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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a></p> </li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a></p> </li> <li> <p><a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a></strong></p> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> <p id="models_2"><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>)</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></p> </li> </ul> </div></div> <h4 id="variational_calculus">Variational calculus</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <h2 id="differential_geometric_version">Differential geometric version</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a>, <a class="existingWikiWord" href="/nlab/show/local+action+functional">local action functional</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/source+form">source form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lepage+form">Lepage form</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principle+of+extremal+action">principle of extremal action</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noether%27s+theorem">Noether's theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/conserved+current">conserved current</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a></p> </li> </ul> </li> </ul> <h2 id="derived_differential_geometric_version">Derived differential geometric version</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+complex">BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/variational+calculus+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#InTermsOfSmoothSpaces'>In terms of smooth spaces</a></li> <ul> <li><a href='#SmoothFunctionals'>Smooth functionals</a></li> <li><a href='#FunctionalDerivative'>Functional derivative</a></li> </ul> <li><a href='#in_terms_of_the_variational_bicomplex'>In terms of the variational bicomplex</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#by_functorial_analysis_and_geometry'>By functorial analysis and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>-geometry</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><strong>Variational calculus</strong> – sometimes called <strong>secondary calculus</strong> – is a version of <a class="existingWikiWord" href="/nlab/show/differential+calculus">differential calculus</a> that deals with local extremization of <a class="existingWikiWord" href="/nlab/show/nonlinear+functionals">nonlinear functionals</a>: extremization of differentiable functions on non-finite dimensional spaces such as <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>, <a class="existingWikiWord" href="/nlab/show/spaces+of+sections">spaces of sections</a> and hence <a class="existingWikiWord" href="/nlab/show/spaces+of+histories">spaces of histories</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> in <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a>.</p> <p>Specifically, it studies the <em><a class="existingWikiWord" href="/nlab/show/critical+points">critical points</a></em> , i.e. the points where the first variational derivative of a functional vanishes, for functionals on spaces of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of <a class="existingWikiWord" href="/nlab/show/jet+bundles">jet bundles</a>. The kinds of equations specifying these critical points are <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a>.</p> <p>This applies to, and is largely motivated from, the study of <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. In <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> the critical points of a specified action functional on the space of field configurations encode the physically observable configurations.</p> <p>There are strong <a class="existingWikiWord" href="/nlab/show/cohomology">cohomological</a> tools for studying variational calculus, such as the <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a> and <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>.</p> <h2 id="InTermsOfSmoothSpaces">In terms of smooth spaces</h2> <p>We discuss some basics of variational calculus of functional in terms of <a class="existingWikiWord" href="/nlab/show/smooth+spaces">smooth spaces</a> and in particular in terms of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>.</p> <h3 id="SmoothFunctionals">Smooth functionals</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a> <a class="existingWikiWord" href="/nlab/show/manifold+with+boundary">manifold with boundary</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Σ</mi><mo>↪</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\partial \Sigma \hookrightarrow \Sigma</annotation></semantics></math>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Smooth</mi><mn>0</mn><mi>Type</mi></mrow><annotation encoding="application/x-tex"> [\Sigma, X] \in Smooth0Type </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a> (a <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>) which is the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, hence such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Sigma, X](U) = C^\infty(U \times \Sigma, X) \,. </annotation></semantics></math></div> <div class="num_defn" id="MappingSpaceWithNonVaryingBoundary"> <h6 