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Swinging Atwood's Machine
The swinging Atwood's machine (SAM) is a mechanism that resembles a simple Atwood's machine except that one of the masses is allowed to swing in a two-dimensional plane, producing a dynamical system that is chaotic for some system parameters and initial conditions. Specifically, it comprises two masses (the pendulum, mass and counterweight, mass ) connected by an inextensible, massless string suspended on two frictionless pulleys of zero radius such that the pendulum can swing freely around its pulley without colliding with the counterweight. The conventional Atwood's machine allows only "runaway" solutions (''i.e.'' either the pendulum or counterweight eventually collides with its pulley), except for M=m. However, the swinging Atwood's machine with M>m has a large parameter space of conditions that lead to a variety of motions that can be classified as terminating or non-terminating, periodic, quasiperiodic or chaotic, bounded or unbounded, singular or non-singular due to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Swinging Atwoods Machine
Swing or swinging may refer to: Apparatus * Swing (seat), a hanging seat that swings back and forth * Pendulum, an object that swings * Russian swing, a swing-like circus apparatus * Sex swing, a type of harness for sexual intercourse * Swing ride, an amusement park ride consisting of suspended seats that rotate like a merry-go-round Arts, entertainment, and media Films * ''Swing'' (1938 film), an American film directed by Oscar Micheaux * ''Swing'' (1999 film), an American film by Nick Mead * ''Swing'' (2002 film), a French film by Tony Gatlif * ''Swing'' (2003 film), an American film by Martin Guigui * ''Swing'' (2010 film), a Hindi short film * ''Swing'' (2021 film), an American film by Michael Mailer Music Styles * Swing (jazz performance style), the sense of propulsive rhythmic "feel" or "groove" in jazz * Swing music, a style of jazz popular during the 1930s–1950s Groups and labels * Swing (Canadian band), a Canadian néo-trad band * Swing (Hong Kong band), a Hong ... [...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] |
Eccentricity (mathematics)
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. More formally two conic sections are similar if and only if they have the same eccentricity. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: * The eccentricity of a circle is zero. * The eccentricity of an ellipse which is not a circle is greater than zero but less than 1. * The eccentricity of a parabola is 1. * The eccentricity of a hyperbola is greater than 1. * The eccentricity of a pair of lines is \infty Definitions Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as . The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Focus (geometry)
In geometry, focuses or foci (), singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an ''n''-ellipse. Conic sections Defining conics in terms of two foci An ellipse can be defined as the locus of points for which the sum of the distances to two given foci is constant. A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, in terms of two different foci, as the locus of points having a fixed ratio of distances to the two foci. A parabola is a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conic Section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a ''focus'', and some particular line, called a ''directrix'', are in a fixed ratio, called the ''eccentricity''. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sam Type B
Sam, SAM or variants may refer to: Places * Sam, Benin * Sam, Boulkiemdé, Burkina Faso * Sam, Bourzanga, Burkina Faso * Sam, Kongoussi, Burkina Faso * Sam, Iran * Sam, Teton County, Idaho, United States, a populated place People and fictional characters * Sam (given name), a list of people and fictional characters with the given name or nickname * Sam (surname), a list of people with the surname ** Cen (surname) (岑), romanized "Sam" in Cantonese ** Shen (surname) (沈), often romanized "Sam" in Cantonese and other languages Religious or legendary figures * Sam (Book of Mormon), elder brother of Nephi * Sām, a Persian mythical folk hero * Sam Ziwa, an uthra (angel or celestial being) in Mandaeism Animals * Sam (army dog) (died 2000) * Sam (horse) (b 1815), British Thoroughbred * Sam (koala) (died 2009), rescued after 2009 bush fires in Victoria, Australia * Sam (orangutan), in the movie ''Dunston Checks In'' * Sam (ugly dog) (1990–2005), voted the world's ugliest dog in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (physics)
In theoretical physics, an invariant is an observable of a physical system which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is closer in scope to the mathematical definition. Invariants of a system are deeply tied to the symmetries imposed by its environment. Invariance is an important concept in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants. Examples In classical and quantum mechanics, invariance of space under translation results in momentum being an invariant and the conservation of momentum, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the conservation of energy. In general, by Noether's theorem, any invariance of a physical system under a continuous symmetry leads to a fundamental conservation law. In crystals, the electron density is peri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sam Type A
Sam, SAM or variants may refer to: Places * Sam, Benin * Sam, Boulkiemdé, Burkina Faso * Sam, Bourzanga, Burkina Faso * Sam, Kongoussi, Burkina Faso * Sam, Iran * Sam, Teton County, Idaho, United States, a populated place People and fictional characters * Sam (given name), a list of people and fictional characters with the given name or nickname * Sam (surname), a list of people with the surname ** Cen (surname) (岑), romanized "Sam" in Cantonese ** Shen (surname) (沈), often romanized "Sam" in Cantonese and other languages Religious or legendary figures * Sam (Book of Mormon), elder brother of Nephi * Sām, a Persian mythical folk hero * Sam Ziwa, an uthra (angel or celestial being) in Mandaeism Animals * Sam (army dog) (died 2000) * Sam (horse) (b 1815), British Thoroughbred * Sam (koala) (died 2009), rescued after 2009 bush fires in Victoria, Australia * Sam (orangutan), in the movie ''Dunston Checks In'' * Sam (ugly dog) (1990–2005), voted the world's ugliest dog in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Harmonic Motion
In physics, complex harmonic motion is a complicated realm based on the simple harmonic motion. The word "complex" refers to different situations. Unlike simple harmonic motion, which is regardless of air resistance, friction, etc., complex harmonic motion often has additional forces to dissipate the initial energy and lessen the speed and amplitude of an oscillation until the energy of the system is totally drained and the system comes to rest at its equilibrium point. Types Damped harmonic motion Introduction Damped harmonic motion is a real oscillation, in which an object is hanging on a spring. Because of the existence of internal friction and air resistance, the system will over time experience a decrease in amplitude. The decrease of amplitude is due to the fact that the energy goes into thermal energy. Damped harmonic motion happens because the spring is not very efficient at storing and releasing energy so that the energy dies out. The damping force is proportio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable System
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian System
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory. Overview Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton to describe the evolution equations of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the initial value problem cannot be solved analytically. One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H(\boldsymbol,\boldsymbol,t), also known as the Hamiltonian. The state of the system, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Of Inertia
The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation. It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis. For bodies constrained to rotate in a plane, only their moment of inertia about an axis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
