dablink This article is about the overall graph theory concept of a Hamiltonianpath. For the specific problem of determining whether a Hamiltonianpath or cycle exists in a given graph, see Hamiltonianpath problem . Image Hamiltonian path.svg right thumb A Hamiltonian cycle in a dodecahedron . Like all platonic solid s, the dodecahedron is Hamiltonian. Image Herschel graph.svg thumb The Herschel graph is the smallest possible polyhedral graph that does not have a Hamiltonian cycle. In the mathematics mathematical field of graph theory , a Hamiltonianpath or traceable path is a path graph theory path in an undirected graph that visits each vertex graph theory vertex exactly once. A Hamiltonian cycle or Hamiltonian circuit is a cycle graph theory cycle in an undirected graph that visits each ... such paths and cycles exist in graphs is the Hamiltonianpath problem , which is NP complete problem NP complete . Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the Icosian game , now also known as Hamilton s puzzle , which involves finding a Hamiltonian cycle ... volume 13 year 1981 . ref Definitions A Hamiltonianpath or traceable path is a path graph theory path that visits each vertex exactly once. A graph that contains a Hamiltonianpath is called a traceable graph . A graph is Hamiltonian connected if for every pair of vertices there is a Hamiltonianpath between the two vertices. A Hamiltonian cycle , Hamiltonian circuit , vertex tour or graph cycle ... 2011 Properties Any Hamiltonian cycle can be converted to a Hamiltonianpath by removing one of its edges, but a Hamiltonianpath can be extended to Hamiltonian cycle only if its endpoints are adjacent ... s graph Hamiltonianpath problem , the computational problem of finding Hamiltonian paths Hypohamiltonian ... , the longest induced path in a hypercube Steinhaus Johnson Trotter algorithm for finding a Hamiltonian ... being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier by Thomas ... more details
dablink This article is about the specific problem of determining whether a Hamiltonianpath or cycle exists in a given graph. For the general graph theory concepts, see Hamiltonianpath . In the mathematics mathematical field of graph theory the Hamiltonianpath problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonianpath or a Hamiltonian cycle exists in a given ... are NP complete . The problem of finding a Hamiltonian cycle or path is in FNP complexity FNP . There is a simple relation between the two problems. The Hamiltonianpath problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G . The Hamiltonian cycle problem is a special case of the travelling ... algorithm for locating Hamiltonian paths is to construct a path abc... and extend it until no longer possible when the path abc...xyz cannot be extended any longer because all neighbours of z already lie in the path, one goes back one step, removing the edge yz and extending the path with a different neighbour of y if no choice produces a Hamiltonianpath, then one takes a further step back, removing the edge xy and extending the path with a different neighbour of x , and so on. This algorithm will certainly find an Hamiltonianpath if any but it runs in exponential time. Some algorithms use a rotation argument as a fast way to get unstuck when a path that cannot be extended, transforming ... his work on solving a 7 vertex instance of the HamiltonianPath Problem using a DNA computing DNA ... path problem using three locations . ref name Guard http www.jbioleng.org content 3 1 11 ref Hamiltonian ... 1045 5 A1.3 GT37&ndash 39, pp. 199&ndash 200. Refend DEFAULTSORT HamiltonianPath Problem Category ... and infinity otherwise. The directed and undirected Hamiltonian cycle problems were two ... Hamiltonian cycle problem remains NP complete for planar graph s and the undirected Hamiltonian ... more details
Hamiltonian may refer to In mathematics after William Rowan Hamilton the term Hamiltonian refers to any energy function defined by a Hamiltonian vector field , a particular vector field on a symplectic manifold more specifically, as an adjective it is used in the phrases Hamiltonian system Hamiltonianpath , in graph theory Hamiltonian cycle, a special case of a HamiltonianpathHamiltonian group , in group theory Hamiltonian control theory Hamiltonian matrix Hamiltonian flow Hamiltonian vector field Quaternions Hamiltonian numbers or quaternions In physics after William Rowan Hamilton Hamiltonian system Hamiltonian mechanics in classical mechanics Hamilton s principle Hamilton Jacobi equation Hamilton Jacobi Bellman equation Hamiltonian quantum mechanics Molecular HamiltonianHamiltonian constraint Hamiltonian fluid mechanics Hamiltonian lattice gauge theory Hamiltonian vector field In Chemistry Molecular Hamiltonian Dyall Hamiltonian In Language Hamiltonian method http www.theamericanscholar.org the new old way of learning languages Other uses Hamiltonian economic program as put forward by the eighteenth century American politician Alexander Hamilton a demonym for a person from any of several places named Hamilton . See also William Rowan Hamilton disambig Category Mathematical disambiguation ar de Hamiltonian es Hamiltoniano fr Hamiltonien gl Hamiltoniano it Hamiltoniano lt Hamiltonianas ... more details
