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 zh ... 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 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 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 In Language after educationist James Hamilton 1769 1831 de James Hamilton Sprachlehrer de wiki 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
Hatnote 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 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 Hamiltonian path 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 Hamiltonian path that is a cycle graph theory cycle . 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 after William Rowan Hamilton who invented the Icosian game , now also known as Hamilton s puzzle , which involves finding a Hamiltonian ..., despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier ... volume 13 year 1981 . ref 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 edge decomposition of a graph into Hamiltonian circuits. This is one of the known but unsolved problems. Clarify ... more details
systems A Hamiltoniansystem may be understood as a fiber bundle E over time R , with the Level set ... 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 ... to quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other ... each with their own degrees of freedom, there are more coordinates. 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. There is a set of differential equation s known as the Hamilton equations which give the time evolution of the system. Hamiltonians can be used to describe ... textbook chapter16 section03.html chapter 16.3 The Hamiltonian title MIT OpenCourseWare website 18.013A ... H mathcal H mathbf q , mathbf p ,t math is the scalar valued Hamiltonian function , and specify ... degrees of freedom the system has, the more complicated its behavior predicted by the solutions , since the degrees of freedom correspond to the configuration of the system i.e. generalized positions ... to obtain qualitative knowledge about the system by approximate analysis of the differential equations ... to a one dimensional 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 provided that there are NO external forces, or additional energy added to the system , which is the sum of kinetic energy ... 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 of dimension math 2 n sup 2 sup n . Properties Let math M be a math 2 n 2 n block matrix given by math ... n n matrices. Then math M is a Hamiltonian matrix provided that the matrices math B and math C are symmetric ... of matrix math D . The 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 ... York Academy of Sciences year 2005 volume 1045 issue 1 pages 291 307 . ref Hamiltonian operators ... V math is called a Hamiltonian operator with respect to math if the form math x, y mapsto Omega ... 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 C. last Waterhouse authorlink William C. Waterhouse doi 10.1016 j.laa.2004.10.003 title The structure of alternating Hamiltonian matrices ..., 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 also Symplectic matrix References Cite book first1 K. R. last1 Meyer first2 G. R. last2 Hall title Introduction to Hamiltonian dynamical systems ... 34 35 isbn 0 387 97637 X Notes references Use dmy dates date September 2010 DEFAULTSORT Hamiltonian ... more details
Quantum mechanically correct form of Hamiltonian function for conservative system journal Phys. Rev ...In atomic, molecular, and optical physics and quantum chemistry , the molecular Hamiltonian is the Hamiltonian quantum mechanics Hamiltonian operator representing the energy of the electron s and Atomic ... Hamiltonian is a sum of several terms its major terms are the kinetic energy kinetic energies of the electrons ... particles. The Hamiltonian that contains only the kinetic energies of electrons and nuclei, and the Coulomb interactions between them, is known as the Coulomb Hamiltonian . From it are missing a number ... Hamiltonian will predict most properties of the molecule, including its shape three dimensional structure , calculations based on the full Coulomb Hamiltonian are very rare. The main reason ... of the Coulomb Hamiltonian first devised by Born Oppenheimer approximation Born and Oppenheimer . The nuclear kinetic energy terms are omitted from the Coulomb Hamiltonian and one considers the remaining Hamiltonian as a Hamiltonian of electrons only. The stationary nuclei enter the problem ... way. Within this framework the molecular Hamiltonian has been simplified to the so called clamped nucleus Hamiltonian , also called electronic Hamiltonian , that acts only on functions of the electronic coordinates. Once the Schr dinger equation of the clamped nucleus Hamiltonian has been solved ... 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 ... motion Hamiltonian . Making the harmonic approximation, we can convert the Hamiltonian into a sum ... fixed frame the Hamiltonian accounts for rotation , translation and vibration of the nuclei. Since Watson introduced in 1968 an important simplification to this Hamiltonian, it is often referred to as Watson s nuclear motion Hamiltonian , but it is also known as the Eckart Hamiltonian . Coulomb ... 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 ... 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 152 2005 139 158 ref References references Category NP complete problems Category Hamiltonian ... more details
system along direction x sub 3 sub . This corresponds to Liouville s theorem Hamiltonian Liouville s theorem , which also applies to Hamiltonian mechanics . quote However, the meaning of Liouville ... 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 , Hamilton s principle states that the evolution of a system math left q 1 left sigma right , cdots ... 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 ... 2 n frac dx k ds math Image Hamiltonian Optics Optical Momentum.png 200px thumb right Optical momentum ... 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 ... int frac dp k dx 3 , dx 3 p k math Image Hamiltonian Optics Rays and Wavefronts.png 200px thumb left ... A mathbf p cdot d mathbf s 0 math Image Hamiltonian Optics Optical Path Length.png 200px thumb right ... math . Image Hamiltonian Optics 2D Phase Space.png 300px thumb right 2D phase space For example, ray ... v . Image Hamiltonian Optics Volume Variation.png 150px thumb left Volume variation This leads ... that the phase space volume is conserved as light travels along an optical system. The volume occupied ... system is represented by a single point in phase space, and the theorem states that the average density ... conservation of etendue shows on the left a diagrammatic two dimensional optical system in which ... 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
Multiple issues unreferenced January 2010 unreferenced December 2009 orphan 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
, the widespread occurrence of rape and infanticide during periods of war indicates Hamiltonian ... means. References reflist 2 sociobiology evolutionary psychology DEFAULTSORT Hamiltonian ... 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 . 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
