Unreferenced stub auto yes date December 2009 About the classical theory Hamiltonian disambiguation Hamiltonian In physics and classical mechanics , a Hamiltoniansystem is a physical system in which force s are momentum Invariant physics invariant . Hamiltonian systems are studied in Hamiltonian mechanics . In mathematics , a Hamiltoniansystem 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 HamiltonianSystem Category Hamiltonian mechanics Classicalmechanics stub ru ... more details
In mathematics, a superintegrable Hamiltoniansystem is a Hamiltoniansystem on a 2 n dimensional symplectic manifold for which the following conditions hold i There exist n k independent integrals F sub i sub of motion. Their level surfaces invariant submanifolds form a fibered manifold math F Z to N F Z math over a connected open subset math N subset mathbb R k math . ii There exist smooth real functions math s ij math on math N math such that the Poisson manifold Poisson bracket of integrals of motion reads math F i,F j s ij circ F math . iii The matrix function math s ij math is of constant corank math m 2n k math on math N math . If math k n math , this is the case of a integrable system completely integrable Hamiltoniansystem . The Mishchenko Fomenko theorem for superintegrable Hamiltonian systems generalizes the Liouville Arnold theorem on action angle coordinates of completely integrable Hamiltoniansystem as follows. Let invariant submanifolds of a superintegrable Hamiltoniansystem be connected compact and mutually diffeomorphic. Then the fibered manifold math F math is a fiber bundle in tori math T m math . Given its fiber math M math , there exists an open neighbourhood math U math of math M math which is a trivial fiber bundle provided with the bundle ... are the Darboux s theorem Darboux coordinates on a symplectic manifold math U math . A Hamiltonian of a superintegrable system depends only on the action variables math I A math which are the Casimir ... theorem for Integrable system completely integrable systems and the Mishchenko Fomenko theorem for the superintegrable ... to a toroidal cylinder math T m r times mathbb R r math . See also Integrable system ... of Hamiltonian systems, Funct. Anal. Appl. 12 1978 113. Bolsinov, A., Jovanovic, B., Noncommutative ... ph 0109031 . Fasso, F., Superintegrable Hamiltonian systems geometry and applications, Acta Appl .... Category Hamiltonian mechanics Category Dynamical systems ... more details
Hamiltonian may refer to In mathematics after William Rowan Hamilton HamiltoniansystemHamiltonian path , in graph theory Hamiltonian cycle, a special case of a Hamiltonian path Hamiltonian 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 HamiltoniansystemHamiltonian 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 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
dablink This article is about the overall graph theory concept of a Hamiltonian path. For the specific problem of determining whether a Hamiltonian path or cycle exists in a given graph, see Hamiltonian path problem . Image Hamiltonian path.svg right thumb A Hamiltonian cycle in a dodecahedron . Like all platonic solid s, the dodecahedron is Hamiltonian. Image Hamilton path.gif right frame A Hamiltonian path black over a graph blue . In the mathematics mathematical field of graph theory , a Hamiltonian ... graph theory vertex exactly once. A Hamiltonian cycle or Hamiltonian circuit is a cycle graph theory ... returns to the starting vertex. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem which is NP complete problem NP complete . Hamiltonian paths and cycles are named ... , which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron . Hamilton solved ... to arbitrary graphs. Definitions A Hamiltonian path or traceable path is a path graph theory path that visits each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph . A graph is Hamiltonian connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle , Hamiltonian circuit , vertex tour or graph ... the start and end, and so is visited twice . A graph that contains a Hamiltonian cycle is called a Hamiltonian graph . Similar notions may be defined for Graph mathematics directed graph s , where ... with arrows and the edges traced tail to head . A Hamiltonian decomposition is an Glossary of graph theory edge decomposition of a graph into Hamiltonian circuits. Examples a complete graph with more than two vertices is Hamiltonian every cycle graph is Hamiltonian every tournament graph theory tournament has an odd number of Hamiltonian paths every platonic solid , considered as a graph, is Hamiltonian every prism is Hamiltonian The Deltoidal hexecontahedron is the only non hamiltonian ... more details
system consisting of one particle of mass m under time independent boundary conditions The Hamiltonian math mathcal H math represents the energy of the system, which is the sum of kinetic energy kinetic ... systems A Hamiltoniansystem may be understood as a fiber bundle E over time R , with the Level ... to define a Hamiltonian vector field Hamiltoniansystem . The function H is known as the Hamiltonian ... by the Hamiltonian flow is commonly called the Hamiltonian mechanics of the Hamiltoniansystem. The symplectic ...Classical mechanics cTopic Formulations Hamiltonian mechanics is a reformulation of classical mechanics ... spaces see Mathematical formalism Mathematical formalism , below . The Hamiltonian method differs ... of freedom of the system , it expresses first order constraints on a 2 n dimensional phase space . ref 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. For a closed system, it is the sum of the kinetic energy kinetic and potential energy in the system ... evolution of the system. Hamiltonians can be used to describe such simple systems as a bouncing ... 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 ... the expressions in step 2 . Calculate the Hamiltonian using the usual definition of H as the Legendre ... explicitly for a system of more than two massive point particles. The finding of conserved quantity ... is the study of integrable system s, where an infinite number of independent conserved quantities can ... mathcal L partial t mathrm d t ,. math The term on the left hand side is just the Hamiltonian that we ... variables understood to represent all N variables of that type. Hamiltonian mechanics ... more details
