The Arrheniusequation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction ... goldbook.iupac.org A00446.html Arrheniusequation IUPAC Goldbook definition ref The equation was first ... processes reactions. A historically useful generalization supported by the Arrheniusequation is that, for many ... Celsius increase in temperature. Overview In short, the Arrheniusequation gives the dependence ... authors define a modified Arrheniusequation , ref http goldbook.iupac.org M03963.html IUPAC Goldbook definition of modified Arrheniusequation ref that makes explicit the temperature dependence of the pre exponential factor. If one allows arbitrary temperature dependence of the prefactor, the Arrhenius ... . Taking the natural logarithm of the Arrheniusequation yields math ln k frac E a R frac 1 T ln A math So, when a reaction has a rate constant that obeys the Arrheniusequation, a plot of ln k versus ... partial ln k partial 1 T right P math Kinetic theory s interpretation of ArrheniusequationArrhenius ... very similar to the Arrheniusequation. Transition state theory Another Arrhenius like expression ... see Levine . See also Accelerated aging Arrhenius plot Eyring equation Q10 temperature coefficient ... in Polyethylene Using Arrheniusequation for calculating species solubility in polymers Category ... in 1889, the Swedish chemist Svante Arrhenius provided a physical justification and interpretation ... temperature kelvin s and activation energy ref http goldbook.iupac.org A00102.html Arrhenius activation ... gas constant . Alternatively, the equation may be expressed as math k A e E a k B T math The only difference ... energy from experimental data becomes singular. The modified equation is usually of the form ... power. Clearly the original Arrhenius expression above corresponds to n 0. Fitted rate ... experiment such as density dependence , there is no obstacle to incisive tests of the Arrhenius ... that again takes the form of an Arrhenius exponential multiplied by a slowly varying function ... more details
Arrhenius may refer to Carl Axel Arrhenius 1757 1824 , Swedish chemist and discoverer of the element yttrium Niklas Arrhenius , Swedish discus thrower Svante Arrhenius 1859 1927 , Swedish physical chemist and 1903 Nobel laureate Arrheniusequation , a formula for modeling the temperature dependence of reaction rate constant s. Arrhenius lunar crater , named for Svante Arrhenius 5697 Arrhenius , main belt asteroid, named for Svante Arrhenius Surname de Arrhenius es Arrhenius desambiguaci n fr Arrhenius no Arrhenius pt Arrhenius desambigua o sv Arrhenius ... more details
Oskar Benjamin Klein known for Arrheniusequation br Dissociation chemistry Theory of ionic dissociation br Acid base reactions Arrhenius definition Acid base theory prizes nowrap Nobel Prize for Chemistry 1903 br Franklin Medal 1920 religion footnotes Svante August Arrhenius 19 February 1859 2 October ... , and one of the founders of the science of physical chemistry . The Arrheniusequation , Moon lunar Impact crater crater Arrhenius lunar crater Arrhenius and the Arrhenius Labs at Stockholm University are named after him. Biography Early years Arrhenius was born on February 19, 1859 at Vik also spelled Wik or Wijk , near Uppsala , Sweden, the son of Svante Gustav and Carolina Thunberg Arrhenius ... that must be overcome before two molecules will react. The Arrheniusequation gives the quantitative ...For the Lunar or Martian meteor craters Arrhenius crater Infobox scientist name Svante Arrhenius image ... position. At the age of three, Arrhenius taught himself to read without the encouragement of his parents ... child prodigy prodigy . In later life, Arrhenius enjoyed using masses of data to discover mathematical ... of salts in water are. Arrhenius explanation was that in forming a solution, the salt dissociates ... had been that ions were produced in the process of electrolysis Arrhenius proposed that, even ... at Uppsala, but Arrhenius sent it to a number of scientists in Europe who were developing .... van t Hoff . They were far more impressed, and Ostwald even came to Uppsala to persuade Arrhenius to join his research team. Arrhenius declined, however, as he preferred to stay in Sweden for a while ... of his ion ionic theory Arrhenius proposed definitions for acid s and Base chemistry bases , in 1884 ... were substances which produce hydroxide ions in solution. Middle period Arrhenius next received a travel ... , and with van t Hoff in Amsterdam . In 1889 Arrhenius explained the fact that most reactions require ... 1900, Arrhenius became involved in setting up the Nobel Institutes and the Nobel Prize s. He was elected ... more details