The Hamiltonian completion problem is to find the minimal number of edges to add to a graph mathematics graph to make it Hamiltonian graph Hamiltonian . The problem is clearly NP hard in general case since its solution gives an answer to the NP complete problem of determining whether a given graph has a Hamiltonian cycle . The associated decision problem of determining whether K edges can be added to a given graph to produce a Hamiltonian graph is NP complete. Moreover, Hamiltonian completion belongs to the APX complexity class , i.e., it is unlikely that efficient constant ratio approximation algorithms exist for this problem. ref Q. S. Wu, C. L. Lu, R. C. T. Lee, http www.springerlink.com content 103cnuhn3aknv262 An Approximate Algorithm for the Weighted HamiltonianPath Completion Problem on a Tree , Lecture Notes in Computer Science , Vol. 1969 2000 Pages 156 167 ref The problem may be solved in polynomial time for certain classes of graphs, including series parallel graph s ref K. Takamizawa, T. Nishizeki, and N. Saito, Linear Time Computability of Combinatorial Problems on Series Parallel Graphs, J. ACM 29 1982 623 641 ref and their generalizations ref N. M. Korneyenko, Combinatorial algorithms on a class of graphs, Discrete Applied Mathematics , v.54 n.2 3, p.215 217, 1994 ref , which include outerplanar graph s, as well as for a line graph of a tree ref Arundhati Raychaudhuri, http portal.acm.org citation.cfm?id 222481&dl GUIDE&coll GUIDE&CFID 16443822&CFTOKEN 97960415 The total interval number of a tree and the Hamiltonian completion number of its line graph , Information .... Meloni, D. Pacciarelli, http portal.acm.org citation.cfm?id 381021 A linear algorithm for the Hamiltonian ... citation.cfm?id 975923&dl GUIDE&coll GUIDE&CFID 13226110&CFTOKEN 18722093 A linear algorithm for the Hamiltonian ... s to make them Hamiltonian. ref David Gamarnik, Maxim Sviridenko, http www.mit.edu gamarnik Papers HamCompletionPublished.pdf Hamiltonian completions of sparse random graphs , Discrete Applied Mathematics ... more details
Unreferenced stub auto yes date December 2009 About the classical theory Hamiltonian disambiguation Hamiltonian In physics and classical mechanics , a Hamiltonian system is a physical system in which force s are momentum Invariant physics invariant . Hamiltonian systems are studied in Hamiltonian mechanics . In mathematics , a Hamiltonian system is a system of differential equation s which can be written in the form of Hamilton s equations . Hamiltonian systems are usually formulated in terms of Hamiltonian vector field s on a symplectic manifold or Poisson manifold . Hamiltonian systems are a special case of dynamical system s. Examples Dynamical billiards Planetary system s Canonical general relativity See also Action angle coordinates Liouville s theorem Hamiltonian Liouville s theorem Integrable system Further Reading Treschev, D., & Zubelevich, O. 2010 . Introduction to the perturbation theory of Hamiltonian systems. Heidelberg Springer Audin, M., & Babbitt, D. G. 2008 . Hamiltonian systems and their integrability. Providence, R.I American Mathematical Society. Zaslavsky, G. M. 2007 . The physics of chaos in Hamiltonian systems. London Imperial College Press. Dickey, L. A. 2003 . Soliton equations and Hamiltonian systems. Advanced series in mathematical physics, v. 26. River Edge, NJ World Scientific. Almeida, A. M. 1992 . Hamiltonian systems Chaos and quantization. Cambridge monographs on mathematical physics. Cambridge u.a. Cambridge Univ. Press. DEFAULTSORT Hamiltonian System Category Hamiltonian mechanics Classicalmechanics stub ru zh ... more details
In mathematics , a Hamiltonian matrix math A is any real math 2 n 2 n matrix mathematics matrix math A that satisfies the condition that math KA is symmetric matrix symmetric , where math K is the skew symmetric matrix math K begin bmatrix 0 & I n I n & 0 end bmatrix math and math I sub n sub is the math n n identity matrix . In other words, math A is Hamiltonian if and only if math KA A T K T KA A T K 0. , math In the vector space of all math 2 n 2 n matrices, Hamiltonian matrices form a subspace ... n n matrices. Then math M is a Hamiltonian matrix provided that the matrices math B and math C are symmetric, and that math 1 A D sup T sup 0 . The matrix transpose transpose of a Hamiltonian matrix is Hamiltonian. The trace linear algebra trace of a Hamiltonian matrix is zero. The commutator of two Hamiltonian matrices is Hamiltonian. The eigenvalues of any Hamiltonian matrix are symmetric about the imaginary axis. The space of all Hamiltonian matrices is a Lie algebra math mathfrak Sp 2n math ... 