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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Hamiltonian mechanics </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/261/#Item_4" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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<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> <h4 id="symplectic_geometry">Symplectic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/almost+symplectic+structure">almost symplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metaplectic+structure">metaplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metalinear+structure">metalinear structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>, <a class="existingWikiWord" href="/nlab/show/n-plectic+form">n-plectic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> <p><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/symplectomorphism+group">symplectomorphism group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+vector+field">symplectic vector field</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+form">Hamiltonian form</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+gradient">symplectic gradient</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a>, <a class="existingWikiWord" href="/nlab/show/moment+map">moment map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+formalism">BRST-BV formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>, <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> </li> </ul> <h2 id="classical_mechanics_and_quantization">Classical mechanics and quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>,</p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong>, <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+form">contact form</a>, <a class="existingWikiWord" href="/nlab/show/Reeb+vector+field">Reeb vector field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+n-algebra">Heisenberg Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a></p> </li> </ul> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/symplectic+geometry+-+contents">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='#definition_in_terms_of_symplectic_geometry'>Definition in terms of symplectic geometry</a></li> <ul> <li><a href='#comments_on_this_definition'>Comments on this definition</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#vortices_on_the_sphere'>Vortices on the sphere</a></li> </ul> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Hamiltonian mechanics is a formulation of <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> in which the basic datum in a mechanical system is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> of the system, which gives the total energy in the system in terms of the <a class="existingWikiWord" href="/nlab/show/position">positions</a> and <a class="existingWikiWord" href="/nlab/show/momentum">momenta</a> of the objects in the system.</p> <p>More abstractly, the Hamiltonian is a function on <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a>, a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> whose coordinates are <span class="newWikiWord">generalised<a href="/nlab/new/generalized+position">?</a></span> positions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">q^i</annotation></semantics></math> and momenta <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math>. (Compare this to <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a>, in which the <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> is a function on <span class="newWikiWord">state space<a href="/nlab/new/state+space">?</a></span>, whose coordinates are generalised positions and <a class="existingWikiWord" href="/nlab/show/velocity">velocities</a>.) So to do Hamiltonian mechanics properly, you must ‘mind your <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>s and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>s’ (blame <a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a> for this pun).</p> <p>To begin with, we often take phase space to be the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> of <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>. (Compare that state space is the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of configuration space.) This comes equipped with a natural <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/differential+form">form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mi mathvariant="normal">d</mi><msub><mi>p</mi> <mi>i</mi></msub><mo>∧</mo><mi mathvariant="normal">d</mi><msup><mi>q</mi> <mi>i</mi></msup><mo>,</mo></mrow><annotation encoding="application/x-tex"> \omega = \sum_i \mathrm{d}p_i \wedge \mathrm{d}q^i ,</annotation></semantics></math></div> <p>or simply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mi mathvariant="normal">d</mi><msub><mi>p</mi> <mi>i</mi></msub><mo>∧</mo><mi mathvariant="normal">d</mi><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\omega = \mathrm{d}p_i \wedge \mathrm{d}q^i</annotation></semantics></math> using the <a class="existingWikiWord" href="/nlab/show/Einstein+summation+convention">Einstein summation convention</a>. This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form is closed, in fact exact, since it is the <a class="existingWikiWord" href="/nlab/show/exterior+derivative">differential</a> of the <a class="existingWikiWord" href="/nlab/show/action+functional">action form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi mathvariant="normal">d</mi><mo stretchy="false">¯</mo></mover><mi>S</mi><mo>=</mo><msub><mi>p</mi> <mi>i</mi></msub><mo>∧</mo><mi mathvariant="normal">d</mi><msup><mi>q</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\bar{\mathrm{d}}S = p_i \wedge \mathrm{d}q^i</annotation></semantics></math>, and therefore it is a <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>.</p> <p>However, it is also possible to take phase space to be <em>any</em> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>, or even any <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>. In any case, phase space itself gives only the <a class="existingWikiWord" href="/nlab/show/kinematics">kinematics</a> (in a momentum-based rather than velocity-based sense); you need the Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> to get the <a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a>.</p> <p>Hamiltonian mechanics was developed originally for <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a>, but it is also the best known formulation of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>; many students of <a class="existingWikiWord" href="/nlab/show/physics">physics</a> (and even more so, students of <a class="existingWikiWord" href="/nlab/show/chemistry">chemistry</a>) learn it only when they study the latter. This sometimes leads to confusion about the essential differences between classical and quantum physics.</p> <h2 id="definition_in_terms_of_symplectic_geometry">Definition in terms of symplectic geometry</h2> <p>Hamiltonian mechanics is best formalized in terms of <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> as described for instance in the monograoph</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Vladimir+Arnold">Vladimir Arnold</a>, <em>Mathemtical Methods of Classical Mechanics</em> Springer.