system of electrons and nuclei with a well defined geometry from the properties of the Coulomb Hamiltonian ...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 ... 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
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 Hamiltonian Path 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 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
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 , 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 Category Loop quantum gravity quantum stub ... more details
Hamiltonian elements as well. Economy It was recently suggested that spiteful motivations may .... References reflist 2 sociobiology evolutionary psychology DEFAULTSORT Hamiltonian Spite Category ... more details
Applied to classical field theory , the familiar symplectic HamiltoniansystemHamiltonian 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 . 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 . 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 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 with respect to math x math we get a term involving math lambda t 1 math on the right ... a costate 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 ... math frac d dt q t frac partial partial p mathcal H math In contrast the Hamiltonian of control theory ... paleale.eecs.berkeley.edu varaiya papers ps.dir NOO.pdf reflist DEFAULTSORT Hamiltonian Control Theory ... more details
saved book title Hamiltonian Mechanics, Quantum Theory, Relativity and Geometry Vol.2 subtitle Symplectic Geometry, TQFT, Algebraic Topology and Algebraic Geometry cover image cover color Hamiltonian Mechanics, Quantum Theory, Relativity and Geometry Vol.2 Symplectic Geometry, TQFT, Algebraic Topology and Algebraic Geometry 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 ... more details
, of a closed quantum system. If the Hamiltonian is time independent, U t form a Stone s theorem ...In quantum mechanics , the Hamiltonian H ,also or , is the Operator physics operator corresponding to the total energy of the system. Its Spectrum of an operator spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time evolution of a system, it is of fundamental importance in most formulations of quantum theory see below . Basic introduction By analogy with classical mechanics , the Hamiltonian is commonly expressed as the sum ... potential energies of a system, in the form math H T V math although this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes . The potential ... term yields math mathbf H frac hbar 2 2m nabla 2 V mathbf r ,t math which allows one to apply the Hamiltonian ... mechanics. However, in the bra ket notation more general formalism of Paul Dirac Dirac , the Hamiltonian ... space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues ... the physical formulation. Schr dinger equation Main Schr dinger equation The Hamiltonian generates the time evolution of quantum states. If math left psi t right rangle math is the state of the system ... Jacobi equation , which is one of the reasons H is also called the Hamiltonian. Given the state ... Unitary matrix unitary operator U commutation relation commutes with the Hamiltonian. To see ... Thus, the expected value of the observable G is conserved for any state of the system. In the case of the free ... Hamilton Hamilton s equations in classical Hamiltonian mechanics have a direct analogy in quantum ... states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time. The instantaneous state of the system at time t , math left psi left t right right rangle ... variables. We can treat them as coordinates which specify the state of the system, like the position ... more details
Image Hamiltonian one 3d.png Image Polar hamiltonian.png Summary by andrej.westermann with grapher Licensing GFDL self with disclaimers migration relicense ... more details
Image US10dollarbill Series 2004A.jpg thumb Alexander Hamilton on the current U.S. ten dollar bill U.S. 10 bill The Hamiltonian economic program was the set of measures that were proposed by American Founding Father and 1st United States Secretary of the Treasury Secretary of the Treasury Alexander Hamilton in three notable reports and implemented by Congress of the United States Congress during George Washington George Washington s first administration. First Report on the Public Credit First Report on Public Credit pertaining to the assumption of federal and state debts and finance of the United States government. Second Report on Public Credit pertaining to the establishment of a National Bank. Report on Manufactures pertaining to the policies to be followed to encourage manufacturing and industry within the United States. Related articles American School economics , for the Hamiltonian American School of economics practiced by the United States from 1790s 1970s rooted in the three Reports, based on tariffs which built the American industrial infrastructure. American System economic plan , for the plan of Henry Clay rooted in the ideas of the three Reports. Category Alexander Hamilton Category Federalist Party pt Programa econ mico de Hamilton ... more details