at a particular temperature. See also Arrheniusequation Eyring equation Category Chemical kinetics Category Plots graphics ar de Arrheniusgraph hu Arrhenius g rbe ...An Arrhenius plot displays the logarithm of kinetic constants math ln k math , ordinate axis plotted against inverse temperature math 1 T math , abscissa . Arrhenius plots are often used to analyze the effect of temperature on the rates of chemical reactions. For a single rate limited thermally activated process, an Arrhenius plot gives a straight line, from which the activation energy and the pre exponential factor can both be determined. style float right Example br Nitrogen dioxide decay center 2 NO sub 2 sub 2 NO O sub 2 sub center Image NO2 Arrhenius k against T.svg thumb Conventional plot br k against T Image NO2 Arrhenius lnk against T 1.svg thumb Arrhenius plot br ln k against 1 T The Arrheniusequation can be given in the form math k A e E a RT math or alternatively math k A e E a k B T math The only difference is the energy units the former form uses energy mole unit mole , which is common in chemistry, while the latter form uses energy directly, which is common in physics. The different units are accounted for in using either math R math Gas constant or Boltzmanns constant math k B math . The former form can be written equivalently as math ln k ln A frac E a R left frac 1 T right math Where math k math Rate constant math A math Pre exponential factor math E a math Activation energy math R math Gas constant math T math Absolute temperature , K When plotted in the manner described above, the value of the y intercept will correspond to math ln A math , and the gradient of the line will be equal to math E a R math . The pre exponential factor, A, is a constant of proportionality that takes into account a number of factors such as the frequency of collision between and the orientation of the reacting particles. The expression math e E a RT math represents the fraction ... more details
Infobox Planet minorplanet yes width 25em bgcolour FFFFC0 apsis name Arrhenius symbol image caption discovery yes discovery ref discoverer Cornelis Johannes van Houten , Ingrid van Houten Groeneveld and Tom Gehrels discovery site Palomar Observatory discovered September 24, 1960 designations yes mp name 5697 alt names 6766 P L named after Svante Arrhenius mp category orbit ref epoch May 14, 2008 aphelion 3.3453724 perihelion 2.9451334 semimajor eccentricity 0.0636259 period 2037.4256990 avg speed inclination 13.69706 asc node 170.73264 mean anomaly 257.77820 arg peri 116.99899 satellites physical characteristics yes dimensions mass density surface grav escape velocity sidereal day axial tilt pole ecliptic lat pole ecliptic lon albedo 0.0774 temperatures temp name1 mean temp 1 max temp 1 temp name2 max temp 2 spectral type abs magnitude 12.0 5697 Arrhenius 6766 P L is a Asteroid belt main belt asteroid discovered on September 24, 1960 by Cornelis Johannes van Houten , Ingrid van Houten Groeneveld and Tom Gehrels at Palomar Observatory . External links http ssd.jpl.nasa.gov sbdb.cgi?sstr 5697 Arrhenius JPL Small Body Database Browser on 5697 Arrhenius MinorPlanets Navigator 5696 Ibsen 5698 Nolde MinorPlanets Footer DEFAULTSORT Arrhenius Category Main Belt asteroids Category Asteroids named for people Category Discoveries by Cornelis Johannes van Houten Category Discoveries by Ingrid van Houten Groeneveld Category Discoveries by Tom Gehrels Category Astronomical objects discovered in 1960 beltasteroid stub fa it 5697 Arrhenius la 5697 Arrhenius hu 5697 Arrhenius pl 5697 Arrhenius pt 5697 Arrhenius uk 5697 vi 5697 Arrhenius yo 5697 Arrhenius ... more details
