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Variational Integrator
Variational integrators are Numerical ordinary differential equation, numerical integrators for Hamiltonian systems derived from the Euler–Lagrange equations of a discretized Hamilton's principle. Variational integrators are momentum-preserving and Symplectic integrator, symplectic. Derivation of a simple variational integrator Consider a mechanical system with a single particle degree of freedom described by the Lagrangian L(t,q,v) = \frac 1 2 m v^2 - V(q), where m is the mass of the particle, and V is a potential. To construct a variational integrator for this system, we begin by forming the discrete Lagrangian. The discrete Lagrangian approximates the action for the system over a short time interval: \begin L_d(t_0, t_1, q_0, q_1) & = \frac \left[ L\left(t_0, q_0, \frac\right) + L\left(t_1, q_1, \frac\right) \right] \\ & \approx \int_^ \, dt\, L(t, q(t), v(t)). \end Here we have chosen to approximate the time integral using the trapezoid method, and we use a linear appro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Numerical Ordinary Differential Equation
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] [Amazon] |
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 William Rowan Hamilton, Hamilton to describe the evolution equation, 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 Three-body problem, planetary movement of three bodies: while there is no closed-form solution to the general problem, Henri Poincaré, 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(\bol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hamilton's Principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the '' differential'' equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories. Mathematical formulation Hamilton's principle states that the true evolution of a system described by generalized coordinates between two specified states and at two specified times and is a stationary point (a point where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Linear Approximation
In mathematics, a linear approximation is an approximation of a general function (mathematics), function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. Definition Given a twice continuously differentiable function f of one real number, real variable, Taylor's theorem for the case n = 1 states that f(x) = f(a) + f'(a)(x - a) + R_2 where R_2 is the remainder term. The linear approximation is obtained by dropping the remainder: f(x) \approx f(a) + f'(a)(x - a). This is a good approximation when x is close enough to since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of f at (a,f(a)). For this reason, this process is also called the tangent line approximation. Linear approximations in this case are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Order Of Accuracy
In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. Consider u, the exact solution to a differential equation in an appropriate normed space (V,, , \ , , ). Consider a numerical approximation u_h, where h is a parameter characterizing the approximation, such as the step size in a finite difference scheme or the diameter of the cells in a finite element method. The numerical solution u_h is said to be nth-order accurate if the error E(h):= , , u-u_h, , is proportional to the step-size h to the nth power: : E(h) = , , u-u_h, , \leq Ch^n where the constant C is independent of h and usually depends on the solution u. Using the big O notation an nth-order accurate numerical method is notated as : , , u-u_h, , = O(h^n) This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in general, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Discrete System
In theoretical computer science, a discrete system is a system with a countable number of states. Discrete systems may be contrasted with continuous systems, which may also be called analog systems. A final discrete system is often modeled with a directed graph and is analyzed for correctness and complexity according to computational theory. Because discrete systems have a countable number of states, they may be described in precise mathematical models. A computer is a finite-state machine that may be viewed as a discrete system. Because computers are often used to model not only other discrete systems but continuous systems as well, methods have been developed to represent real-world continuous systems as discrete systems. One such method involves sampling a continuous signal at discrete time intervals. See also *Digital control *Finite-state machine *Frequency spectrum *Mathematical model *Sample and hold *Sample rate * Sample time *Z-transform In mathematics and signal pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Leapfrog Integration
In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form \ddot x = \frac = A(x), or equivalently of the form \dot v = \frac = A(x), \qquad \dot x = \frac = v, particularly in the case of a dynamical system of classical mechanics. The method is known by different names in different disciplines. In particular, it is similar to the velocity Verlet method, which is a variant of Verlet integration. Leapfrog integration is equivalent to updating positions x(t) and velocities v(t)=\dot x(t) at different interleaved time points, staggered in such a way that they "leapfrog" over each other. Leapfrog integration is a second-order method, in contrast to Euler integration, which is only first-order, yet requires the same number of function evaluations per step. Unlike Euler integration, it is stable for oscillatory motion, as long as the time-step \Delta t is constant, and \Delta t < 2/\omega. Using Yoshida coefficien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lie Group Integrator
A Lie group integrator is a numerical integration method for differential equations built from coordinate-independent operations such as Lie group actions on a manifold. They have been used for the animation and control of vehicles in computer graphics and control systems/artificial intelligence research. These tasks are particularly difficult because they feature nonholonomic constraints. See also * Euler integration * Lie group * Numerical methods for ordinary differential equations * Parallel parking problem * Runge–Kutta methods * Variational integrator Variational integrators are Numerical ordinary differential equation, numerical integrators for Hamiltonian systems derived from the Euler–Lagrange equations of a discretized Hamilton's principle. Variational integrators are momentum-preserving a ... References {{applied-math-stub Numerical analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
