direction opposite to the displacement.

The potential energy stored in a simple harmonic oscillator at position x is

Damped harmonic oscillator

In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases in proportion to the acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic oscillator there is in addition a frictional force which is always in a direction to oppose the motion. In many vibrating systems the frictional force Ff can be modeled as being proportional to the velocity v of the object: $F_{f} = −cv$, where c is called the viscous damping coefficient.

The balance of forces (Newton's second law) for damped harmonic oscillators is then $$ F = - kx - c\frac{\mathrm{d}x}{\mathrm{d}t} = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2},$$ which can be rewritten into the form $$ \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = 0, $$ where
 * $\omega_0 = \sqrt{\frac k m}$ is called the "undamped angular frequency of the oscillator",
 * is called the "damping ratio".

The value of the damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be:
 * Overdamped (ζ > 1): The system returns (exponentially decays) to steady state without oscillating. Larger values of the damping ratio ζ return to equilibrium more slowly.
 * Critically damped (ζ = 1): The system returns to steady state as quickly as possible without oscillating (although overshoot can occur if the initial velocity is nonzero).  This is often desired for the damping of systems such as doors.
 * Underdamped (ζ < 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. The angular frequency of the underdamped harmonic oscillator is given by $\omega_1 = \omega_0\sqrt{1 - \zeta^2},$ the exponential decay of the underdamped harmonic oscillator is given by $$\lambda = \omega_0\zeta.$$

The Q factor of a damped oscillator is defined as $$Q = 2\pi \times \frac{\text{energy stored}}{\text{energy lost per cycle}}.$$

Q is related to the damping ratio by $Q = \frac{1}{2\zeta}.$

Driven harmonic oscillators
Driven harmonic oscillators are damped oscillators further affected by an externally applied force F(t).

Newton's second law takes the form $$F(t) - kx - c\frac{\mathrm{d}x}{\mathrm{d}t}=m\frac{\mathrm{d}^2x}{\mathrm{d}t^2}. $$

It is usually rewritten into the form $$ \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{F(t)}{m}. $$

This equation can be solved exactly for any driving force, using the solutions z(t) that satisfy the unforced equation $$ \frac{\mathrm{d}^2z}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}z}{\mathrm{d}t} + \omega_0^2 z = 0,$$

and which can be expressed as damped sinusoidal oscillations: $$z(t) = A e^{-\zeta \omega_0 t} \sin \left( \sqrt{1 - \zeta^2} \omega_0 t + \varphi \right), $$ in the case where $ζ ≤ 1$. The amplitude A and phase φ determine the behavior needed to match the initial conditions.

Step input
In the case $ζ < 1$ and a unit step input with $x(0) = 0$: $$ \frac{F(t)}{m} = \begin{cases} \omega _0^2 & t \geq 0 \\ 0 & t < 0 \end{cases}$$ the solution is $$ x(t) = 1 - e^{-\zeta \omega_0 t} \frac{\sin \left( \sqrt{1 - \zeta^2} \omega_0 t + \varphi \right)}{\sin(\varphi)},$$

with phase φ given by

$$\cos \varphi = \zeta.$$

The time an oscillator needs to adapt to changed external conditions is of the order $τ = 1/(ζω_{0})$. In physics, the adaptation is called relaxation, and τ is called the relaxation time.

In electrical engineering, a multiple of τ is called the settling time, i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The term overshoot refers to the extent the response maximum exceeds final value, and undershoot refers to the extent the response falls below final value for times following the response maximum.

Sinusoidal driving force

In the case of a sinusoidal driving force: $$ \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{1}{m} F_0 \sin(\omega t),$$ where $$F_0$$ is the driving amplitude, and $$\omega$$ is the driving frequency for a sinusoidal driving mechanism. This type of system appears in AC-driven RLC circuits (resistor–inductor–capacitor) and driven spring systems having internal mechanical resistance or external air resistance.

The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude $$F_0$$, driving frequency $$\omega$$, undamped angular frequency $$\omega_0$$, and the damping ratio $$\zeta$$.

The steady-state solution is proportional to the driving force with an induced phase change $$\varphi$$: $$ x(t) = \frac{F_0}{m Z_m \omega} \sin(\omega t + \varphi),$$ where

is the absolute value of the impedance or linear response function, and $$ \varphi = \arctan\left(\frac{2\omega \omega_0\zeta}{\omega^2 - \omega_0^2} \right) + n\pi$$

is the phase of the oscillation relative to the driving force. The phase value is usually taken to be between −180° and 0 (that is, it represents a phase lag, for both positive and negative values of the arctan argument).

For a particular driving frequency called the resonance, or resonant frequency $\omega_r = \omega_0 \sqrt{1 - 2\zeta^2}$, the amplitude (for a given $$F_0$$) is maximal. This resonance effect only occurs when $$\zeta < 1 / \sqrt{2}$$, i.e. for significantly underdamped systems. For strongly underdamped systems the value of the amplitude can become quite large near the resonant frequency.

