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Liouville Dynamical System
In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy ''T'' and potential energy ''V'' can be expressed in terms of the ''s'' generalized coordinates ''q'' as follows: : T = \frac \left\ \left\ : V = \frac The solution of this system consists of a set of separably integrable equations : \frac\, dt = \frac = \frac = \cdots = \frac where ''E = T + V'' is the conserved energy and the \gamma_ are constants. As described below, the variables have been changed from ''qs'' to φs, and the functions ''us'' and ''ws'' substituted by their counterparts ''χs'' and ''ωs''. This solution has numerous applications, such as the orbit of a small planet about two fixed stars under the influence of Newtonian gravity. The Liouville dynamical system is one of several things named after Joseph Liouville, an eminent French mathematician. Example of bicentric orbits In classical mechanics, Euler's three-body problem describ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility). The earliest development of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on foundational works of Sir Isaac Newton, and the mathematical methods invented by Gottfried Wilhelm Leibniz, Joseph-Louis Lagrange, Leonhard Euler, and other contemporaries, in the 17th century to describe the motion of bodies under the influence of a system of forces. Later, more abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron's mass is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum ( spin) of a half-integer value, expressed in units of the reduced Planck constant, . Being fermions, no two electrons can occupy the same quantum state, per the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: They can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \left(t, \sqrt \right) \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Leg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parametric Solution
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrisation) of the object. For example, the equations :\begin x &= \cos t \\ y &= \sin t \end form a parametric representation of the unit circle, where ''t'' is the parameter: A point (''x'', ''y'') is on the unit circle if and only if there is a value of ''t'' such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: :(x, y)=(\cos t, \sin t). Parametric representations are generally nonunique (see the "Examples in two dimensions" section bel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Coordinates
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively, on the x-axis of the Cartesian coordinate system. Basic definition The most common definition of elliptic coordinates (\mu, \nu) is : x = a \ \cosh \mu \ \cos \nu : y = a \ \sinh \mu \ \sin \nu where \mu is a nonnegative real number and \nu \in , 2\pi On the complex plane, an equivalent relationship is : x + iy = a \ \cosh(\mu + i\nu) These definitions correspond to ellipses and hyperbolae. The trigonometric identity : \frac + \frac = \cos^ \nu + \sin^ \nu = 1 shows that curves of constant \mu form ellipses, whereas the hyperbolic trigonometric identity : \frac - \frac = \cosh^ \mu - \sinh^ \mu = 1 shows that curves of constant \nu form hyperbolae. Scale factors In an orthogonal coordinate system the lengths of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bonnet's Theorem
In classical mechanics, Bonnet's theorem states that if ''n'' different force fields each produce the same geometric orbit (say, an ellipse of given dimensions) albeit with different speeds ''v''1, ''v''2,...,''v''''n'' at a given point ''P'', then the same orbit will be followed if the speed at point ''P'' equals : v_ = \sqrt History This theorem was first derived by Adrien-Marie Legendre in 1817, but it is named after Pierre Ossian Bonnet. Derivation The shape of an orbit is determined only by the centripetal forces at each point of the orbit, which are the forces acting perpendicular to the orbit. By contrast, forces ''along'' the orbit change only the speed, but not the direction, of the velocity. Let the instantaneous radius of curvature at a point ''P'' on the orbit be denoted as ''R''. For the ''k''th force field that produces that orbit, the force normal to the orbit ''F''''k'' must provide the centripetal force A centripetal force (from Latin ''centrum'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler Problem
In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be either attractive or repulsive. The problem is to find the position or speed of the two bodies over time given their masses, positions, and velocities. Using classical mechanics, the solution can be expressed as a Kepler orbit using six orbital elements. The Kepler problem is named after Johannes Kepler, who proposed Kepler's laws of planetary motion (which are part of classical mechanics and solved the problem for the orbits of the planets) and investigated the types of forces that would result in orbits obeying those laws (called ''Kepler's inverse problem''). For a discussion of the Kepler problem specific to radial orbits, see Radial trajectory. General relativity provides more accurate solutions to the two-body problem, especiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihydrogen Cation
The dihydrogen cation or hydrogen molecular ion is a cation (positive ion) with formula . It consists of two hydrogen nuclei ( protons) sharing a single electron. It is the simplest molecular ion. The ion can be formed from the ionization of a neutral hydrogen molecule . It is commonly formed in molecular clouds in space, by the action of cosmic rays. The dihydrogen cation is of great historical and theoretical interest because, having only one electron, the equations of quantum mechanics that describe its structure can be solved in a relatively straightforward way. The first such solution was derived by Ø. Burrau in 1927, just one year after the wave theory of quantum mechanics was published. Physical properties Bonding in can be described as a covalent one-electron bond, which has a formal bond order of one half. The ground state energy of the ion is -0.597 Hartree. Isotopologues The dihydrogen cation has six isotopologues, that result from replacement of o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic Nucleus
The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden experiments, Geiger–Marsden gold foil experiment. After the discovery of the neutron in 1932, models for a nucleus composed of protons and neutrons were quickly developed by Dmitri Ivanenko and Werner Heisenberg. An atom is composed of a positively charged nucleus, with a cloud of negatively charged electrons surrounding it, bound together by electrostatic force. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. Protons and neutrons are bound together to form a nucleus by the nuclear force. The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium. These dimensions are much smaller than the diameter of the atom itself (nucleus + electron cloud), by a factor of about 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, one of the four fundamental interactions (also called forces) of nature. Electric fields are important in many areas of physics, and are exploited in electrical technology. In atomic physics and chemistry, for instance, the electric field is the attractive force holding the atomic nucleus and electrons together in atoms. It is also the force responsible for chemical bonding between atoms that result in molecules. The electric field is defined as a vector field that associates to each point in space the electrostatic (Coulomb) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star
A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make them appear as fixed points of light. The most prominent stars have been categorised into constellations and asterisms, and many of the brightest stars have proper names. Astronomers have assembled star catalogues that identify the known stars and provide standardized stellar designations. The observable universe contains an estimated to stars. Only about 4,000 of these stars are visible to the naked eye, all within the Milky Way galaxy. A star's life begins with the gravitational collapse of a gaseous nebula of material composed primarily of hydrogen, along with helium and trace amounts of heavier elements. Its total mass is the main factor determining its evolution and eventual fate. A star shines for most of its active life due t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and 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 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, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