id="definition">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mo>≔</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><msub><mo>×</mo> <mrow><mo stretchy="false">[</mo><mo>∂</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></msub><mo>♭</mo><mo stretchy="false">[</mo><mo>∂</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [\Sigma, X]_{\partial \Sigma} \coloneqq [\Sigma, X] \times_{[\partial \Sigma,X]} \flat [\partial \Sigma,X] </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> in smooth spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>♭</mo><mo stretchy="false">[</mo><mo>∂</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mo>∂</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ [\Sigma,X]_{\partial \Sigma} &\to& \flat [\partial \Sigma, X] \\ \downarrow && \downarrow \\ [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X] } \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p>the bottom morphism is the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>∂</mo><mi>Σ</mi><mo>↪</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\partial \Sigma \hookrightarrow \Sigma, X]</annotation></semantics></math> of configurations to the boundary;</p> </li> <li> <p>the right vertical morphism is the <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Disc</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Disc \dashv \Gamma)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> on smooth spaces.</p> </li> </ul> </div> <div class="num_prop" id="PlotsOfMappingSpaceWithNonVaryingBoundary"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[\Sigma, X]_{\partial \Sigma}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> whose underlying set is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\Sigma,X)</annotation></semantics></math> and whose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-plots for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> are smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon U \times \Sigma \to X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi(-,s) \colon U \to X</annotation></semantics></math> is the constant function for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mo>∂</mo><mi>Σ</mi><mo>↪</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">s \in \partial \Sigma \hookrightarrow \Sigma</annotation></semantics></math>.</p> </div> <div class="num_defn" id="SmoothFunctional"> <h6 id="definition_2">Definition</h6> <p>A <strong>functional</strong> on the mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma, X]</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of smooth spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mo>→</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R} \,. </annotation></semantics></math></div></div> <h3 id="FunctionalDerivative">Functional derivative</h3> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo lspace="verythinmathspace">:</mo><mi>ℝ</mi><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> \mathbf{d} \colon \mathbb{R} \to \Omega^1 </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> incarnated as a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/smooth+spaces">smooth spaces</a> from the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> to the smooth <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/differential+1-forms">differential 1-forms</a>.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>The <strong><a class="existingWikiWord" href="/nlab/show/functional+derivative">functional derivative</a></strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>S</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{d}S \in \Omega^1([\Sigma,X]_{\partial \Sigma}) </annotation></semantics></math></div> <p>of a functional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, def. <a class="maruku-ref" href="#SmoothFunctional"></a>, is simply its <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> as a homomorphism of smooth spaces, hence the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>S</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mover><mo>→</mo><mi>S</mi></mover><mi>ℝ</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{d} S \colon [ \Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,. </annotation></semantics></math></div></div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>This means that for each smooth parameter space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> and for each smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-parameterized collection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \phi \colon U \times \Sigma \to X </annotation></semantics></math></div> <p>of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math>, constant in the parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> in some neighbourhood of the boundary of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>ϕ</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S_\phi \colon [\Sigma,X]_{\partial \Sigma}(U) \to C^\infty(U,\mathbb{R}) </annotation></semantics></math></div> <p>is the function that sends each such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-collection of configurations