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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 Magnus expansion ： ウィキペディア英語版 Magnus expansion In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first order homogeneous linear differential equation for a linear operator. In particular it furnishes the fundamental matrix of a system of linear ordinary differential equations of order with varying coefficients. The exponent is built up as an infinite series whose terms involve multiple integrals and nested commutators. == Magnus approach and its interpretation == Given the coefficient matrix , one wishes to solve the initial value problem associated with the linear ordinary differential equation :$Y^(t)=A(t)Y(t),\backslash qquad\backslash qquad\; Y(t\_0)=Y\_$ for the unknown -dimensional vector function . When ''n'' = 1, the solution simply reads :$Y(t)=\; \backslash exp\; \backslash left(\; \backslash int\_^A(s)\backslash ,ds\; \backslash right)\; Y\_.$ This is still valid for ''n'' > 1 if the matrix satisfies for any pair of values of ''t'', ''t''1 and ''t''2. In particular, this is the case if the matrix is independent of . In the general case, however, the expression above is no longer the solution of the problem. The approach introduced by Magnus to solve the matrix initial value problem is to express the solution by means of the exponential of a certain matrix function , :$Y(t)=\backslash exp\; \backslash left(\; \backslash Omega\; (t,t\_0)\backslash right)\; \backslash ,\; Y\_0\; ~,$ which is subsequently constructed as a series expansion, ::$\backslash Omega(t)=\backslash sum\_^\backslash Omega\_(t),$ where, for simplicity, it is customary to write for and to take ''t''0 = 0. Magnus appreciated that, since , using a Poincaré−Hausdorff matrix identity, he could relate the time-derivative of to the generating function of Bernoulli numbers and the adjoint endomorphism of , ::$\backslash Omega\; \text{'}\; =\; \backslash frac\; ~\; A~,$ to solve for recursively in terms of , "in a continuous analog of the CBH expansion", as outlined in a subsequent section. The equation above constitutes the Magnus expansion or Magnus series for the solution of matrix linear initial value problem. The first four terms of this series read :$$ \begin \Omega_1(t) &= \int_0^t A(t_1)\,dt_1, \\ \Omega_2(t) &= \frac\int_0^t dt_1 \int_0^ dt_2\ \left( A(t_1),A(t_2)\right ), \\ \Omega_3(t) &= \frac \int_0^t dt_1 \int_0^d t_2 \int_0^ dt_3 \Bigl(\left+\left\Bigr), \\ \Omega_4(t) &= \frac \int_0^t dt_1 \int_0^d t_2 \int_0^ dt_3 \int_0^ dt_4 \Bigl(\left,A_4\right] \\ &\quad+\left\right] +\left\right] +\left\right]\Bigr) \end where is the matrix commutator of ''A'' and ''B''. These equations may be interpreted as follows: coincides exactly with the exponent in the scalar ( = 1) case, but this equation cannot give the whole solution. If one insists in having an exponential representation (Lie group), the exponent needs to be corrected. The rest of the Magnus series provides that correction systematically: or parts of it are in the Lie algebra of the Lie group of the evolution. In applications, one can rarely sum exactly the Magnus series and one has to truncate it to get approximate solutions. The main advantage of the Magnus proposal is that, very often, the truncated series still shares with the exact solution important qualitative properties, at variance with other conventional perturbation theories. For instance, in classical mechanics the symplectic character of the time evolution is preserved at every order of approximation. Similarly the unitary character of the time evolution operator in quantum mechanics is also preserved (in contrast, e.g., to the Dyson series solving the same problem). 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「Magnus expansion」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.