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About: Canonical transformation
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This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics)." /> <meta property="og:site_name" content="DBpedia" /> <!-- /OpenGraph--> </head> <body about="http://dbpedia.org/resource/Canonical_transformation"> <!-- navbar --> <nav class="navbar navbar-expand-md navbar-light bg-light fixed-top align-items-center"> <div class="container-xl"> <a class="navbar-brand" href="http://wiki.dbpedia.org/about" title="About DBpedia" style="color: #2c5078"> <img class="img-fluid" src="/statics/images/dbpedia_logo_land_120.png" alt="About DBpedia" /> </a> <button class="navbar-toggler" type="button" data-bs-toggle="collapse" data-bs-target="#dbp-navbar" aria-controls="dbp-navbar" aria-expanded="false" aria-label="Toggle navigation"> <span class="navbar-toggler-icon"></span> </button> <div class="collapse 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Entity of Type: <a href="http://dbpedia.org/class/yago/Abstraction100002137">Abstraction100002137</a>, </span> <span class="text-nowrap">from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, </span> <span class="text-nowrap">within Data Space: <a href="http://dbpedia.org">dbpedia.org</a></span> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> <p class="lead">In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p, t) → (Q, P, t) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).</p> </div> </div> </div> </section> <!-- page-header --> <!-- property-table --> <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract"><small>dbo:</small>abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ca" >En mecànica hamiltoniana, una transformada canònica és un canvi de coordenades canònicament que preserva la forma canònica de les equacions de Hamilton, fins i tot quan la pròpia forma del hamiltonià no queda invariant. Les transformacions canòniques resulten útils en l'ús de l'equació de Hamilton-Jacobi i del , entre d'altres. Com que la mecànica lagrangiana es basa en coordenades generalitzades, les transformacions de coordenades no afecten les equacions de Lagrange i, per tant, no affecten les si modifiquem la quantitat de moviment de forma simultània mitjançant la transformada de Legendre: Per tant, les transformacions de coordenades (també anomenades transformacions puntuals) són un tipus particular de transformació canònica. Existeixen altres classes de transformacions canòniques. Podem construir transformacions més generals que involucren també la quantitat de moviment i el temps, de tipus: Les transformacions canòniques que no inclouen el temps de forma explícita s'anomenen transformacions canòniques restringides.</span><small> (ca)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="de" >In der klassischen Mechanik bezeichnet man eine aktive Transformation des Phasenraums als kanonisch, wenn sie wesentliche Aspekte der Dynamik invariant lässt. Die Invarianz der hamiltonschen Gleichungen ist dabei ein notwendiges, jedoch nicht hinreichendes Kriterium. Notwendig und hinreichend ist die Invarianz der Poisson-Klammern, ein weiteres notwendiges Kriterium ist die Invarianz des Phasenraumvolumens. Ziel dabei ist, die neue Hamilton-Funktion möglichst zu vereinfachen, im Idealfall sogar unabhängig von einer oder mehreren Variablen zu machen. In dieser Funktion sind kanonische Transformationen der Ausgangspunkt zum Hamilton-Jacobi-Formalismus. Kanonische Transformationen können aus sogenannten erzeugenden Funktionen konstruiert werden. Wichtige Beispiele kanonischer Transformationen sind Transformationen des Phasenraums, die von Transformationen des Konfigurationsraums induziert werden – sogenannte Punkttransformationen –, sowie der kanonische Fluss bei festgehaltener Zeitkonstanten, also Transformationen des Phasenraums, die durch Fortschreiten der Dynamik um eine konstante Zeitdifferenz entstehen. Die erzeugende Funktion in letzterem Fall ist die Hamiltonsche Prinzipalfunktion und entspricht gerade der Wirkung zwischen den beiden Zeitpunkten, aufgefasst als Funktion der alten und neuen Koordinaten.