The transient solutions are the same as the unforced ($$F_0 = 0$$) damped harmonic oscillator and represent the systems response to other events that occurred previously. The transient solutions typically die out rapidly enough that they can be ignored.

Parametric oscillators
A parametric oscillator is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force. A familiar example of parametric oscillation is "pumping" on a playground swing. A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing and squatting, in rhythm with the swing's oscillations. The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance frequency $$\omega$$ and damping $$\beta$$.

Parametric oscillators are used in many applications. The classical varactor parametric oscillator oscillates when the diode's capacitance is varied periodically. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics, waveguide/YAG based parametric oscillators operate in the same fashion. The designer varies a parameter periodically to induce oscillations.

Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range. Thermal noise is minimal, since a reactance (not a resistance) is varied. Another common use is frequency conversion, e.g., conversion from audio to radio frequencies. For example, the Optical parametric oscillator converts an input laser wave into two output waves of lower frequency ($$\omega_s, \omega_i$$).

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing, since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the instability phenomenon.

Universal oscillator equation
The equation $$\frac{\mathrm{d}^2q}{\mathrm{d} \tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = 0$$ is known as the universal oscillator equation, since all second-order linear oscillatory systems can be reduced to this form. This is done through nondimensionalization.

If the forcing function is $f(t) = cos(ωt) = cos(ωt_{c}τ) = cos(ωτ)$, where $ω = ωt_{c}$, the equation becomes $$\frac{\mathrm{d}^2q}{\mathrm{d} \tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = \cos(\omega \tau).$$

The solution to this differential equation contains two parts: the "transient" and the "steady-state".

Transient solution
The solution based on solving the ordinary differential equation is for arbitrary constants c1 and c2

$$q_t (\tau) = \begin{cases} e^{-\zeta\tau} \left( c_1 e^{\tau \sqrt{\zeta^2 - 1}} + c_2 e^{- \tau \sqrt{\zeta^2 - 1}} \right) & \zeta > 1 \text{ (overdamping)} \\ e^{-\zeta\tau} (c_1+c_2 \tau) = e^{-\tau}(c_1+c_2 \tau) & \zeta = 1 \text{ (critical damping)} \\ e^{-\zeta \tau} \left[ c_1 \cos \left(\sqrt{1-\zeta^2} \tau\right) + c_2 \sin\left(\sqrt{1-\zeta^2} \tau\right) \right] & \zeta < 1 \text{ (underdamping)} \end{cases}$$

The transient solution is independent of the forcing function.

Steady-state solution
Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution: $$\frac{\mathrm{d}^2 q}{\mathrm{d}\tau^2} + 2 \zeta \frac{\mathrm{d}q}{\mathrm{d}\tau} + q = \cos(\omega \tau) + i\sin(\omega \tau) = e^{ i \omega \tau}.$$

Supposing the solution is of the form $$q_s(\tau) = A e^{i (\omega \tau + \varphi) }. $$

Its derivatives from zeroth to second order are $$q_s = A e^{i (\omega \tau + \varphi) }, \quad \frac{\mathrm{d}q_s}{\mathrm{d} \tau} = i \omega A e^{i (\omega \tau + \varphi) }, \quad \frac{\mathrm{d}^2 q_s}{\mathrm{d} \tau^2} = -\omega^2 A e^{i (\omega \tau + \varphi) } .$$

Substituting these quantities into the differential equation gives $$-\omega^2 A e^{i (\omega \tau + \varphi)} + 2 \zeta i \omega A e^{i(\omega \tau + \varphi)} + A e^{i(\omega \tau + \varphi)} = (-\omega^2 A + 2 \zeta i \omega A + A) e^{i (\omega \tau + \varphi)} = e^{i \omega \tau}.$$

Dividing by the exponential term on the left results in $$-\omega^2 A + 2 \zeta i \omega A + A = e^{-i \varphi} = \cos\varphi - i \sin\varphi.$$

Equating the real and imaginary parts results in two independent equations $$A (1 - \omega^2) = \cos\varphi, \quad 2 \zeta \omega A = -\sin\varphi.$$

Amplitude part
Squaring both equations and adding them together gives $$\left. \begin{aligned} A^2 (1-\omega^2)^2 &= \cos^2\varphi \\ (2 \zeta \omega A)^2 &= \sin^2\varphi \end{aligned} \right\} \Rightarrow A^2[(1 - \omega^2)^2 + (2 \zeta \omega)^2] = 1.$$

Therefore, $$A = A(\zeta, \omega) = \sgn \left( \frac{-\sin\varphi}{2 \zeta \omega} \right) \frac{1}{\sqrt{(1 - \omega^2)^2 + (2 \zeta \omega)^2}}.$$

Compare this result with the theory section on resonance, as well as the "magnitude part" of the RLC circuit. This amplitude function is particularly important in the analysis and understanding of the frequency response of second-order systems.