Niklas Arrhenius born September 10, 1982 in Provo, Utah is a competitor in the discus throw who won the Swedish competition in this event in 2004 and 2006. He was also Sweden s discus competitor at the Athletics at the 2008 Summer Olympics Men s discus throw 2008 Summer Olympics . Arrhenius is the son of Anders Arrhenius who was a professional shot put competitor in Sweden. Niklas younger brother, Leif Arrhenius is also a thrower. Arrhenius was raised in Utah but has dual citizenship. He attended Brigham Young University where he was on the track and field team. Arrhenius is a The Church of Jesus Christ of Latter day Saints Latter day Saint . He served as an LDS missionary in the Sweden Stockholm Mission LDS Church Mission . Achievements AchievementTable colspan 5 Representing SWE 2006 2006 European Athletics Championships European Championships Gothenburg, Sweden 21st 2006 European Athletics Championships Men s discus throw 56.62 m 2008 Athletics at the 2008 Summer Olympics Olympic Games Beijing , PR China 32nd Athletics at the 2008 Summer Olympics Men s discus throw 58.22 m Personal bests Discus Throw 65.77 m 2007 Shot Put 19.75 2010 , 19.91 m indoor 2004 While competing for Mountain View High School Utah Mountain View High School in Orem, Utah , Niklas was the United States high school national records in track and field National High School Record Holder for the discus for eight years, with a throw of 234 3 breaking the previous record by nearly nine feet. References iaaf name id 176723 http mormontimes.com MITN sports.php?id 1824 Mormon Times , August 25th, 2008 http mormontimes.com MITN sports.php?id 1221 Mormon Times , June 2nd, 2008 Persondata Metadata see Wikipedia Persondata . NAME Arrhenius, Niklas ALTERNATIVE NAMES SHORT DESCRIPTION Athletics sport competitor ... PLACE OF DEATH DEFAULTSORT Arrhenius, Niklas Category 1982 births Category Living people Category ... national record holder sweden athletics bio stub fi Niklas Arrhenius sv Niklas Arrhenius ... more details
Cleanup date July 2011 Histinfo about equations in mathematics the chemistry term chemical equation Image First Equation Ever.png thumb right 300px The first use of an equals sign, equivalent to 14 x 15 71 in modern notation. From The Whetstone of Witte by Robert Recorde 1557 . An equation is a mathematics ... of two expression mathematics expressions . ref cite web url http dictionary.reference.com browse equation title Equation work Dictionary.com publisher Dictionary.com, LLC accessdate 2009 11 24 ref In modern ... of the knowns is called Equation solving solving the equation . In an equation with a single unknown, a value of that unknown for which the equation is true is called a solution or root of the equation ... equation is an equation involving only algebraic expressions in the unknowns. These are further classified by Degree mathematics degree . A linear equation is an algebraic equation of degree one. A Polynomial Polynomial equations polynomial equation is an equation in which a polynomial is set equal to another polynomial. A transcendental equation is an equation involving a transcendental function of one of its variables. A functional equation is an equation in which the unknowns are Function mathematics functions rather than simple quantities. A differential equation is an equation involving derivative s. An integral equation is an equation involving integral s. A Diophantine equation is an equation where the unknowns are required to be integer s. A quadratic equation Identities One use of equations ... 3 isbn 0 691 11822 1 ref In this case, they can be equation solving solved to find the values that satisfy the equality. For example, consider the following. math x 2 x 0 ,. math The equation is true only for two values of x , the solutions of the equation. In this case, the solutions are math x 0 math and math x 1 math . Many mathematicians ref name Nahin reserve the term equation exclusively for the second ... x 1 , math is an equation with solutions math x 0 math and math x 1 math . Whether a statement is meant ... more details