to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-collection of their values under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>S</mi><msub><mo stretchy="false">)</mo> <mi>ϕ</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{d}S)_\phi \in \Omega^1(U) </annotation></semantics></math></div> <p>is the smooth <a class="existingWikiWord" href="/nlab/show/differential+1-form">differential 1-form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>S</mi><msub><mo stretchy="false">)</mo> <mi>ϕ</mi></msub><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{d}S)_\phi = \mathbf{d}(S(\phi)) \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\Sigma = [0,1] \hookrightarrow \mathbb{R}</annotation></semantics></math> be the standard <a class="existingWikiWord" href="/nlab/show/interval">interval</a>. Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>t</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L(-,-) \mathbf{d}t \in \Omega^1([0,1], C^\infty(\mathbb{R}^2)) </annotation></semantics></math></div> <p>and consider the functional</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mover><mo>→</mo><mi>γ</mi></mover><mi>X</mi><mo stretchy="false">)</mo><mo>↦</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><mi>L</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mover><mi>γ</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S \colon ([0,1] \stackrel{\gamma}{\to} X) \mapsto \int_{0}^1 L(\gamma(t), \dot \gamma(t)) d t \,. </annotation></semantics></math></div> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">U = \mathbb{R}</annotation></semantics></math> and any smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-parameterized collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>u</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{\gamma_{u} \colon \Sigma \to X\}_{u \in I}</annotation></semantics></math> the functional derivative takes the value</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mi>S</mi> <mrow><msub><mi>γ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>u</mi></mrow></mfrac><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><mi>L</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>dt</mi><mo>)</mo></mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>u</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><mi>γ</mi></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mrow><mi>d</mi><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>u</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><mover><mi>γ</mi><mo>˙</mo></mover></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mrow><mo>∂</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><mo>)</mo></mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>u</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><mi>γ</mi></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mrow><mi>d</mi><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>u</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><mover><mi>γ</mi><mo>˙</mo></mover></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><mo>)</mo></mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>u</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><mi>γ</mi></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><mover><mi>γ</mi><mo>˙</mo></mover></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mfrac><mrow><mo>∂</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>u</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{d}S_{\gamma_{(-)}} & = \left( \frac{d}{d u} \int_0^1 L(\gamma_u(t), \dot \gamma_u(t)) dt \right) \mathbf{d}u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{d \gamma_u(t)}{d u} + \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{\partial \dot \gamma_u(t)}{\partial u} \right) \mathbf{d} u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{d \gamma_u(t)}{d u} + \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{\partial }{\partial t}\frac{\partial \gamma_u(t)}{\partial u} \right) \mathbf{d} u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) - \frac{\partial}{\partial t}\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \right) \frac{\partial \gamma_u(s)}{\partial u} \mathbf{d}u \end{aligned} \,. </annotation></semantics></math></div> <p>Here we used <a class="existingWikiWord" href="/nlab/show/integration+by+parts">integration by parts</a> and used that the boundary term vanishes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mrow><mo>(</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mover><mi>γ</mi><mo>˙</mo></mover></mrow></mfrac><mi>L</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mrow><mo>∂</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><mo>)</mo></mrow><mi>d</mi><mi>s</mi></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mover><mi>γ</mi><mo>˙</mo></mover></mrow></mfrac><mi>L</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mrow><mo>∂</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mover><mi>γ</mi><mo>˙</mo></mover></mrow></mfrac><mi>L</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>u</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mrow><mo>∂</mo><msub><mi>γ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \int_{0}^1 \frac{\partial}{\partial t} \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(s), \dot \gamma_u(s)) \frac{\partial \gamma_u(s)}{\partial u} \right) d s & = \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(1), \dot \gamma_u(1)) \frac{\partial \gamma_u(1)}{\partial u} \right) - \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(0), \dot \gamma_u(0)) \frac{\partial \gamma_u(0)}{\partial u} \right) \\ & = 0 \end{aligned} </annotation></semantics></math></div> <p>since by prop. <a class="maruku-ref" href="#PlotsOfMappingSpaceWithNonVaryingBoundary"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma_{(-)}(1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma_{(-)}(0)</annotation></semantics></math> are constant.