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 ... relation between the problems of finding a Hamiltonian path and a Hamiltonian cycle. In one direction, 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 . Thus, finding a Hamiltonian ... than finding a Hamiltonian cycle. In the other direction, a graph G has a Hamiltonian cycle ... vertices, one connected to u and one connected to v , has a Hamiltonian path. Therefore, by trying this replacement for all edges incident to some chosen vertex of G , the Hamiltonian cycle problem can be solved by at most n Hamiltonian path computations, where n is the number of vertices in the graph. The Hamiltonian cycle problem is also a special case of the travelling salesman problem , obtained ... There are n different sequences of vertices that might be Hamiltonian paths in a given n vertex ... the problem of counting the number of Hamiltonian cycles to a simpler counting problem, of counting ..., he showed how to solve the Hamiltonian cycle problem in arbitrary n vertex graphs by a Monte Carlo ... three, a careful backtracking search can find a Hamiltonian cycle if one exists in time O 1.251 sup ... COCOON 2007 volume 4598 year 2007 . ref Because of the difficulty of solving the Hamiltonian ... of computing. For instance, Leonard Adleman showed that the Hamiltonian path problem may be solved ... Complexity The problem of finding a Hamiltonian cycle or path is in FNP complexity FNP the analogous decision problem is to test whether a Hamiltonian cycle or path exists. The directed and undirected ... 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
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 ... in 1838. ref J. Liouville, Journ. de Math., 3, 349 1838 . ref Consider a dynamical system with canonical ... ,d n p math that the system will be found in the infinitesimal phase space volume math d nq ,d n p math ... to Hamilton s equations for the system. This equation demonstrates the conservation of density ... 0, math where math H 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 ... time translations, and the generator mathematics generator or Noether charge of the symmetry is the Hamiltonian ... Avogadro 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 ... states equation is satisfied by math rho math equal to any function of the Hamiltonian math H math ... also referred to as Liouville s theorem. In Hamiltonian mechanics , the phase space is a differentiable ... is invariant 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 ... manifold is invariant under the Hamiltonian flows. The symplectic structure is represented as a 2 ... form is zero along every Hamiltonian vector field. In fact, the symplectic structure itself ... wircq eng.html ihf invariant Hamiltonian formalism , the theorem about existence of symplectic ... more details
all particles and their corresponding potentials the result is that the Hamiltonian of the system is the sum ... operator . It is the time evolution operator , or propagator , of a closed quantum system. If the Hamiltonian ...In quantum mechanics , the Hamiltonian is the Operator physics operator corresponding to the total energy of the system. It is usually denoted by H , also or . 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. Introduction main Operator physics The Hamiltonian is the sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with the system. For different situations and or number of particles, the Hamiltonian is different since it includes the sum .... The Schr dinger Hamiltonian One particle By analogy with classical mechanics , the Hamiltonian ... kinetic and potential energy potential energies of a system, in the form math hat H hat T hat V ... partial 2 partial z 2 math Although this is not the technical definition of the Hamiltonian mechanics Hamiltonian in classical mechanics , it is the form it most commonly takes. Combining these together ... which allows one to apply the Hamiltonian to systems described by a wave function r , t . This is the approach ... energy function, now a function of the spatial configuration of the system and time a particular set ... to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian a mix of the gradients for two particles math frac hbar 2 2M nabla ... kinetic energy. Terms of this form are known as mass polarization terms , and appear in the Hamiltonian ..., the potential of the system is the sum of the separate potential energy for each particle, ref ... r 1,t V bold r 2,t cdots V bold r N,t math The general form of the Hamiltonian in this case is math ... 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 it Matrice anti hamiltoniana ... 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 plaquettes 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
expand Italian date December 2011 Campo vettoriale hamiltoniano In mathematics and physics , a Hamiltonian vector field on a symplectic manifold is a vector field , defined for any energy function or Hamiltonian ... , 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 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 ... X sub H sub , called the Hamiltonian vector field with the Hamiltonian H , by requiring that for every ... authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying ... product . Then the Hamiltonian vector field with Hamiltonian H takes the form math Chi H left ... The assignment f X sub f sub is linear map linear , so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields. Suppose that q sup 1 sup , ..., q ... t q t ,p t is an integral curve of the Hamiltonian vector field X sub H sub if and only if it is a solution ... i frac partial H partial q i . math The Hamiltonian H is constant along the integral curves, that is, H t is actually independent of t . This property corresponds to the conservation of energy in Hamiltonian ... form 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 ... holds math X f,g X f,X g , math where the right hand side represents the Lie bracket of the Hamiltonian ... 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
unref date March 2012 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. See also Federalist Party , Hamilton s political party, which supported his program and pushed most of them through Congress American School economics , for the Hamiltonian American School of ecomomics practiced by the United States from 1790s 1970s rooted in the three Reports, based on tariffs which built the American industrial infrastructure. Category Alexander Hamilton Category Federalist Party Category American political philosophy pt Programa econ mico de Hamilton ... more details
s. Repeated indices imply the use of the summation convention . Hamiltonian approach to the geodesic equations Geodesics can be understood to be the Hamiltonian flow s of a special Hamiltonian vector field defined on the cotangent space of the manifold. The Hamiltonian is constructed from the metric ... differential equations taking the form of the Hamiltonian Jacobi equations by introducing additional ... n sub . Then introduce the Hamiltonian vector field Hamiltonian as math H x,p frac 1 2 g ab x p a p ... of the geodesic flow onto the manifold M . This is a Hamiltonian flow , and that the Hamiltonian ... section 33 . Category Symplectic geometry Category Hamiltonian mechanics Category Geodesic ... more details
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