Phase part
To solve for $φ$, divide both equations to get $$\tan\varphi = -\frac{2 \zeta \omega}{1 - \omega^2} = \frac{2 \zeta \omega}{\omega^2 - 1} \implies \varphi \equiv \varphi(\zeta, \omega) = \arctan \left( \frac{2 \zeta \omega}{\omega^2 - 1} \right ) + n\pi.$$

This phase function is particularly important in the analysis and understanding of the frequency response of second-order systems.

Full solution
Combining the amplitude and phase portions results in the steady-state solution $$q_s(\tau) = A(\zeta,\omega) \cos(\omega \tau + \varphi(\zeta, \omega)) = A\cos(\omega \tau + \varphi).$$

The solution of original universal oscillator equation is a superposition (sum) of the transient and steady-state solutions: $$q(\tau) = q_t(\tau) + q_s(\tau).$$

Equivalent systems
Harmonic oscillators occurring in a number of areas of engineering are equivalent in the sense that their mathematical models are identical (see universal oscillator equation above). Below is a table showing analogous quantities in four harmonic oscillator systems in mechanics and electronics. If analogous parameters on the same line in the table are given numerically equal values, the behavior of the oscillators – their output waveform, resonant frequency, damping factor, etc. – are the same.

Application to a conservative force
The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, behaves as a simple harmonic oscillator.

A conservative force is one that is associated with a potential energy. The potential-energy function of a harmonic oscillator is $$V(x) = \tfrac{1}{2} k x^2.$$

Given an arbitrary potential-energy function $$V(x)$$, one can do a Taylor expansion in terms of $$x$$ around an energy minimum ($$x = x_0$$) to model the behavior of small perturbations from equilibrium.

$$V(x) = V(x_0) + V'(x_0) \cdot (x - x_0) + \tfrac{1}{2} V''(x_0) \cdot (x - x_0)^2 + O(x - x_0)^3.$$

Because $$V(x_0)$$ is a minimum, the first derivative evaluated at $$x_0$$ must be zero, so the linear term drops out: $$V(x) = V(x_0) + \tfrac{1}{2} V''(x_0) \cdot (x - x_0)^2 + O(x - x_0)^3.$$

The constant term $V(x_{0})$ is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved: $$V(x) \approx \tfrac{1}{2} V''(0) \cdot x^2 = \tfrac{1}{2} k x^2.$$

Thus, given an arbitrary potential-energy function $$V(x)$$ with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.

Simple pendulum

Assuming no damping, the differential equation governing a simple pendulum of length $$l$$, where $$g$$ is the local acceleration of gravity, is

If the maximal displacement of the pendulum is small, we can use the approximation $$\sin\theta \approx \theta$$ and instead consider the equation $$\frac{d^2\theta}{dt^2} + \frac{g}{l}\theta = 0.$$

The general solution to this differential equation is $$\theta(t) = A \cos\left(\sqrt{\frac{g}{l}} t + \varphi \right),$$ where $$A$$ and $$\varphi$$ are constants that depend on the initial conditions. Using as initial conditions $$\theta(0) = \theta_0$$ and $$\dot{\theta}(0) = 0$$, the solution is given by $$\theta(t) = \theta_0 \cos\left(\sqrt{\frac{g}{l}} t\right),$$ where $$\theta_0$$ is the largest angle attained by the pendulum (that is, $$\theta_0$$ is the amplitude of the pendulum). The period, the time for one complete oscillation, is given by the expression

which is a good approximation of the actual period when $$\theta_0$$ is small. Notice that in this approximation the period $$\tau$$ is independent of the amplitude $$\theta_0$$. In the above equation, $$\omega$$ represents the angular frequency.

Spring/mass system

When a spring is stretched or compressed by a mass, the spring develops a restoring force. Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length: $$F(t) = -kx(t),$$ where F is the force, k is the spring constant, and x is the displacement of the mass with respect to the equilibrium position. The minus sign in the equation indicates that the force exerted by the spring always acts in a direction that is opposite to the displacement (i.e. the force always acts towards the zero position), and so prevents the mass from flying off to infinity.

By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation: $$ F(t) = -kx(t) = m \frac{\mathrm{d}^2}{\mathrm{d} t^2} x(t) = ma, $$ the latter being Newton's second law of motion.

If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by $$ x(t) = A \cos \left( \sqrt{\frac{k}{m}} t \right).$$

Given an ideal massless spring, $$m$$ is the mass on the end of the spring. If the spring itself has mass, its effective mass must be included in $$m$$.

Energy variation in the spring–damping system
In terms of energy, all systems have two types of energy: potential energy and kinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which is then transferred into kinetic energy. The potential energy within a spring is determined by the equation $ U = \frac{1}{2}kx^2. $

When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximal potential energy, the kinetic energy of the mass is zero. When the spring is released, it tries to return to equilibrium, and all its potential energy converts to kinetic energy of the mass.

title: Harmonic oscillator