Infobox Television episode Title The Equation Series Fringe TV series Fringe Image Caption Season 1 Episode 8 Airdate November 18, 2008 Production 3T7657 Writer J. R. Orci br David H. Goodman Director Gwyneth Horder Payton Guests William Sadler actor William Sadler as Dr. Bruce Sumner Randall Duk Kim as Dashiell Kim Gillian Jacobs as Joanne Ostler Charlie Tahan as Ben Stockton Adam Grupper as Andrew Stockton Chance Kelly as Mitchell Loeb Kate Hodge as Abby Stockton Michael Cerveris as the List of Fringe ... of Fringe episodes List of Fringe episodes Prev In Which We Meet Mr. Jones Next The Dreamscape The Equation ... of some sort working on an unfinished equation. To discover the child s whereabouts, Olivia ... matter. Production The Equation was written by supervising producer J. R. Orci and co executive producer ... fearnetreview However, Wax continued that The Equation seemed rather pedestrian because nothing ... s Travis Fickett rated The Equation 7.5 10, called it a solid episode despite a few perceived plotholes ... web url http tv.ign.com articles 931 931841p1.html title Fringe The Equation Review first Travis last ... the equation title Fringe The Equation first Jane last Boursaw publisher AOL TV date 2008 11 19 accessdate ... . ref name cinemablend cite web url http www.cinemablend.com television TV Recap Fringe The Equation 13514.html title TV Recap Fringe The Equation first Erin last Dougherty publisher Cinema Blend ... articles the equation,13370 title The Equation first Noel last Murray publisher A.V. Club date 2008 ... a bit hard to swallow . ref cite web url http www.ugo.com tv fringe 108 the equation review title Fringe 1.08 The Equation Review first last publisher UGO Networks date 2008 11 19 accessdate 2011 05 26 ref References reflist 2 External links Wikiquote Fringe The Equation .5B1.08.5D The Equation http www.fox.com fringe recaps season 1 episode 8 The Equation at Fox Broadcasting Company Fox IMDb episode 1248548 Tv.com episode 1236569 Fringe Fringe episodes DEFAULTSORT Equation, The Category Fringe ... more details
refimprove date August 2008 Lt. Carl Axel Arrhenius 1757 1824 was a Swedish chemist. He is most widely known as the discoverer of the element Yttrium . Arrhenius was born in Stockholm . He was interested in mineralogy and chemistry after he met Peter Jacob Hjelm at the Swedish Royal Mint laboratory. Arrhenius was a lieutenant at the Svea artilleriregemente stationed in Vaxholm he took part in the campaign in Finland in 1788. He was promoted to Feldzeugmeister and Lieutenant Colonel at the Svea artilleriregemente and was handed the command in 1816 of the manufacture of powder in the kingdom. His chemistry studies started at the Royal Mint s Kungliga Myntet laboratory, where he studied the characteristics of powder as an artillery officer. During his visit to Paris in 1787 88 he met Antoine Lavoisier , the father of modern chemistry , and upon his return to Sweden became an ardent defender of the revolutionary teachings in chemistry promoted by Antoine Lavoisier. During his time in Vaxholm he also visited the feldspar mine in Ytterby on the island of Resar n near Vaxholm. He found a dark mineral which he named ytterbite and sent to Johan Gadolin at the University of bo for further analysis. Arrhenius was a member of the Royal Swedish Academy of War Sciences from 1799 and of the Royal Swedish Academy of Sciences from 1817. External links http elements.vanderkrogt.net element.php?sym Y Yttrium BR Persondata Metadata see Wikipedia Persondata . NAME Arrhenius, Carl Axel ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH 1757 PLACE OF BIRTH sweden DATE OF DEATH 1824 PLACE OF DEATH DEFAULTSORT Arrhenius, Carl Axel Category 1757 births Category 1824 deaths Category People from Stockholm Category Swedish chemists Category Members of the Royal Swedish Academy of Sciences Sweden scientist stub chemist stub de Carl Axel Arrhenius it Carl Axel Arrhenius pt Carl Axel Arrhenius sv Carl Axel Arrhenius vi Carl Axel Arrhenius ... more details