of classical systems, involves describing a classical system using Hamiltonian mechanics. Classical ...About Liouville s theorem in Hamiltonian mechanics Liouville s theorem disambiguation Refimprove date ... theorem in classical statistical mechanics statistical and Hamiltonian mechanics . It asserts that the phase space phase space distribution function is constant along the trajectories of the system that is that the density of system points in the vicinity of a given system point travelling ... of phase space distribution function . Consider a dynamical system with canonical coordinates math ... distribution math rho p,q math determines the probability math rho p,q ,d nq ,d n p math that the system ... for the system. This equation demonstrates the conservation of density in phase space which ... math is the Hamiltonian, and Hamilton s equations have been used. That is, viewing the motion through phase space as a fluid flow of system points, the theorem that the convective derivative of the density ..., and the generator mathematics generator or Noether charge of the symmetry is the Hamiltonian. Physical ... s number , for a laboratory scale system . Setting math frac partial rho partial t 0 math gives an equation for the stationary states of the system and can be used to find the density of Microstate ... equation is satisfied by math rho math equal to any function of the Hamiltonian math H math in particular ... referred to as Liouville s theorem. In Hamiltonian mechanics , the phase space is a differentiable manifold ... under the Hamiltonian flow . More generally, one can describe the necessary and sufficient condition under which a smooth measure is invariant under a flow. The Hamiltonian case then becomes ... is invariant under the Hamiltonian flows. The symplectic structure is represented as a 2 form , given ... form is zero along every Hamiltonian vector field. In fact, the symplectic structure itself is preserved ... wircq eng.html ihf invariant Hamiltonian formalism , the theorem about existence of symplectic structure ... more details
Summary view of hamiltonian 1 as represented with n 1, n 1 made with grapher Image Polar hamiltonian.png Licensing GFDL self with disclaimers migration relicense ... 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 maths stub ... more details
dablink This article is about the specific problem of determining whether a Hamiltonian path or cycle exists in a given graph. For the general graph theory concepts, see Hamiltonian path . In the mathematics mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path 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 Hamiltonian path 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 ... 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 ... algorithm for locating hamiltonian paths is to construct a path abc... and extend it until no longer ... neighbour of y if no choice produces a hamiltonian path, then one takes a further step back, removing ... will certainly find an hamiltonian path if any but it runs in exponential time. Some algorithms ... , while the subpath op...xyz is rotated . br This argument alone does not guarantee that a hamiltonian ... his work on solving a 7 vertex instance of the Hamiltonian Path Problem using a DNA computing DNA ... that a DNA computing bacterial computer can be used to solve a simple Hamiltonian path problem using ... Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the Icosian game , now also known as Hamilton s puzzle , to find a Hamiltonian cycle in the edge graph of the dodecahedron ... salesman problem Tait s conjecture External links Hamiltonian Page http alife.ccp14.ac.uk memetic www.densis.fee.unicamp.br moscato Hamilton.html Hamiltonian cycle and path problems, their generalizations ... more details
Quantum Hamiltonian n 1, n 1 by andrej.westermann with grapher Image Polar hamiltonian.png source andrej.westermann September 29 2006 andrej.westermann 20 19, 29 September 2006 UTC GFDL self with disclaimers migration relicense ... more details
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 field is a geometric manifestation of Hamilton s equations in classical mechanics . The integral curve 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 ... 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 of f and g . Definition Suppose ... the Hamiltonian vector field with the Hamiltonian H , by requiring that for every vector field Y on M , the identity math mathrm d H Y omega X H,Y , math must hold. Note Some authors define the Hamiltonian ... the symplectic form is expressed as math omega sum i mathrm d q i wedge mathrm d p i. math Then the Hamiltonian vector field with Hamiltonian H takes the form math Chi H left frac partial H partial ... 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 q 1, ldots ,q n,p 1, ldots ... curve of the Hamiltonian vector field X sub H sub if and only if it is a solution of the Hamilton ... i . math The Hamiltonian H is constant along the integral curves, that is, math H gamma t math is actually independent of t . This property corresponds to the conservation of energy in Hamiltonian mechanics ... 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 leads to a skew symmetric , bilinear ... 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 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
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