1 pages 291 307 . ref Hamiltonian operators Let math V be a vector space, equipped with a symplectic form math . A linear map math A V mapsto V math is called a Hamiltonian operator with respect to math ... , such that math is written as math sum i e i wedge e n i math . A linear operator is Hamiltonian with respect to math if and only if its matrix in this basis is Hamiltonian. ref citation first William ... of alternating Hamiltonian matrices journal Linear Algebra and its Application volume 396 year 2005 pages 385 390 . ref From this definition, the following properties are apparent. A square of a Hamiltonian matrix is skew Hamiltonian matrix skew Hamiltonian . An exponential of a Hamiltonian matrix is symplectic matrix symplectic , and a logarithm of a symplectic matrix is Hamiltonian. See ... Introduction to Hamiltonian dynamical systems and the math N body problem publisher Springer Science ... September 2010 DEFAULTSORT Hamiltonian Matrix Category Matrices fr Matrice hamiltonienne ... more details
A mathbf p cdot d mathbf s 0 math Image Hamiltonian Optics Optical Path Length.png 200px thumb right ... Netherlands, 2011 ISBN 978 0792375821 ref and Hamiltonian optics ref name IntroductionHO H. A. Buchdahl, An Introduction to Hamiltonian Optics , Dover Publications, 1993 ISBN 978 0486675978 ref are two ... mechanics and Hamiltonian mechanics . Hamilton s principle main Hamilton s principle In physics ... math . A different approach to solving this problem consists in defining a Hamiltonian taking a Legendre ... a new set of differential equations Hamiltonian mechanics Deriving Hamilton s equations can be derived ... as in Hamiltonian mechanics, only with time t now replaced by a general parameter &sigma ... s, while Euler Lagrange s equations are second order. Lagrangian and Hamiltonian optics The general ... that the optical length of the path followed by light between two fixed points, A and B , is an extremum ... now that light travels along the x sub 3 sub axis, the path of a light ray may be parametrized ... x 2 2 math is the optical Lagrangian and math dot x k dx k dx 3 math . The optical path length OPL ... refractive index as a function of position along the path between points A and B . The Euler ... 2 n frac dx k ds math Image Hamiltonian Optics Optical Momentum.png 200px thumb right Optical momentum ... index optic the path of the light ray is curved and vector p is tangent to the light ray. The expression for the optical path length can also be written as a function of the optical momentum. Having ... math and the expression for the optical path length is math S int L , dx 3 int mathbf p cdot d mathbf s math Hamilton s equations Similarly to what happens in Hamiltonian mechanics , also in optics the Hamiltonian ..., only p sub 3 sub changes from p sub 3 A sub to p sub 3 B sub . Image Hamiltonian Optics Refraction.png ... mathbf i cdot mathbf n right mathbf n math Rays and wavefronts From the definition of optical path length ... int frac dp k dx 3 , dx 3 p k math Image Hamiltonian Optics Rays and Wavefronts.png 200px thumb left ... more details
unreferenced date January 2010 Unreferenced stub auto yes date December 2009 Orphan date December 2009 In quantum chemistry , the Dyall Hamiltonian is a modified Hamiltonian quantum mechanics Hamiltonian with two electron nature. It can be written as follows math hat mathcal H D hat mathcal H D i hat mathcal H D v C math math hat mathcal H D i sum i rm core epsilon i E ii sum r rm virt epsilon r E rr math math hat mathcal H D v sum ab rm act h ab rm eff E ab frac 1 2 sum abcd rm act left langle ab left. right cd right rangle left E ac E bd delta bc E ad right math math C 2 sum i rm core h ii sum ij rm core left 2 left langle ij left. right ij right rangle left langle ij left. right ji right rangle right 2 sum i rm core epsilon i math math h ab rm eff h ab sum j left 2 left langle aj left. right bj right rangle left langle aj left. right jb right rangle right math where labels math i,j, ldots math , math a,b, ldots math , math r,s, ldots math denote core, active and virtual orbitals see Complete active space respectively, math epsilon i math and math epsilon r math are the orbital energies of the involved orbitals, and math E mn math operators are the spin traced operators math a dagger m alpha a n alpha a dagger m beta a n beta math . These operators commute with math S 2 math and math S z math , therefore the application of these operators on a spin pure function produces again a spin pure function. The Dyall Hamiltonian behaves like the true Hamiltonian inside the CAS space, having the same eigenvalues and eigenvectors of the true Hamiltonian projected onto the CAS space. Category Quantum chemistry Chem stub it Hamiltoniano di Dyall ... more details
