|
Hamiltonian (other)
Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program, the economic policies advocated by Alexander Hamilton, the first United States Secretary of the Treasury See also * Alexander Hamilton (1755 or 1757–1804), American statesman and one of the Founding Fathers of the US * Hamilton (other) * List of things named after William Rowan Hamilton {{Short de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. Overview Phase space coordinates (p,q) and Hamiltonian H Let (M, \mathcal L) be a mechanical system with the configuration space M and the smooth Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant t, the Legendre transformat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program, the economic policies advocated by Alexander Hamilton, the first United States Secretary of the Treasury See also * Alexander Hamilton (1755 or 1757–1804), American statesman and one of the Founding Fathers of the US * Hamilton (other) Hamilton may refer to: People * Hamilton (name), a common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyall Hamiltonian
In quantum chemistry, the Dyall Hamiltonian is a modified Hamiltonian with two-electron nature. It can be written as follows: :\hat^ = \hat^_i + \hat^_v + C :\hat^_i = \sum_^ \varepsilon_i E_ + \sum_r^ \varepsilon_r E_ :\hat^_v = \sum_^ h_^ E_ + \frac \sum_^ \left\langle ab \left.\ cd \right\rangle \left(E_ E_ - \delta_ E_ \right) :C = 2 \sum_^ h_ + \sum_^ \left( 2 \left\langle ij \left.\ ij\right\rangle - \left \langle ij \left.\ ji\right\rangle \right) - 2 \sum_^ \varepsilon_i :h_^ = h_ + \sum_j \left( 2 \left\langle aj \left.\ bj \right\rangle - \left\langle aj \left.\ jb \right\rangle \right) where labels i,j,\ldots, a,b,\ldots, r,s,\ldots denote core, active and virtual orbitals (see Complete active space In quantum chemistry, a complete active space is a type of classification of molecular orbitals. Spatial orbitals are classified as belonging to three classes: * ''core'', always hold two electrons * ''active'', partially occupied orbitals * ''vi ...) respectively, \var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Hamiltonian
In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity. The elementary parts of a molecule are the nuclei, characterized by their atomic numbers, ''Z'', and the electrons, which have negative elementary charge, −''e''. Their interaction gives a nuclear charge of ''Z'' + ''q'', where , with ''N'' equal to the number of electrons. Electrons and nuclei are, to a very good approximation, point charges and point masses. The molecular Hamiltonian is a sum of several terms: its major terms are the kinetic energies of the electrons and the Coulomb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (control Theory)
The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. Inspired by, but distinct from, the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian. Problem statement and definition of the Hamiltonian Consider a dynamical system of n first-order differential equations :\dot(t) = \mathbf(\mathbf(t),\mathbf(t),t) where \mathbf(t) = \left x_(t), x_(t), \ldots, x_(t) \right denotes a vector of state variables, and \mathbf(t) = \left u_(t), u_(t), \ldots, u_(t) \right a vector of control variables. Once initial conditions \mathbf(t_) = \mathbf_ and controls \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Path
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are 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 in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Group
In group theory, a Dedekind group is a group ''G'' such that every subgroup of ''G'' is normal. All abelian groups are Dedekind groups. A non-abelian Dedekind group is called a Hamiltonian group. The most familiar (and smallest) example of a Hamiltonian group is the quaternion group of order 8, denoted by Q8. Dedekind and Baer have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a direct product of the form , where ''B'' is an elementary abelian 2-group, and ''D'' is a torsion abelian group with all elements of odd order. Dedekind groups are named after Richard Dedekind, who investigated them in , proving a form of the above structure theorem (for finite groups). He named the non-abelian ones after William Rowan Hamilton, the discoverer of quaternions. In 1898 George Miller delineated the structure of a Hamiltonian group in terms of its order and that of its subgroups. For instance, he shows "a Hamilton group of order 2''a'' has quater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Economic Program
The Hamiltonian economic program was the set of measures that were proposed by American Founding Father and first United States Secretary of the Treasury, Secretary of the Treasury Alexander Hamilton in four notable reports and implemented by the US Congress during George Washington's first term. They outlined a coherent program of national mercantilism government-assisted economic development…… * 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 (1790). Hamilton included his plan to tax distilled spirits among other domestic goods to boost revenue. He thought that a tax on spirits would be the least objectionable way to make money, as it could be philosophically equated to a pigovian or sin tax. However, his new tax set off the Whiskey Rebellion which highlighted separation in social classes as rural Pennsylvania farmers fought against the government. Eventual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Hamilton
Alexander Hamilton (January 11, 1755 or 1757July 12, 1804) was an American military officer, statesman, and Founding Father who served as the first United States secretary of the treasury from 1789 to 1795. Born out of wedlock in Charlestown, Nevis, Hamilton was orphaned as a child and taken in by a prosperous merchant. He pursued his education in New York before serving as an artillery officer in the American Revolutionary War. Hamilton saw action in the New York and New Jersey campaign, served for years as an aide to General George Washington, and helped secure American victory at the Siege of Yorktown. After the war, Hamilton served as a delegate from New York to the Congress of the Confederation. He resigned to practice law and founded the Bank of New York. In 1786, Hamilton led the Annapolis Convention to replace the Articles of Confederation with the Constitution of the United States, which he helped ratify by writing 51 of the 85 installments of ''The Federalist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamilton (other)
Hamilton may refer to: People * Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname ** The Duke of Hamilton, the premier peer of Scotland ** Lord Hamilton (other), several Scottish, Irish and British peers, and some members of the judiciary, who may be referred to simply as ''Hamilton'' ** Clan Hamilton, an ancient Scottish kindred * Alexander Hamilton (1755–1804), first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States * Lewis Hamilton, a British Formula One driver *William Rowan Hamilton (1805–1865), Irish physicist, astronomer, and mathematician for whom ''Hamiltonian mechanics'' is named * Hamílton (footballer) (born 1980), Togolese footballer Places Australia * Hamilton, New South Wales, suburb of Newcastle * Hamilton Hill, Western Australia, suburb of Perth * Hamilton, South Australia * Hamilton, Tasmania * Hamilton, Victoria Queens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
