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Double Pendulum
In physics and mathematics, in the area of dynamical systems, a double pendulum, also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamical systems, dynamic behavior with a butterfly effect, strong sensitivity to initial conditions. The motion of a double pendulum is governed by a pair of coupled ordinary differential equations and is chaos theory, chaotic. Analysis and interpretation Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be simple pendulums or compound pendulums (also called complex pendulums) and the motion may be in three dimensions or restricted to one vertical plane. In the following analysis, the limbs are taken to be identical compound pendulums of length and mass , and the motion is restricted to two dimensions. In a compound pendulum, the mass is distributed along its length. If the d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Configuration Space (physics)
In classical mechanics, the parameters that define the configuration of a system are called '' generalized coordinates,'' and the space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system. Notice that this is a notion of "unrestricted" configuration space, i.e. in which different point particles may occupy the same position. In mathematics, in particular in topology, a notion of "restricted" configuration space is mostly used, in which the diagonals, representing "colliding" particles, are removed. Examples A particle in 3D space The position of a single particle moving in ordinary Euclidean 3-space is defined by the vector q=(x,y,z), and therefore its ''configuration space'' is Q=\mathbb^3. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Double Pendulum Flip Time 2021
Double, The Double or Dubble may refer to: Mathematics and computing * Multiplication by 2 * Double precision, a floating-point representation of numbers that is typically 64 bits in length * A double number of the form x+yj, where j^2=+1 * A 2-tuple, or ordered list of two elements, commonly called an ordered pair, denoted (a,b) * Double (manifold), in topology Food and drink * A drink order of two shots of hard liquor in one glass * A "double decker", a hamburger with two patties in a single bun Games * Double, action in games whereby a competitor raises the stakes ** , in contract bridge ** Doubling cube, in backgammon ** Double, doubling a blackjack bet in a favorable situation ** Double, a bet offered by UK bookmakers which combines two selections * Double, villain in the video game ''Mega Man X4'' * A kart racing game '' Mario Kart: Double Dash'' * An arcade action game ''Double Dragon'' Sports * Double (association football), the act of a winning a division and primary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Methods For Ordinary Differential Equations
Numerical methods for ordinary differential equations are methods used to find Numerical analysis, numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution. Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved. The problem A firs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge–Kutta Methods
In numerical analysis, the Runge–Kutta methods ( ) are a family of Explicit and implicit methods, implicit and explicit iterative methods, List of Runge–Kutta methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The Runge–Kutta method The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: : \frac = f(t, y), \quad y(t_0) = y_0. Here y is an unknown function (scalar or vector) of time t, which we would like to approximate; we are told that \frac, the rate at which y changes, is a function of t and of y itself. At the initial time t_0 the corresponding y value is y_0. The function f and the initial conditions t_0, y_0 are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure ('' quadrature'' or ''squaring''), as in the quadrature of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Form Expression
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. Commonly, the basic functions that are allowed in closed forms are ''n''th root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions. The ''closed-form problem'' arises when new ways are introduced for specifying mathematical objects, such as limits, series, and integrals: given an object specified with such tools, a natural problem is to find, if possible, a ''closed-form expression'' of this object; that is, an expression of this object in terms of previous ways of specifying it. Example: roots of polynomials The quadratic formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Lagrange Equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote the time derivative. In addition to the normal ( Leibniz's) notation, :\frac A very common short-hand notation used, especially in physics, is the 'over-dot'. I.E. :\dot (This is called Newton's notation) Higher time derivatives are also used: the second derivative with respect to time is written as :\frac with the corresponding shorthand of \ddot. As a generalization, the time derivative of a vector, say: : \mathbf v = \left v_1,\ v_2,\ v_3, \ldots \right is defined as the vector whose components are the derivatives of the components of the original vector. That is, : \frac = \left \frac,\frac ,\frac , \ldots \right . Use in physics Time derivatives are a key concept in physics. For example, for a changing position x, its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Notation
In differential calculus, there is no single standard notation for differentiation. Instead, several notations for the derivative of a function or a dependent variable have been proposed by various mathematicians, including Leibniz, Newton, Lagrange, and Arbogast. The usefulness of each notation depends on the context in which it is used, and it is sometimes advantageous to use more than one notation in a given context. For more specialized settings—such as partial derivatives in multivariable calculus, tensor analysis, or vector calculus—other notations, such as subscript notation or the ∇ operator are common. The most common notations for differentiation (and its opposite operation, antidifferentiation or indefinite integration) are listed below. Leibniz's notation The original notation employed by Gottfried Leibniz is used throughout mathematics. It is particularly common when the equation is regarded as a functional relationship between dependent and independe ... [...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] |