Classical mechanics cTopic Formulations Hamiltonian mechanics is a reformulation of classical mechanics ... spaces see Mathematical formalism Mathematical formalism , below . The Hamiltonian method differs ... Citation last1 LaValle first1 Steven M. chapter 13.4.4 Hamiltonian mechanics chapter url http planning.cs.uiuc.edu ... as understood through Hamiltonian mechanics, as well as its connection to other areas of science. Simplified overview of uses The value of the Hamiltonian is the total energy of the system being described ... 16.3 The Hamiltonian title MIT OpenCourseWare website 18.013A accessdate February 2007 ref The Hamilton ... math mathcal H p,q,t math is the so called Hamiltonian, or scalar valued Hamiltonian function. Thus ... system consisting of one particle of mass m under time independent boundary conditions The Hamiltonian ... in step 2 . Calculate the Hamiltonian using the usual definition of H as the Legendre transformation ... t mathrm d t ,. math The term on the left hand side is just the Hamiltonian that we have defined ... understood to represent all N variables of that type. Hamiltonian mechanics aims to replace the generalized ... . The Hamiltonian is the Legendre transformation Legendre transform of the Lagrangian math mathcal ... momenta into this equation and matching coefficients, we obtain the equations of motion of Hamiltonian ... not occur in the Hamiltonian, the corresponding momentum is conserved, and that coordinate can be ignored ... The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in the theory of classical mechanics, and for formulations of quantum mechanics. Geometry of Hamiltonian systems A Hamiltonian ... symplectic form , and this latter function is the Hamiltonian. Generalization to quantum mechanics ... is the Hamiltonian. To find out the rules for evaluating a Poisson bracket without resorting to differential ... Any smooth function smooth real valued function H on a symplectic manifold can be used to define a Hamiltonian vector field Hamiltonian system . The function H is known as the Hamiltonian or the energy ... more details
No footnotes date April 2009 In loop quantum gravity , dynamics such as time evolutions of fields are controlled by the Hamiltonian constraint . The identity of the Hamiltonian constraint is a major open question in quantum gravity , as is extracting of physical observables from any such specific constraint. The Thomas Thiemann Thiemann Operator physics operator has been proposed as such a constraint. Although this operator defines a complete and consistent quantum theory, doubts have been raised as to the physical reality of this theory due to inconsistencies with classical general relativity the quantum constraint algebra closes, but it is not isomorphic to the classical constraint algebra of GR, which is seen as circumstantial evidence of inconsistencies definitely not a proof of inconsistencies , and so variants have been proposed. External links http relativity.livingreviews.org open?pubNo lrr 1998 1&page node27.html Overview by Carlo Rovelli http arxiv.org abs gr qc 9606088 Thiemann s paper in Physics Letters http arxiv.org pdf gr qc 9710008 Good information on LQG Category Loop quantum gravity quantum stub ... more details
In atomic, molecular, and optical physics as well as in quantum chemistry , molecular Hamiltonian is the name given to the Hamiltonian quantum mechanics Hamiltonian representing the energy of the electron ..., point charge s and point masses. The molecular Hamiltonian is a sum of several terms its major terms ... interactions between the two kinds of charged particles. The Hamiltonian that contains only the kinetic ... Hamiltonian . From it are missing a number of small terms, most of which are due to electronic ... Schr dinger equation associated with the Coulomb Hamiltonian will predict most properties ... Hamiltonian are very rare. The main reason is that its Schr dinger equation is very difficult ... of molecular wavefunctions are based on the separation of the Coulomb Hamiltonian first devised ... from the Coulomb Hamiltonian and one considers the remaining Hamiltonian as a Hamiltonian of electrons ... the electrons move in a quantum mechanical way. Within this framework the molecular Hamiltonian has been simplified to the so called clamped nucleus Hamiltonian , also called electronic Hamiltonian ... nucleus Hamiltonian has been solved for a sufficient number of constellations of the nuclei ... Oppenheimer approximation the part of the full Coulomb Hamiltonian that depends on the electrons is replaced by the potential energy surface. This converts the total molecular Hamiltonian into another Hamiltonian that acts only on the nuclear coordinates. In the case of a breakdown of the Born ... displacements. This gives the harmonic nuclear motion Hamiltonian . Making the harmonic approximation, we can convert the Hamiltonian into a sum of uncoupled one dimensional harmonic oscillator Hamiltonians ... with the molecule. Formulated with respect to this body fixed frame the Hamiltonian accounts for rotation ... to this Hamiltonian, it is often referred to as Watson s nuclear motion Hamiltonian , but it is also known as the Eckart Hamiltonian . Coulomb Hamiltonian The algebraic form of many observables&mdash ... more details
