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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Variational integrator ： ウィキペディア英語版 Variational integrator Variational integrators are numerical integrators for Hamiltonian systems derived from the Euler–Lagrange equations of a discretized Hamilton's principle. Variational integrators are momentum-preserving and symplectic. ==Derivation of a Simple Variational Integrator== Consider a mechanical system with a single particle degree of freedom described by the Lagrangian : $L(t,q,v)\; =\; \backslash frac\; m\; v^2\; -\; V(q)$, 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: : $L\_d\backslash left(t\_0,\; t\_1,\; q\_0,\; q\_1\backslash right)\; =\; \backslash frac\; \backslash left(L\backslash left(t\_0,\; q\_0,\; \backslash frac\backslash right)\; +\; L\backslash left(t\_1,\; q\_1,\; \backslash frac\backslash right)\; \backslash right)\; \backslash approx\; \backslash int\_^\; dt\backslash ,\; L(t,\; q(t),\; v(t))$. Here we have chosen to approximate the time integral using the trapezoid method, and we use a linear approximation to the trajectory, : $q(t)\; \backslash approx\; \backslash frac\; \backslash left(\; t\; -\; t\_0\; \backslash right)\; +\; q\_0$ between $t\_0$ and $t\_1$, resulting in a constant velocity $v\; \backslash approx\; \backslash left(q\_1\; -\; q\_0\; \backslash right)/\backslash left(t\_1\; -\; t\_0\; \backslash right)$. Different choices for the approximation to the trajectory and the time integral give different variational integrators. The order of accuracy of the integrator is controlled by the accuracy of our approximation to the action; since : $L\_d\backslash left(\; t\_0,\; t\_1,\; q\_0,\; q\_1\; \backslash right)\; =\; \backslash int\_^\; dt\backslash ,\; L(t,q(t),v(t))\; +\; \backslash mathcal\backslash left(t\_1\; -\; t\_0\backslash right)^3$, our integrator will be second-order accurate. Evolution equations for the discrete system can be derived from a stationary-action principle. The discrete action over an extended time interval is a sum of discrete Lagrangians over many sub-intervals: : $S\_d\; =\; L\_d\backslash left(t\_0,\; t\_1,\; q\_0,\; q\_1\; \backslash right)\; +\; L\_d\backslash left(\; t\_1,\; t\_2,\; q\_1,\; q\_2\; \backslash right)\; +\; \backslash ldots$. The principle of stationary action states that the action is stationary with respect to variations of coordinates that leave the endpoints of the trajectory fixed. So, varying the coordinate $q\_1$, we have : $\backslash frac\; =\; 0\; =\; \backslash frac\; L\_d\backslash left(t\_0,\; t\_1,\; q\_0,\; q\_1\; \backslash right)\; +\; \backslash frac\; L\_d\backslash left(\; t\_1,\; t\_2,\; q\_1,\; q\_2\; \backslash right)$. Given an initial condition $(q\_0,\; q\_1)$, and a sequence of times $(t\_0,t\_1,t\_2)$ this provides a relation that can be solved for $q\_2$. The solution is : $q\_2\; =\; q\_1\; +\; \backslash frac\; \backslash left(\; q\_1\; -\; q\_0\; \backslash right)\; -\; \backslash frac\; \backslash fracV\backslash left(q\_1\backslash right)$. We can write this in a simpler form if we define the discrete momenta, : $p\_0\; \backslash equiv\; -\backslash frac\; L\_d\backslash left(\; t\_0,\; t\_1,\; q\_0,\; q\_1\; \backslash right)$ and : $p\_1\; \backslash equiv\; \backslash frac\; L\_d\backslash left(\; t\_0,\; t\_1,\; q\_0,\; q\_1\; \backslash right)$. Given an initial condition $(q\_0,p\_0)$, the stationary action condition is equivalent to solving the first of these equations for $q\_1$, and then determining $p\_1$ using the second equation. This evolution scheme gives : $q\_1\; =\; q\_0\; +\; \backslash frac\; p\_0\; -\; \backslash frac\; \backslash frac\; V\backslash left(\; q\_0\; \backslash right)$ and : $p\_1\; =\; m\; \backslash frac\; -\; \backslash frac\; \backslash frac\; V\backslash left(q\_1\backslash right)$. This is a leapfrog integration scheme for the system; two steps of this evolution are equivalent to the formula above for $q\_2$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Variational integrator」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.