lunar crater data latitude 55.6 N or S N longitude 91.3 E or W E diameter 40 km depth Unknown colong 269 eponym Svante Arrhenius Arrhenius is a moon lunar impact crater that is located just on the Far side Moon far side of the Moon , near the southwest limb. In this location the vicinity of the crater can be viewed during favorable libration s, although it is viewed from on edge. To the south southeast is the worn crater Blanchard crater Blanchard , and De Roy crater De Roy lies further to the west. The outer wall of Arrhenius has been somewhat worn and eroded due to a history of minor impacts, leaving the rim rounded and low. There is a knotch in the rim to the north northwest, and an outward bulge along the southeast face. A small craterlet lies across the southwestern rim. The inner floor is relatively flat and free of features of interest. The mid point lacks a central peak. Satellite craters By convention these features are identified on lunar maps by placing the letter on the side of the crater mid point that is closest to Arrhenius. class wikitable width 25 style background eeeeee Arrhenius width 25 style background eeeeee Latitude width 25 style background eeeeee Longitude width 25 style background eeeeee Diameter align center J align center 57.6 S align center 88.3 W align center 18 km The following craters have been renamed by the International Astronomical Union IAU . Arrhenius P &mdash See Blanchard crater . References Lunar crater references Category Impact craters on the Moon Moon crater stub da Arrhenius m nekrater de Arrhenius Mondkrater fr Arrhenius crat re lunaire sv Arrhenius m nkrater ... more details
and therefore the empirical Arrheniusequation is preferred with a phenomenological interpretation ...Citation style The Eyring equation also known as Eyring Polanyi equation in chemical kinetics relates the reaction rate to temperature . It was developed almost simultaneously in 1935 by Henry Eyring , Meredith Gwynne Evans M.G. Evans and Michael Polanyi . This equation follows from the transition state theory aka , activated complex theory and is trivially equivalent to the empirical Arrheniusequation which are both readily derived from statistical thermodynamics in the kinetic theory kinetic theory of gases . ref Chapman & Enskog 1939 ref General form The general form of the Eyring Polanyi equation somewhat resembles the Arrheniusequation math k frac k mathrm B T h mathrm e frac Delta G Dagger RT math where G sup sup is the Gibbs free energy Gibbs energy of activation, k sub B sub is Boltzmann s constant , and h is Planck s constant . It can be rewritten as math k left frac k mathrm B T h right mathrm exp left frac Delta S ddagger R right mathrm exp left frac Delta H ddagger RT right math To find the linear form of the Eyring Polanyi equation math ln frac k T frac Delta H ddagger R cdot frac 1 T ln frac k mathrm B h frac Delta S ddagger R math where math k math reaction rate constant math T math absolute temperature math Delta H ddagger math enthalpy of activation math R math gas constant math k mathrm B math Boltzmann constant math h math Planck s constant math Delta S ddagger math entropy of activation A certain chemical reaction is performed at different temperatures ... kappa math as a prefactor in the Eyring equation above. This value is usually taken to be unity i.e. ... equation at the University of Regensburg http www jmg.ch.cam.ac.uk tools magnus eyring.html Online tool to calculate the reaction rate from an energy barrier in kJ mol using the Eyring equation DEFAULTSORT Eyring Equation Category Chemical kinetics Category Equations Category Physical chemistry ... more details
Bernoulli equation may refer to Bernoulli differential equation Bernoulli s equation , in fluid dynamics. Euler Bernoulli beam equation , in solid mechanics disambig zh ... more details
Characteristic equation may refer to Characteristic equation calculus , used to solve linear differential equations Characteristic equation, a Characteristic polynomial Characteristic equation characteristic polynomial equation in linear algebra used to find eigenvalues of a matrix Characteristic equation, a polynomial used to solve a recurrence relation Theorem recurrence relation mathdab ... more details
Equation editor may refer to Formula editor Read this for the comparison chart for major mathematical equation editors Microsoft Equation Editor MathType MathMagic equation editor Category Formula editors dab A long comment added to the page to prevent it being listed on Special Shortpages. Generated via Template Longcomment. ... more details