indicates Hamiltonian elements as well. Infanticide is a biologically spiteful action in that it costs ... DEFAULTSORT Hamiltonian Spite Category Evolution Category Selection Category Human behavior Category ... more details
Summary Information Description Figure used to show that the optical pat length is constant between wavefronts. Source Own work Date 2011 05 21 Author Own work Permission other versions No Licensing self cc by sa 3.0 Copy to Wikimedia Commons bot Fbot priority true ... more details
The Path may refer to The Path album The Path album , a 2003 studio album by Show Of Hands The Path book The Path book , collection of short essayes by Konosuke Matsushita The Path comics The Path comics , an American comic book series by CrossGen Entertainment The Path video game The Path video game , a psychological horror art PC game See also Path disambiguation disambig fr The Path it The Path ... more details
About uses of path and pathway the acronym PATHPATH disambiguation PATH wiktionarypar path pathway TOC right Path , pathway or PATH may refer to Path Course navigation , the intended path of a vehicle over the surface of the Earth Trail , hiking trail , footpath , or bridle path See also Track disambiguation Footpath disambiguation Sidewalk running along the edge of a road, in some varieties of English Bicycle path or bikeway way Golden Path Dune , a metaphysical theme from Frank Herbert s Dune novels Path Vol.2 is a 2000 single by Apocalyptica from their album Cult Mathematics Path graph theory , a sequence of vertices of a graph Path topology , a continuous function Computing Path computing , in computer file systems, the human readable address of a resource PATH variable , an environment variable specifying a list of directories where executable programs are located Path social network , a social networking enabled photo sharing and messaging service Clipping path , a computer image outlining option to remove background and create transparency Control flow path, a possible execution sequence in a program often depicted as a sequence of edges in a control flow graph The st connectivity problem is sometimes known as the path problem. Pathway Biology Genetic pathway , a group of genes interacting to form an aggregate biological function Metabolic pathway , a series of chemical reactions within a cell Signal transduction Signalling pathway , a series of interactions eg from cell receptors to affect gene expression. Neural pathway , a neural tract connecting one part of the nervous system with another Dopaminergic pathways , neural pathways in the brain which transmit the neurotransmitter dopamine Music The Pathway , second album by Officium Triste released on Displeased ... Brothers of France PATH disambiguation , disambiguation page for the acronym PATH The Path disambiguation disambiguation cs Path da Sti de PATH fr Path ko nl Pad pt Caminho simple Path ... more details
About the acronym PATH other uses of pathPath disambiguation PathPATH may refer to Port Authority Trans Hudson , a subway system linking Manhattan, New York with locations in northern New Jersey PATH Atlanta , trail building organization Georgia, USA PATH Toronto , a network of underground pedestrian tunnels in Toronto, Ontario, Canada Partners for Advanced Transit and Highways , a research organization operated by the University of California The Performance Assessment Tool For Quality Improvement In Hospitals , a performance assessment system designed by the World Health Organization to support hospitals in defining quality improvement strategies, questioning their own results and translating them into actions for improvement. Positive Alternatives to Homosexuality , a coalition of ex gay organizations Program for Appropriate Technology in Health , an international, nonprofit organization based in Seattle, Washington, USA Projects for Assistance in Transition from Homelessness , to support service delivery to individuals with serious mental illnesses who are homeless or at risk of becoming homeless Potomac Appalachian Transmission Highline , proposed electrical line PATH variable , a computer operating system environment variable specifying a list of directories where executable programs are located disambig ... more details