</span><small> (de)</small></span></li> <li><span class="literal"><span property="dbo:abstract" lang="en" >In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p, t) → (Q, P, t) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics). Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if we simultaneously change the momentum by a Legendre transformation into Therefore, coordinate transformations (also called point transformations) are a type of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include the time explicitly are called restricted canonical transformations (many textbooks consider only this type). For clarity, we restrict the presentation here to calculus and classical mechanics. Readers familiar with more advanced mathematics such as cotangent bundles, exterior derivatives and symplectic manifolds should read the related symplectomorphism article. (Canonical transformations are a special case of a symplectomorphism.) However, a brief introduction to the modern mathematical description is included at the end of this article.</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="es" >En mecánica hamiltoniana, una transformación canónica es un cambio de coordenadas canónicamente conjugadas que preserva la forma canónica de las ecuaciones de Hamilton, aun cuando la propia forma del Hamiltoniano no queda invariante. Las transformaciones canónicas resultan útiles en el enfoque de Hamilton-Jacobi de la mecánica clásica (como medio de calcular magnitudes conservadas) y en el uso del teorema de Liouville (que constituye la base de la mecánica estadística clásica). Por claridad, este artículo se restringe a un resumen básico de su uso común en mecánica clásica. El tratamiento avanzado basado en el fibrado cotangente, la derivación exterior y topología simpléctica se resume en el artículo sobre simplectomorfismos. De hecho las transformaciones canónicas son un tipo especial de simplectomorfismo. Sin embargo, este artículo contiene una breve introducción matemática a este enfoque moderno más avanzado.</span><small> (es)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="fr" >En mécanique hamiltonienne, une transformation canonique est un changement des coordonnées canoniques (q, p, t) → (Q, P, t) qui conserve la forme des équations de Hamilton, sans pour autant nécessairement conserver le Hamiltonien en lui-même. Les transformations canoniques sont utiles pour les équations de Hamilton-Jacobi (une technique utile pour calculer les quantités conservées) et le théorème de Liouville (à la base de la mécanique statistique classique). La mécanique lagrangienne étant basée sur les coordonnées généralisées, les transformations des coordonnées q → Q n'affectent pas les équations de Lagrange, et donc pas la forme des équations de Hamilton, si l'on change en même temps le moment par une transformée de Legendre en : Ainsi, les changements de coordonnées sont des sortes de transformations canoniques. Néanmoins, la classe des transformations canoniques est bien plus grande, car les coordonnées généralisées de départ, les moments et même le temps peuvent être combinés pour former de nouvelles coordonnées généralisées et de nouveaux moments. Les transformations canoniques n'impliquant pas explicitement le temps sont appelées transformations canoniques restreintes (de nombreux ouvrages se limitent à ce type de transformations).</span><small> (fr)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ko" >정준 변환 또는 바른틀 변환(canonical transformation)이란 해밀턴 역학에서 해밀턴 방정식의 형태를 보존하는 일반화 좌표의 을 말한다. 해밀턴 방정식의 형태를 보존한다는 말은, 다시 말해서 변환전의 좌표값과 변환후의 좌표값으로 치환함으로써 동일한 해밀토니안을 얻을 수 있다는 것을 말한다.</span><small> (ko)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ja" >ハミルトン形式の解析力学において、正準変換（せいじゅんへんかん、英: canonical transformation）とは、正準変数を新たなハミルトンの運動方程式を満たす新しい正準変数に写す。正準変換の下では、正準変数である一般化座標と一般化運動量は互いに混ざり合うことができ、等価な役割を果たす。また、正準変換はポアソン括弧を不変に保つ性質を持つ。幾何学的な観点からは、相空間をシンプレクティック多様体として見做した場合、基本 2形式を保つシンプレクティック同相写像に対応する。</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="it" >In meccanica razionale si chiamano trasformazioni canoniche quelle trasformazioni delle variabili generalizzate usate per descrivere un sistema attraverso le equazioni di Hamilton, che mantengono la forma delle equazioni di Hamilton. Il problema è quello di trovare una particolare trasformazione canonica (un diffeomorfismo) tale che le equazioni di Hamilton assumano una forma semplice per la loro risoluzione.