</li> </ul> <p>A classical Hamiltonian <a class="existingWikiWord" href="/nlab/show/mechanical+system">mechanical system</a> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((X,\omega), H)</annotation></semantics></math> consisting of a</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math></p> </li> <li> <p>and a <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H \in C^\infty(X)</annotation></semantics></math>.</p> </li> </ul> <p>Here</p> <ul> <li> <p><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> is the <strong><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></strong> of the physical system;</p> </li> <li> <p>a curve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma : \mathbb{R} \to X</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/trajectory">trajectory</a></strong> of the physical system in time;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\omega)</annotation></semantics></math> defines the <strong><a class="existingWikiWord" href="/nlab/show/kinematics">kinematics</a></strong> of the system;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is the <strong><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a></strong> that defined the <strong><a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a></strong> of the system.</p> </li> </ul> <p>The <strong>dynamics</strong> is encoded by declaring that those trajectories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma : \mathbb{R} \to X</annotation></semantics></math> are the physically realized trajectories that satisfy the equation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>H</mi><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>γ</mi><mo>′</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d H = \omega(\gamma', -) </annotation></semantics></math></div> <p>The components of this are <strong><a class="existingWikiWord" href="/nlab/show/Hamilton%27s+equations">Hamilton's equations</a></strong>.</p> <p>In more detail this equation means that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">t \in \mathbb{R}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/differential+form">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><mi>d</mi><mi>H</mi><msub><mo stretchy="false">)</mo> <mrow><mi>γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> (d H)_{\gamma(t)} </annotation></semantics></math></div> <p>and the 1-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mrow><mi>γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mi>γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega_{\gamma(t)}((\frac{d}{d t}\gamma(t), -) </annotation></semantics></math></div> <p>coincide.</p> <h3 id="comments_on_this_definition">Comments on this definition</h3> <p>At first, this formulation of Hamilton’s mechanics is just that, an equivalent reformulation. But as any reformulation in more abstract terms, it serves to</p> <ol> <li> <p>clarify a structure</p> </li> <li> <p>allow more powerful thinking about that structure</p> </li> <li> <p>and eventually it bears in it the seed for further developments pointing beyond this structure</p> </li> </ol> <p>Regarding the first point: this formulation of Hamiltonian mechanics makes clear what the meaning of Hamilton’s equations is for systems topologically more interesting than the example <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">X = \mathbb{R}^{2 n}</annotation></semantics></math> that many introductory physics texts concentrate on-</p> <p>Regarding the second point: the differential calculus formulation lends itself much more to high-powered arguments than the traditional component-ridden presentation. Of course the latter may still be the preferred method for some concrete computations.</p> <p>Regarding the second point: after Hamilton’s times people started thinking about what <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of a classical system should mean. One successful formalization is that of <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> which takes a symplectic manifold with Hamiltonian function on it as input datum.</p> <p>The impact that this idea of quantization from symplectic geometry eventually had is hard to underestimate. In the hands of <a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a> and his school it led to <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a>s, <a class="existingWikiWord" href="/nlab/show/Courant+algebroid">Courant algebroid</a>s and other higher <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theoretic structures</a>. In the hands of <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a> it led to the theorem on formal <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> and the vast machinery nowadays associated with that.</p> <h3 id="examples">Examples</h3> <p>The symplectic-geometry description of Hamiltonian mechanics is especially well-suited to describe topologically nontrivial phase spaces that are not <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a>s.</p> <h4 id="vortices_on_the_sphere">Vortices on the sphere</h4> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> vortices on the sphere as finite dimensional limit of 2D <a class="existingWikiWord" href="/nlab/show/equation+of+motion">Euler equations</a>: the phase space of the system of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> vortices is not a <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> but is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><msup><mo stretchy="false">)</mo> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(S^2)^n</annotation></semantics></math> .</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamilton%27s+equations">Hamilton's equations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamilton-Jacobi+equation">Hamilton-Jacobi equation</a></p> </li> <li> <p>traditional <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> and <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> are naturally embedding into <a class="existingWikiWord" href="/nlab/show/local+prequantum+field+theory">local prequantum field theory</a> by the notion of <a class="existingWikiWord" href="/nlab/show/prequantized+Lagrangian+correspondences">prequantized Lagrangian correspondences</a></p> </li> </ul> <h2 id="references">References</h2> <p>Named after <a class="existingWikiWord" href="/nlab/show/William+Rowan+Hamilton">William Rowan Hamilton</a>.</p> <p>Formulation in <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Vladimir+Arnol%27d">Vladimir Arnol'd</a>, <em><a class="existingWikiWord" href="/nlab/show/Mathematical+methods+of+classical+mechanics">Mathematical methods of classical mechanics</a></em>, Graduate Texts in Mathematics <strong>60</strong>, Springer (1978) [<a href="https://doi.org/10.1007/978-1-4757-1693-1">doi:10.1007/978-1-4757-1693-1</a>]</li> </ul> <p>A motivation for formulating Hamiltonian mechanics in terms of <a class="existingWikiWord" href="/nlab/show/symplectic+manifolds">symplectic manifolds</a>:</p> <ul> <li>Henry Cohn, <em><a href="http://math.mit.edu/~cohn/Thoughts/symplectic.html">Why symplectic geometry is the natural setting for classical mechanics</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 2, 2023 at 22:04:49. 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