Infobox film name On the Path image On the Path.jpg image size caption director Jasmila bani producer Damir Ibrahimovic writer Jasmila bani starring Mirjana Karanovi music cinematography Christine A. Maier editing distributor released Film date 2010 2 18 60th Berlin International Film Festival Berlinale 2010 2 20 Bosnia and Herzegovina runtime country Bosnia and Herzegovina language Bosnian budget On the Path lang bs Na putu is a 2010 Bosnian and Herzegovinan drama film directed by Jasmila bani . Plot Luna and Amar are a young Bosniaks Bosnian couple living in Sarajevo. Both have traumatic memories from the Bosnian War of the 1990 s. Luna had seen her parents killed by an anti Muslim militia in Bijeljina , and had come to Sarajevo with her grandparents as a child refugee. Amar had served as a soldier in the war and lost his brother. At present, however, they have apparently built up a successful life she as an air hostess with B&H Airlines , he as an air traffic controller at the Sarajevo International Airport . When she comes back from a flight they make love passionately and go to have a good time at a local nightclub. Though identifying as Islam in Bosnia and Herzegovina Muslim s in the context of Bosnia s ethnic set up, religion plays no part in their life. In fact, Amar drinks alcoholic drinks a bit too much which is forbidden by Islam and it is this which begins to put their relationship under strain. First of all, Amar loses his job for being drunk at work. Luna is very worried and has little hope of realizing her fragile dream of having a child with Amar. But her fears for their future increase when Amar takes on a well paid job in a Muslim community hours away from where they live. Only after quite some time has elapsed during which they have had no contact ... and Amar together on the path to a lifetime of happiness. Cast Zrinka Cvite i Leon Lu ev Mirjana ... accessdate 2011 01 01 ref References reflist External links imdb title 1156531 DEFAULTSORT On The Path ... more details
saved book title Hamiltonian Mechanics and Mathematics subtitle cover image cover color Hamiltonian Mechanics and Mathematics Basic Concepts Classical mechanics Dynamical system definition Dynamical system Equations of motion Canonical transformation Canonical transformations Generalized coordinates Phase space Hamiltonian mechanics William Rowan Hamilton Hamilton s principle Hamiltonian mechanics Hamiltonian vector field Hamilton Jacobi equation Hamilton Jacobi equations Lie bracket of vector fields Euler Lagrange equation Euler Lagrange equations Lagrangian mechanics Legendre transformation Legendre transformations Convex conjugate Legendre Fenchel transformations Poisson bracket Poisson algebra Poisson manifold Vector space Differential Geometry and Molecular Mechanics Differential geometry Symplectic vector space Symplectic manifold Symplectic group Almost complex manifold Symplectic matrix Symplectic representation Symplectic sum Symplectic geometry Symplectomorphism Symplectomorphisms Algebraic geometry Category theory Molecular Dynamics and Integrators Dynamical system Symplectic integrator Molecular dynamics Molecular modelling Relativity Theory Einstein Hilbert action General relativity Einstein field equations Solutions of the Einstein field equations Spherical coordinate system Maxwell s equations in curved spacetime Riemannian manifold Riemannian manifolds Pseudo Riemannian manifold Pseudo Riemannian manifolds Quantum Theory in Feynman s Formulation and Hamiltonian formalism inadequacies Quantum mechanics Commutator Commutators Canonical quantization Moyal bracket Path integral formulation Dirac bracket Quantum field theory Jacobi identity Lie algebra Lie group Lie groups Lie theory Lie groupoid Lie groupoids Lie algebroid Lie algebroids R algebroid Algebraic topology Double groupoid Double groupoids Higher dimensional algebra Poisson superalgebra Quantum Symmetry and TQFT Symmetry Chiral symmetry Loop quantum cosmology Quantum cohomology Topological quantum ... more details
In linear algebra , skew Hamiltonian matrices are special Matrix mathematics matrices which correspond to skew symmetric bilinear form s on a symplectic vector space . Let V be a vector space , equipped with a Symplectic vector space symplectic form math Omega math . Such a space must be even dimensional. A linear map math A V mapsto V math is called a skew Hamiltonian operator with respect to math Omega math if the form math x, y mapsto Omega A x , y math is skew symmetric. Choose a basis math e 1, ... e 2n math in V , such that math Omega math is written as math sum i e i wedge e n i math . Then a linear operator is skew Hamiltonian with respect to math Omega math if and only if its matrix A satisfies math A T J J A math , where J is the skew symmetric matrix math J begin bmatrix 0 & I n I n & 0 end bmatrix math and I sub n sub is the math n times n math identity matrix . ref name waterhouse William C. Waterhouse , http linkinghub.elsevier.com retrieve pii S0024379504004410 The structure of alternating Hamiltonian matrices , Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385 390 ref Such matrices are called skew Hamiltonian . The square of a Hamiltonian matrix is skew Hamiltonian. The converse is also true every skew Hamiltonian matrix can be obtained as the square of a Hamiltonian matrix. ref name waterhouse ref Heike Fa bender, D. Steven Mackey, Niloufer Mackey and Hongguo Xu http www.icm.tu bs.de hfassben papers hamsqrt.pdf Hamiltonian Square Roots of Skew Hamiltonian Matrices, Linear Algebra and its Applications 287, pp. 125 159, 1999 ref Notes references Category Matrices Category Linear algebra Linear algebra stub ... more details