</span><small> (it)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ru" >В гамильтоновой механике каноническое преобразование (также контактное преобразование) — это преобразование канонических переменных, не меняющее общий вид уравнений Гамильтона для любого гамильтониана. Канонические преобразования могут быть введены и в квантовом случае как не меняющие вид уравнений Гейзенберга. Они позволяют свести задачу с определённым гамильтонианом к задаче с более простым гамильтонианом как в классическом, так и в квантовом случае. Канонические преобразования образуют группу.</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="uk" >Канонічні перетворення — заміна узагальнених координат та узагальнених імпульсів класичної механічної системи на інші, при якій зберігається вигляд основних рівнянь гамільтонової механіки — рівнянь Гамільтона. У гамільтоновій механіці стан механічної системи задається узагальненими координатами та імпульсами , які вважаються незалежними змінними, та функцією Гамільтона . Рівняння Гамільтона мають вигляд При переході до нових змінних та форма запису рівнянь Гамільтона загалом не зберігається. Однак серед усіх таких переходів існує клас, який зберігає рівняння Гамільтона в незмінному вигляді при деякій новій функції Гамільтона . Такі перетворення називаються канонічними.</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="zh" >在哈密頓力學裏，正則變換（canonical transformation）是一種正則坐標的改變，，而同時維持哈密頓方程的形式，雖然哈密頓量可能會改變。正則變換是哈密頓-亞可比方程式與刘维尔定理的基礎。</span><small> (zh)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageID"><small>dbo:</small>wikiPageID</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><span property="dbo:wikiPageID" datatype="xsd:integer" >514534</span><small> (xsd:integer)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageLength"><small>dbo:</small>wikiPageLength</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><span property="dbo:wikiPageLength" datatype="xsd:nonNegativeInteger" >25216</span><small> (xsd:nonNegativeInteger)</small></span></li> 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style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ko" >정준 변환 또는 바른틀 변환(canonical transformation)이란 해밀턴 역학에서 해밀턴 방정식의 형태를 보존하는 일반화 좌표의 을 말한다. 해밀턴 방정식의 형태를 보존한다는 말은, 다시 말해서 변환전의 좌표값과 변환후의 좌표값으로 치환함으로써 동일한 해밀토니안을 얻을 수 있다는 것을 말한다.</span><small> (ko)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ja" >ハミルトン形式の解析力学において、正準変換（せいじゅんへんかん、英: canonical transformation）とは、正準変数を新たなハミルトンの運動方程式を満たす新しい正準変数に写す。正準変換の下では、正準変数である一般化座標と一般化運動量は互いに混ざり合うことができ、等価な役割を果たす。また、正準変換はポアソン括弧を不変に保つ性質を持つ。幾何学的な観点からは、相空間をシンプレクティック多様体として見做した場合、基本 2形式を保つシンプレクティック同相写像に対応する。</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="it" >In meccanica razionale si chiamano trasformazioni canoniche quelle trasformazioni delle variabili generalizzate usate per descrivere un sistema attraverso le equazioni di Hamilton, che mantengono la forma delle equazioni di Hamilton. Il problema è quello di trovare una particolare trasformazione canonica (un diffeomorfismo) tale che le equazioni di Hamilton assumano una forma semplice per la loro risoluzione.</span><small> (it)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ru" >В гамильтоновой механике каноническое преобразование (также контактное преобразование) — это преобразование канонических переменных, не меняющее общий вид уравнений Гамильтона для любого гамильтониана. Канонические преобразования могут быть введены и в квантовом случае как не меняющие вид уравнений Гейзенберга. Они позволяют свести задачу с определённым гамильтонианом к задаче с более простым гамильтонианом как в классическом, так и в квантовом случае. Канонические преобразования образуют группу.</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="zh" >在哈密頓力學裏，正則變換（canonical transformation）是一種正則坐標的改變，，而同時維持哈密頓方程的形式，雖然哈密頓量可能會改變。正則變換是哈密頓-亞可比方程式與刘维尔定理的基礎。