|
Liouville's Theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical mechanics, statistical and Hamiltonian mechanics. It asserts that ''the phase space, phase-space distribution function is constant along the Trajectory, trajectories of the system''—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability. Liouville's theorem applies to conservative systems, that is, systems in which the effects of friction are absent or can be ignored. The general mathematical formulation for such systems is the measure-preserving dynamical system. Liouville's theorem applies when there are degrees of freedom that can be interpreted as positions and momenta; not all measure-preserving dynamical systems have these, but Hamiltonian systems do. The general se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Scribner's Sons
Charles Scribner's Sons, or simply Scribner's or Scribner, is an American publisher based in New York City that has published several notable American authors, including Henry James, Ernest Hemingway, F. Scott Fitzgerald, Kurt Vonnegut, Marjorie Kinnan Rawlings, Stephen King, Robert A. Heinlein, Thomas Wolfe, George Santayana, John Clellon Holmes, Don DeLillo, and Edith Wharton. The firm published ''Scribner's Magazine'' for many years. More recently, several Scribner titles and authors have garnered Pulitzer Prize, Pulitzer Prizes, National Book Award, National Book Awards and other merits. In 1978, the company merged with Atheneum Books, Atheneum and became The Scribner Book Companies. It merged into Macmillan Inc., Macmillan in 1984. Simon & Schuster bought Macmillan in 1994. By this point, only the trade book and reference book operations still bore the original family name. After the merger, the Macmillan and Atheneum adult lists were merged into Scribner's, and the Scribn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called '' canonical transformations'', which map canonical coordinate systems into other canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by q_i and p_i, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself \mathcal H =\mathcal H(q, p, t) as one of the new canonical momentum coordinates. In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convective Derivative
Convection is single or multiphase fluid flow that occurs spontaneously through the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convection is unspecified, convection due to the effects of thermal expansion and buoyancy can be assumed. Convection may also take place in soft solids or mixtures where particles can flow. Convective flow may be transient (such as when a multiphase mixture of oil and water separates) or steady state (see convection cell). The convection may be due to gravitational, electromagnetic or fictitious body forces. Heat transfer by natural convection plays a role in the structure of Earth's atmosphere, its oceans, and its mantle. Discrete convective cells in the atmosphere can be identified by clouds, with stronger convection resulting in thunderstorms. Natural convection also plays a role in stellar physics. Convection is often categorise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conserved Current
In physics a conserved current is a current, j^\mu, that satisfies the continuity equation \partial_\mu j^\mu=0. The continuity equation represents a conservation law, hence the name. Indeed, integrating the continuity equation over a volume V, large enough to have no net currents through its surface, leads to the conservation law \fracQ = 0\;,where Q = \int_V j^0 dV is the conserved quantity. In gauge theories the gauge fields couple to conserved currents. For example, the electromagnetic field couples to the conserved electric current. Conserved quantities and symmetries Conserved current is the flow of the canonical conjugate of a quantity possessing a continuous translational symmetry. The continuity equation for the conserved current is a statement of a ''conservation law''. Examples of canonical conjugate quantities are: *Time and energy - the continuous translational symmetry of time implies the conservation of energy *Space and momentum - the continuous translationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuity Equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations. Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy in the universe is fixed. This statement does not rule out the possibility that a quantity of energy could disappear from one point while simultaneously appearing at another point. A stronger statement is that energy is ''locally'' conserved: energy can neither be created nor destroyed, ''nor'' can it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergence Theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region". The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advanced Classical Mechanics/Liouville's Theorem
The Advanced Party (), otherwise known as the Advanced Association () was a liberal and centrist Zionist political association in Mandatory Palestine founded by several urban liberal Zionists. The party was founded in order to represent the voice of Tel Aviv liberals and Zionists in the election to the Yishuv's Assembly of Representatives in 1920. The party placed sixth in the election, coming in behind their rural General Zionist counterrpart, Hitahdut HaIkarim. The party represents first formal General Zionist political party to be founded, and as one of the earliest political ancestors of the modern-day Likud. History Under the British Mandate, the Yishuv (Jewish community), established a network of political and administrative institutions, among them the Assembly of Representatives.Itamar Rabinovich & Jehuda Reinharz (2008''Israel in the Middle East''Brandeis University Press, p84 To ensure that small groups were properly represented, a system of proportional representation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American mechanical engineer and scientist who made fundamental theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of Statistical ensemble (mathematical physics), ensembles of the possible states of a physical system composed of many particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he created modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period) and described the Gibbs phenomenon in the theory of Fourier analysis. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamilton's Equations
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. 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 configuration space M and 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 inst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