</p> </div> <h2 id="in_terms_of_the_variational_bicomplex">In terms of the variational bicomplex</h2> <p>In the special case that the functional to be varied comes from a <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a>, then its variational derivative is the image under <a class="existingWikiWord" href="/nlab/show/transgression+of+variational+differential+forms">transgression</a> of the <a class="existingWikiWord" href="/nlab/show/vertical+derivative">vertical derivative</a> in the <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a> of differential forms on the given <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a>.</p> <p>(…)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a>, <a class="existingWikiWord" href="/nlab/show/shell">shell</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/source+form">source form</a>, <a class="existingWikiWord" href="/nlab/show/evolutionary+vector+field">evolutionary vector field</a>, <a class="existingWikiWord" href="/nlab/show/evolutionary+derivative">evolutionary derivative</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/secondary+differential+calculus">secondary differential calculus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrange+multiplier">Lagrange multiplier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+sequence">variational sequence</a></p> </li> </ul> <h2 id="references">References</h2> <p>Exposition of variational calculus in terms of <a class="existingWikiWord" href="/nlab/show/jet+bundles">jet bundles</a> and <a class="existingWikiWord" href="/nlab/show/Lepage+forms">Lepage forms</a> and aimed at examples from <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is in</p> <ul> <li id="MusilovaHronek16"><a class="existingWikiWord" href="/nlab/show/Jana+Musilov%C3%A1">Jana Musilová</a>, <a class="existingWikiWord" href="/nlab/show/Stanislav+Hronek">Stanislav Hronek</a>, <em>The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories</em>, Communications in Mathematics, Volume 24, Issue 2 (Dec 2016) (<a href="https://doi.org/10.1515/cm-2016-0012">doi.org/10.1515/cm-2016-0012</a>)</li> </ul> <p>Fundamental texts on variational calculus include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ian+Anderson">Ian Anderson</a>, <em>The variational bicomplex</em>, (<a class="existingWikiWord" href="/nlab/files/AndersonVariationalBicomplex.pdf" title="pdf">pdf</a>)</p> </li> <li> <p>Hubert Goldschmidt, <a class="existingWikiWord" href="/nlab/show/Shlomo+Sternberg">Shlomo Sternberg</a>, <em>The Hamilton-Cartan formalism in the calculus of variations</em>, Annales de l’institut Fourier 23 no. 1 (1973), p. 203-267 <a href="http://www.numdam.org/item?id=AIF_1973__23_1_203_0">numdam</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Olver">Peter Olver</a>, <em>Applications of Lie groups to differential equations</em>, Springer; <em>Equivalence, invariants, and symmetry</em>, Cambridge Univ. Press 1995.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Demeter+Krupka">Demeter Krupka</a>, <em>Introduction to global variational geometry</em>, 2015</p> </li> <li> <p>Olga Krupková, <em>The geometry of ordinary variational equations</em>, Springer 1997, 251 p.</p> </li> <li> <p>Robert Hermann, <em>Some differential-geometric aspects of the Lagrange variational problem</em>, Illinois J. Math. <strong>6</strong>, 1962, 634–673 <a href="http://www.ams.org/mathscinet-getitem?mr=145457">MR145457</a> <a href="http://projecteuclid.org/euclid.ijm/1255632711">euclid</a>; <em>Differential geometry and the calculus of variations</em>, Acad. Press 1968</p> </li> <li> <p>J. Jost, X. Li-Jost, <em>Calculus of variations</em>, CUP 1998</p> </li> <li id="Zuckerman"> <p><a class="existingWikiWord" href="/nlab/show/Gregg+Zuckerman">G. J. Zuckerman</a>, <em>Action Principles and Global Geometry</em> , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. (<a class="existingWikiWord" href="/nlab/files/ZuckermanVariation.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Zuckerman’s ideas are used in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <em>Poisson algebras for non-linear field theories in the Cahiers topos</em>, <a href="http://arxiv.org/abs/1602.00708">arxiv/1602.00708</a></li> </ul> <p>Examples: <a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Jost">Jürgen Jost</a>, <em>Variational problems from physics and geometry</em>, <a href="http://www.mis.mpg.de/fileadmin/jjost/variational_problems_from_physics_and_geometry.pdf">pdf</a></p> <ul> <li>J. J. Duistermaat, <em>On the Morse index in variational calculus</em>, Adv. Math. <strong>21</strong> (1976), 2, 173–195 <a href="http://www.maths.ed.ac.uk/~aar/papers/duistermaat.pdf">pdf</a>.