In mathematics , a summation equation or discrete integral equation is an equation in which an unknown function mathematics function appears under a summation sign. The theories of summation equations and integral equation s can be unified as integral equations on time scales ref http web.maths.unsw.edu.au cct tis tomasia IJDE rev.pdf Volterra integral equations on time scales Basic qualitative and quantitative results with applications to initial value problems on unbounded domains , Tomasia Kulik, Christopher C. Tisdell, September 3, 2007 ref using time scale calculus . A summation equation compares to a difference equation as an integral equation compares to a differential equation . The Volterra summation equation is math x t f t sum i m n k t, s, x s math where x is the unknown function, and s, a, t are integers, and f, k are known functions. References references http scholar.google.com scholar?q 22discrete integral equations 22 OR 22summation equations 22 OR 22discrete integral equation 22 OR 22summation equation 22 Summation equations or discrete integral equations Category Integral equations ... more details
In mathematics, the term exact equation can refer either of the following Exact differential equation Closed and exact differential forms Exact differential form disambig ... more details
HH equation may refer to Henderson Hasselbach equation Hodgkin Huxley model disambig Long comment to avoid being listed on short pages ... more details
Stokes equation may refer to the Airy equation the equations of Stokes flow , a linearised form of the Navier Stokes equations in the limit of small Reynolds number Stokes law disambiguation ... more details
An independent equation is an equation in a system of simultaneous equations which cannot be derived algebraically from the other equations. See also Linear algebra Indeterminate system Independent variable References Unreferenced date June 2008 Category Linear algebra Linear algebra stub ... more details
Unreferenced date December 2009 In mathematics , LHS is informal shorthand for the left hand side of an equation . Similarly, RHS is the right hand side . Each is solely a name for a term as part of an expression and they are in practice interchangeable, since equality mathematics equality is equivalence relation symmetric . This abbreviation is seldom if ever used in print it is very informal. More generally, these terms may apply to an inequation or inequality mathematics inequality . In the inequality case , there is no symmetry. The right hand side is everything on the right side of a test operator in an Expression mathematics expression . Conversely, the left hand side is everything on the left side. Some examples The equation on the right side right part of the sign is the right side of the equation and the left of the is the left side left part of equation. br br Take x 5 y 8 where x 5 would be the left hand side and y 8 would be the right hand side Homogeneous and inhomogeneous equations In solving mathematical equations, particularly linear simultaneous equations , differential equation s and integral equation s, the terminology homogeneous is often used for equations with the RHS set equal to zero. The corresponding inhomogeneous or nonhomogeneous equation then has the RHS ... operator L , with the difference being that between the equation Lf 0, to be solved for a function f , and the equation Lf g , with g a fixed function, to solve again for f . The point of the terminology appears for L a linear operator . Then any solution of the inhomogeneous equation may have a solution of the homogeneous equation added to it, and still remain a solution. For example in mathematical physics , the homogeneous equation may correspond to a physical theory formulated in empty space , while the inhomogeneous equation asks for more realistic solutions with some matter, or charged ..., though. See also equal sign DEFAULTSORT Sides Of An Equation Category Mathematical terminology es ... more details