Applied to classical field theory , the familiar symplectic Hamiltonian system Hamiltonian formalism takes the form of instantaneous Hamiltonian formalism on an infinite dimensional phase space, where canonical coordinates are field functions at some instant of time. ref Gotay, M., A multisymplectic framework for classical field theory and the calculus of variations. II. Space time decomposition, in Mechanics, Analysis and Geometry 200 Years after Lagrange North Holland, 1991 . ref This Hamiltonian formalism is applied to quantization of fields, e.g., in quantum gauge theory . The true Hamiltonian counterpart of classical first order Lagrangian classical field theory field theory is covariant Hamiltonian formalism where canonical momenta math p mu i math correspond to derivatives of fields with respect to all world coordinates math x mu math . ref Giachetta, G., Mangiarotti, L., Gennadi Sardanashvily Sardanashvily, G. , Advanced Classical Field Theory , World Scientific, 2009, ISBN 9789812838957. ref Covariant Hamilton equations are equivalent to the Euler Lagrange equations in the case of hyperregular Lagrangians. Covariant Hamiltonian field theory is developed in the Hamilton De Donder ref Krupkova, O., Hamiltonian field theory, J. Geom. Phys. 43 2002 93. ref , polysymplectic ref Giachetta, G., Mangiarotti, L., Gennadi Sardanashvily Sardanashvily, G. , Covariant Hamiltonian equations for field theory, J. Phys. A32 1999 6629 http xxx.lanl.gov abs hep th 9904062 arXiv hep th 9904062 . ref , multisymplectic ref Echeverria Enriquez, A., Munos Lecanda, M., Roman Roy, N., Geometry of multisymplectic Hamiltonian first order field theories, J. Math. Phys. 41 2002 7402. ref and math k math symplectic ref Rey, A., Roman Roy, N. Saldago, M., Gunther s formalism math k math symplectic .... 46 2005 052901. ref variants. A phase space of covariant Hamiltonian field theory is a finite ... autonomous mechanics Hamiltonian non autonomous mechanics is formulated as covariant Hamiltonian field ... more details
The Hamiltonian of Optimal control optimal control theory was developed by Lev Semyonovich Pontryagin L. S. Pontryagin as part of his Pontryagin s minimum principle minimum principle . It was inspired by, but is distinct from, the Hamiltonian mechanics Hamiltonian of classical mechanics. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to minimize the Hamiltonian. For details see Pontryagin s minimum principle . Notation and Problem statement A control math u t math is to be chosen so as to minimize the objective function math J u Psi x T int T 0 L x,u,t dt math The system state math x t math evolves according to the state equations math dot x f x,u,t qquad x 0 x 0 quad t in 0,T math the control must satisfy the constraints math a le u t le b quad t in 0,T math Definition of the Hamiltonian math H x, lambda,u,t lambda T t f x,u,t L x,u,t , math where math lambda t math is a vector of Costate equations costate variables of the same dimension as the state variables math x t math . For information on the properties of the Hamiltonian, see Pontryagin s minimum principle . The Hamiltonian in discrete time When the problem is formulated in discrete time, the Hamiltonian is defined as math H x, lambda,u,t lambda T t 1 f x,u,t L x,u,t , math and the costate equations are math lambda t frac partial H partial x math Note that the discrete time Hamiltonian at time math t math involves the costate variable at time math t 1. math ref Varaiya, Chapter 6 ref This small detail is essential so that when we differentiate ... equation which is not a backwards difference equation . The Hamiltonian of control compared to the Hamiltonian of mechanics William Rowan Hamilton defined the Hamiltonian mechanics Hamiltonian as a function ... d dt q t frac partial partial p mathcal H math In contrast the Hamiltonian of control theory as defined ... varaiya papers ps.dir NOO.pdf reflist DEFAULTSORT Hamiltonian Control Theory Category ... more details
