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Symplectic Integrator
In mathematics, a symplectic integrator (SI) is a Numerical ordinary differential equations, numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, particle accelerator, accelerator physics, plasma physics, quantum physics, and celestial mechanics. Introduction Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read \dot p = -\frac \quad\mbox\quad \dot q = \frac, where q denotes the position coordinates, p the momentum coordinates, and H is the Hamiltonian. The set of position and momentum coordinates (q,p) are called canonical coordinates. (See Hamiltonian mechanics for more background.) The time evolution of Hamilton's equations is a symplectomorphism, meaning that it conserves the symplectic 2-form dp \wedge dq. A numerical sche ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In regular perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Energy Drift
In computer simulations of mechanical systems, energy drift is the gradual change in the total energy of a closed system over time. According to the laws of mechanics, the energy should be a constant of motion and should not change. However, in simulations the energy might fluctuate on a short time scale and increase or decrease on a very long time scale due to numerical integration artifacts that arise with the use of a finite time step . This is somewhat similar to the flying ice cube problem, whereby numerical errors in handling equipartition of energy can change vibrational energy into translational energy. More specifically, the energy tends to increase exponentially; its increase can be understood intuitively because each step introduces a small perturbation to the true velocity , which (if uncorrelated with , which will be true for simple integration methods) results in a second-order increase in the energy E = \sum m \mathbf^2 = \sum m \mathbf_\text^2 + \sum m \, \delta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle-in-cell
In plasma physics, the particle-in-cell (PIC) method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles (or fluid elements) in a Lagrangian frame are tracked in continuous phase space, whereas moments of the distribution such as densities and currents are computed simultaneously on Eulerian (stationary) mesh points. PIC methods were already in use as early as 1955, even before the first Fortran compilers were available. The method gained popularity for plasma simulation in the late 1950s and early 1960s by Buneman, Dawson, Hockney, Birdsall, Morse and others. In plasma physics applications, the method amounts to following the trajectories of charged particles in self-consistent electromagnetic (or electrostatic) fields computed on a fixed mesh. Technical aspects For many types of problems, the classical PIC method invented by Buneman, Dawson, Hockney, Birdsall, Morse and others is relatively intu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baker–Campbell–Hausdorff Formula
In mathematics, the Baker–Campbell–Hausdorff formula gives the value of Z that solves the equation e^X e^Y = e^Z for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for Z in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in X and Y and iterated commutators thereof. The first few terms of this series are: Z = X + Y + \frac ,Y+ \frac ,[X,Y + \frac [Y,[Y,X + \cdots\,, where "\cdots" indicates terms involving higher Commutator#Identities_(ring_theory)">commutators of X and Y. If X and Y are sufficiently small elements of the Lie algebra \mathfrak g of a Lie group G, the series is convergent. Meanwhile, every element g sufficiently close to the identity in G can be expressed as g = e^X for a small X in \mathfrak g. Thus, we can say that ''near the identity'' the group multiplication in G—written as e^X e^Y = e^Z—can be expressed in purely Lie alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Verlet Integration
Verlet integration () is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Jean Baptiste Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics. It was also used by P. H. Cowell and A. C. C. Crommelin in 1909 to compute the orbit of Halley's Comet, and by Carl Størmer in 1907 to study the trajectories of electrical particles in a magnetic field (hence it is also called Størmer's method). The Verlet integrator provides good numerical stability, as well as other properties that are important in physical systems such as time reversibility and preservation of the symplectic form on phase space, at no significant additional computational cost over the simple Euler method. Basic Størmer–Verlet For a second-order differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Euler Method
In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method for solving Hamilton's equations, a system of ordinary differential equations that arises in classical mechanics. It is a symplectic integrator and hence it yields better results than the standard Euler method. Origin The method has been discovered and forgotten many times, dating back to Newton's ''Principiae'', as recalled by Richard Feynman in his ''Feynman Lectures'' (Vol. 1, Sec. 9.6) In modern times, the method was rediscovered in a 1956 preprint by René De Vogelaere that, although never formally published, influenced subsequent work on higher-order symplectic methods. Setting The semi-implicit Euler method can be applied to a pair of differential equations of the form :\begin &= f(t,v) \\ &= g(t,x), \end where ''f'' and ''g'' are given functions. Here, ''x'' and ''v'' may be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Exponential
In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map (Lie theory), exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real number, real or complex number, complex matrix (mathematics), matrix. The exponential of , denoted by or , is the matrix given by the power series e^X = \sum_^\infty \frac X^k where X^0 is defined to be the identity matrix I with the same dimensions as X, and . The series always converges, so the exponential of is well-defined. Equivalently, e^X = \lim_ \left(I + \frac \right)^k for integer-valued , where is the identity matrix. Equivalently, given by the solution to the differential equation \frac d e^ = X e^, \quad e^ = I When is an diagonal matrix then will be an diagonal matr ... [...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] |
Potential Energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity or those in a spring. The term ''potential energy'' was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to the ancient Greek philosopher Aristotle's concept of Potentiality and Actuality, ''potentiality''. Common types of potential energy include gravitational potential energy, the elastic potential energy of a deformed spring, and the electric potential energy of an electric charge and an electric field. The unit for energy in the International System of Units (SI) is the joule (symbol J). Potential energy is associated with forces that act on a body in a way that the total Work (physics), work done by these forces on the body depends only on the initial and final positions of the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinetic Energy
In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Robert and Halliday, David (1960) ''Physics'', Section 7-5, Wiley International Edition The kinetic energy of an object is equal to the work, or force ( F) in the direction of motion times its displacement ( s), needed to accelerate the object from rest to its given speed. The same amount of work is done by the object when decelerating from its current speed to a state of rest. The SI unit of energy is the joule, while the English unit of energy is the foot-pound. In relativistic mechanics, \fracmv^2 is a good approximation of kinetic energy only when ''v'' is much less than the speed of light. History and etymology The adjective ''kinetic'' has its roots in the Greek word κίνησις ''kinesis'', meaning "motion". The dichoto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