Refimprove date January 2010 In mathematics , an algebraic equation , also called polynomial equation over a given Field mathematics field is an equation of the form math P Q math where P and Q are possibly ... 2 frac x 3 3 xy 2 y 2 frac 1 7 math is an algebraic equation over the rationals. Two equations are equivalent if they have the same set of Equation solutions . In particular the equation math P Q math ... to the study of polynomials. An algebraic equation over the rationals can always be converted to an equivalent ... 3 7 and grouping its terms in the first member, the algebraic equation above becomes the algebraic equation math 42y 4 21xy 14x 3 42xy 2 42y 2 6 0 math Although the equation math e T x 2 frac 1 T xy sin T z 2 0 math is not an algebraic equation in four variables x , y , z and T over the rational numbers because sine , exponentiation and 1 T are not polynomial functions . It is an algebraic equation ... T 3 3 frac T 5 5 frac T 7 7 cdots math 1 T and 2 are all elements of Q T . As for any equation, the solutions of an equation are the values of the variables for which the equation is true, but for algebraic ... of the algebraic equation P 0 are the roots of the polynomial P . When solving an equation, it is important to specify in which Set mathematics set the solutions are allowed. For example, for an equation .... In this case the equation is a diophantine equation . One may also look for solutions in the field of complex numbers the fundamental theorem of algebra asserts that a non constant equation has always ... has found the solution of the Cubic function equation of degree 3 and Lodovico Ferrari has solved the Quartic function equation of degree 4 . Finally Niels Henrik Abel has proved in 1824 that the quintic equationequation of degree 5 and the equations of higher degree are not always solvable using radicals. Galois theory , named after variste Galois , were introduced to give criteria deciding if an equation is solvable using radicals. References MathWorld title Algebraic Equation urlname AlgebraicEquation ... more details
An adjoint equation is a linear differential equation , usually derived from its primal equation using integration by parts . Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equation. Methods based on solution of adjoint equations are used in wing shape optimization , fluid flow control and uncertainty quantification . References reflist cite journal last Jameson first Antony title Aerodynamic Design via Control Theory journal Journal of Scientific Computing volume 3 issue 3 year 1988 DEFAULTSORT Adjoint Equation Category Differential calculus ru ... more details
In mathematics , more specifically in the study of dynamical system s and differential equation s, a Li nard equation ref Li nard, A. 1928 Etude des oscillations entretenues, Revue g n rale de l lectricit 23 , pp. 901 912 and 946 954. ref is a second order differential equation, named after the French physicist Alfred Marie Li nard . During the development of radio and vacuum tube technology, Li nard equations were intensely studied as they can be used to model oscillating circuit s. Under certain additional assumptions Li nard s theorem guarantees the uniqueness and existence of a limit cycle for such a system. Definition Let f and g be two continuously differentiable functions on R , with g an odd function and f an even function then the second order ordinary differential equation of the form math d 2x over dt 2 f x dx over dt g x 0 math is called the Li nard equation . Li nard system The equation can be transformed into an equivalent two dimensional system of ordinary differential equation s. We define math F x int 0 x f xi d xi math math x 1 x , math math x 2 dx over dt F x math then math begin bmatrix dot x 1 dot x 2 end bmatrix mathbf h x 1, x 2 begin bmatrix x 2 F x 1 g x 1 end bmatrix math is called a Li nard system . Alternatively, since Li nard equation itself also belongs to autonomous differential equation , the substitution math v dx over dt math leads the Li nard equation to a first order differential equation math v dv over dx f x v g x 0 math which belongs to Abel equation of the second kind . ref http eqworld.ipmnet.ru en solutions ode ode0317.pdf Li nard equation at eqworld . ref ref http eqworld.ipmnet.ru en solutions ode ode0125.pdf Abel equation of the second ... dt x 0 math is a Li nard equation. Li nard s theorem A Li nard system has a unique and Stability ... also Autonomous differential equation Abel equation of the second kind Footnotes reflist External links PlanetMath title LienardSystem urlname LienardSystem DEFAULTSORT Lienard equation Category Dynamical ... more details
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