</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ca" >En mecànica hamiltoniana, una transformada canònica és un canvi de coordenades canònicament que preserva la forma canònica de les equacions de Hamilton, fins i tot quan la pròpia forma del hamiltonià no queda invariant. Les transformacions canòniques resulten útils en l'ús de l'equació de Hamilton-Jacobi i del , entre d'altres. Per tant, les transformacions de coordenades (també anomenades transformacions puntuals) són un tipus particular de transformació canònica. Les transformacions canòniques que no inclouen el temps de forma explícita s'anomenen transformacions canòniques restringides.</span><small> (ca)</small></span></li> <li><span class="literal"><span property="rdfs:comment" lang="en" >In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p, t) → (Q, P, t) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="de" >In der klassischen Mechanik bezeichnet man eine aktive Transformation des Phasenraums als kanonisch, wenn sie wesentliche Aspekte der Dynamik invariant lässt. Die Invarianz der hamiltonschen Gleichungen ist dabei ein notwendiges, jedoch nicht hinreichendes Kriterium. Notwendig und hinreichend ist die Invarianz der Poisson-Klammern, ein weiteres notwendiges Kriterium ist die Invarianz des Phasenraumvolumens. Ziel dabei ist, die neue Hamilton-Funktion möglichst zu vereinfachen, im Idealfall sogar unabhängig von einer oder mehreren Variablen zu machen. In dieser Funktion sind kanonische Transformationen der Ausgangspunkt zum Hamilton-Jacobi-Formalismus. Kanonische Transformationen können aus sogenannten erzeugenden Funktionen konstruiert werden.</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="es" >En mecánica hamiltoniana, una transformación canónica es un cambio de coordenadas canónicamente conjugadas que preserva la forma canónica de las ecuaciones de Hamilton, aun cuando la propia forma del Hamiltoniano no queda invariante. Las transformaciones canónicas resultan útiles en el enfoque de Hamilton-Jacobi de la mecánica clásica (como medio de calcular magnitudes conservadas) y en el uso del teorema de Liouville (que constituye la base de la mecánica estadística clásica).</span><small> (es)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="fr" >En mécanique hamiltonienne, une transformation canonique est un changement des coordonnées canoniques (q, p, t) → (Q, P, t) qui conserve la forme des équations de Hamilton, sans pour autant nécessairement conserver le Hamiltonien en lui-même. Les transformations canoniques sont utiles pour les équations de Hamilton-Jacobi (une technique utile pour calculer les quantités conservées) et le théorème de Liouville (à la base de la mécanique statistique classique).</span><small> (fr)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="uk" >Канонічні перетворення — заміна узагальнених координат та узагальнених імпульсів класичної механічної системи на інші, при якій зберігається вигляд основних рівнянь гамільтонової механіки — рівнянь Гамільтона. У гамільтоновій механіці стан механічної системи задається узагальненими координатами та імпульсами , які вважаються незалежними змінними, та функцією Гамільтона . Рівняння Гамільтона мають вигляд</span><small> (uk)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label"><small>rdfs:</small>label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ca" >Transformació canònica</span><small> (ca)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="de" >Kanonische Transformation</span><small> (de)</small></span></li> <li><span class="literal"><span property="rdfs:label" lang="en" >Canonical transformation</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="es" >Transformación canónica</span><small> (es)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="fr" >Transformation canonique</span><small> (fr)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="it" >Trasformazione canonica</span><small> (it)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ja" >正準変換</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ko" >정준변환</span><small> (ko)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ru" >Каноническое преобразование</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="zh" >正則變換</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="uk" >Канонічне перетворення</span><small> (uk)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#sameAs"><small>owl:</small>sameAs</a> </td><td 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