expand Italian date December 2011 In mathematics and physics , a Hamiltonian vector field on a symplectic manifold is a vector field , defined for any energy function or Hamiltonian . Named after the physicist and mathematician William Rowan Hamilton Sir William Rowan Hamilton , a Hamiltonian vector ... s of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphism s of a symplectic manifold arising from the flow mathematics flow of a Hamiltonian vector field are known as canonical transformation s in physics and Hamiltonian symplectomorphism s in mathematics. Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold . The Lie bracket of two Hamiltonian vector fields corresponding to functions f and g on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket ... vector field X sub H sub , called the Hamiltonian vector field with the Hamiltonian H , by requiring .... Note Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful ... q i wedge mathrm d p i. math Then the Hamiltonian vector field with Hamiltonian H takes the form math ... Properties The assignment math f mapsto X f math is linear map linear , so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields. Suppose that math ... math gamma t q t ,p t math is an integral curve of the Hamiltonian vector field X sub H sub if and only ... dot p i frac partial H partial q i . math The Hamiltonian H is constant along the integral curves ... of energy in Hamiltonian mechanics . More generally, if two functions F and H have a zero Poisson ... s theorem . Symplectic form math omega math is preserved by Hamiltonian flow or equivalently, Lie derivative math mathcal L X H omega 0 math Poisson bracket The notion of a Hamiltonian vector field ... the Lie bracket of the Hamiltonian vector fields with Hamiltonians f and g . As a consequence ... more details
Hamiltonian fluid mechanics is the application of Hamiltonian mechanics Hamiltonian methods to fluid mechanics . This formalism can only apply to non dissipative fluids. Irrotational barotropic flow Take the simple example of a barotropic , inviscid vorticity free fluid. Then, the conjugate fields are the mass density field &rho and the velocity potential &phi . The Poisson bracket is given by math varphi vec x , rho vec y delta d vec x vec y math and the Hamiltonian by math mathcal H int mathrm d d x left frac 1 2 rho vec nabla varphi 2 e rho right , math where e is the internal energy density, as a function of &rho . For this barotropic flow, the internal energy is related to the pressure p by math e frac 1 rho p , math where an apostrophe , denotes differentiation with respect to &rho . This Hamiltonian structure gives rise to the following two equations of motion math begin align frac partial rho partial t & frac delta mathcal H delta varphi vec nabla cdot rho vec v , frac partial varphi partial t & frac delta mathcal H delta rho frac 1 2 vec v cdot vec v e , end align math where math vec v stackrel mathrm def nabla varphi math is the velocity and is vorticity free . The second equation leads to the Euler equations math frac partial vec v partial t vec v cdot nabla vec v e nabla rho frac 1 rho nabla p math after exploiting the fact that the vorticity is zero math vec nabla times vec v vec 0 . math See also Luke s variational principle References cite journal journal Annual Review of Fluid Mechanics volume 20 pages 225 256 year 1988 doi 10.1146 annurev.fl.20.010188.001301 title Hamiltonian Fluid Mechanics author R. Salmon bibcode 1988AnRFM..20..225S cite journal doi 10.1016 S0065 2687 08 60429 X title Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics author T. G. Shepherd year 1990 journal Advances in Geophysics volume 32 pages 287 338 Category Fluid dynamics Category Hamiltonian mechanics Category Dynamical systems ... more details
In physics , Hamiltonian lattice gauge theory is a calculational approach to gauge theory and a special case of lattice gauge theory in which the space is discretized but time is not. The Hamiltonian quantum mechanics Hamiltonian is then re expressed as a function of degrees of freedom defined on a d dimensional lattice. Following Wilson, the spatial components of the vector potential are replaced with Wilson line s over the edges, but the time component is associated with the vertices. However, the temporal gauge is often employed, setting the electric potential to zero. The eigenvalue s of the Wilson line operator mathematics operator s U e where e is the oriented edge in question take on values on the Lie group G. It is assumed that G is compact group compact , otherwise we run into many problems. The conjugate operator to U e is the electric field E e whose eigenvalues take on values in the Lie algebra math mathfrak g math . The Hamiltonian receives contributions coming from the plaquette s the magnetic contribution and contributions coming from the edges the electric contribution . Hamiltonian lattice gauge theory is exactly dual to a theory of spin network s. This involves using the Peter Weyl theorem . In the spin network basis, the spin network states are eigenstate s of the operator math Tr E e 2 math . quantum stub References Hamiltonian formulation of Wilson s lattice gauge theories, John Kogut and Leonard Susskind , Phys. Rev. D 11, 395&ndash 408 1975 Category Lattice models pt Teoria de ret culo gauge hamiltoniano ... more details
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