</li> </ul> <p>Some new results are in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/E.+Getzler">E. Getzler</a>, <em>A Darboux theorem for Hamiltonian operators in the formal calculus of variations</em>, Duke Math. J. <strong>111</strong>, n. 3 (2002), 535-560, <a href="http://www.ams.org/mathscinet-getitem?mr=1885831">MR2003e:32026</a>, <a href="http://dx.doi.org/10.1215/S0012-7094-02-11136-3">doi</a></li> <li>Alberto De Sole, Victor G. Kac, <em>The variational Poisson cohomology</em>, <a href="http://arxiv.org/abs/1106.0082">arxiv/1106.0082</a></li> </ul> <p>Geometric extremization problems (e.g. minimal surfaces), see also <a class="existingWikiWord" href="/nlab/show/geometric+measure+theory">geometric measure theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/J%C3%BCrgen+Jost">Jürgen Jost</a>, <em>The geometric calculus of variations: a short survey and a list of open problems</em>, Exposition. Math. <strong>6</strong> (1988), no. 2, 111–143, <a href="http://www.ams.org/mathscinet-getitem?mr=938179">MR89h:58036</a></li> <li>H. Federer, <em>Geometric measure theory</em>, Springer 1969(especially appendices to Russian transl.)</li> <li>Frederick J., Jr. Almgren, Almgren’s big regularity paper (book form of a 1970s preprint)</li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/BV+formalism">BV formalism</a>:</p> <ul> <li>Arthemy V. Kiselev, <em>The geometry of variations in Batalin-Vilkovisky formalism</em>, <a href="http://arxiv.org/abs/1312.1262">http://arxiv.org/abs/1312.1262</a></li> </ul> <p>Other references</p> <ul> <li>J. C. P. Bus, <em>The Lagrange multiplier rule on manifolds and optimal control of nonlinear systems</em>, SIAM J. Control and Optimization <strong>22</strong>, 5, 1984, 740-757 <a href="http://oai.cwi.nl/oai/asset/2552/2552A.pdf">pdf</a></li> </ul> <p>In the <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>-perspective:</p> <ul id="Vitagliano"> <li>L. Vitagliano, <em>Secondary calculus and the covariant phase space</em>, <a href="https://diffiety.mccme.ru/preprint/2008/02-08.pdf">pdf</a></li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> as a <a class="existingWikiWord" href="/nlab/show/convenient+category+of+spaces">convenient category</a> for <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> of <a class="existingWikiWord" href="/nlab/show/Lagrangian+quantum+field+theory">Lagrangian</a> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Grigorios+Giotopoulos">Grigorios Giotopoulos</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Field Theory via Higher Geometry I: <a class="existingWikiWord" href="/schreiber/show/Smooth+Sets+of+Fields">Smooth Sets of Fields</a></em> [<a href="https://arxiv.org/abs/2312.16301">arXiv:2312.16301</a>]</li> </ul> <h3 id="by_functorial_analysis_and_geometry">By functorial analysis and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>-geometry</h3> <p>See also references at <a class="existingWikiWord" href="/nlab/show/diffiety">diffiety</a>.</p> <p>A formalism for variational calculus based on <a class="existingWikiWord" href="/nlab/show/functorial+analysis">functorial analysis</a> (with a precise relation with functional analytic methods and jet formalism) and a long list of examples of variational problems arising in <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a> and <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> are collected in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Fr%C3%A9d%C3%A9ric+Paugam">Frédéric Paugam</a>, <em>Towards the mathematics of quantum field theory</em> (<a href="http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf">pdf</a>)</li> </ul> <p>The formulation of variational calculus in terms of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> is mentioned for instance in section 1.65 of</p> <ul id="PIZ"> <li><a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <em>Diffeology</em> (<a href="http://math.huji.ac.il/~piz/documents/Diffeology.pdf#page=64">pdf</a>)</li> </ul> <ul id="Paugam"> <li><a class="existingWikiWord" href="/nlab/show/Fr%C3%A9d%C3%A9ric+Paugam">Frédéric Paugam</a>, <em>Histories and observables in covariant field theory</em> (<a href="http://arxiv.org/abs/1010.3210">arXiv:1010.3210</a>), sec. 2.4</li> </ul> <p>following section 2.3.20 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Beilinson">Alexander Beilinson</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Drinfeld">Vladimir Drinfeld</a>, <em><a class="existingWikiWord" href="/nlab/show/Chiral+Algebras">Chiral Algebras</a></em></li> </ul> <p>For variational calculus in <a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a> see survey</p> <ul> <li>Vítor Neves, <em>Nonstandard calculus of variations, a survey</em>, <a href="http://www2.mat.ua.pt/vneves/nsa/CalcVar-vitor6.pdf">pdf</a></li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/analysis">analysis</a>, <a class="category_link" href="/nlab/all_pages/physics">physics</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on February 19, 2025 at 03:44:06. 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