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| 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 {{mvar|n}} with varying coefficients. The exponent is built up as an infinite series whose terms involve multiple integrals and nested commutators.
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| == Magnus approach and its interpretation ==
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| Given the {{math|''n'' × ''n''}} coefficient matrix {{math|''A''(''t'')}}, one wishes to solve the [[initial value problem]] associated with the linear ordinary differential equation
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| :<math>Y^{\prime}(t)=A(t)Y(t),\qquad\qquad Y(t_0)=Y_{0}</math>
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| for the unknown {{mvar|n}}-dimensional vector function {{math|''Y''(''t'')}}. | |
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| When ''n'' = 1, the solution simply reads
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| :<math>Y(t)= \exp \left( \int_{t_0}^{t}A(s)\,ds \right) Y_{0}.</math> | |
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| This is still valid for ''n'' > 1 if the matrix {{math|''A''(''t'')}} satisfies {{math|''A''(''t''<sub>1</sub>) ''A''(''t''<sub>2</sub>) {{=}} ''A''(''t''<sub>2</sub>) ''A''(''t''<sub>1</sub>)}} for any pair of values of ''t'', ''t''<sub>1</sup> and ''t''<sub>2</sup>. In particular, this is the case if the matrix {{mvar|A}} is independent of {{mvar|t}}. In the general case, however, the expression above is no longer the solution of the problem.
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| 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 {{math|''n'' × ''n''}} matrix function {{math|''Ω''(''t'',''t''<sub>0</sub>)}},
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| :<math> Y(t)=\exp \left( \Omega (t,t_0)\right) \, Y_0 ~,</math>
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| which is subsequently constructed as a [[Series (mathematics)|series]] expansion,
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| ::<math>\Omega(t)=\sum_{k=1}^{\infty}\Omega_{k}(t),</math>
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| where, for simplicity, it is customary to write {{math|''Ω''(''t'')}} for {{math|''Ω''(''t'',''t''<sub>0</sub>)}} and to take ''t''<sub>0</sub> = 0.
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| Magnus appreciated that, since {{math|(<sup>''d''</sup>⁄<sub>''dt''</sub> ''e<sup>Ω</sup>'') ''e<sup>−Ω</sup>'' {{=}} ''A''(''t'')}}, using a [[Baker–Campbell–Hausdorff formula#Selected tractable cases| Poincaré−Hausdorff]] matrix identity,
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| he could relate the time-derivative of {{mvar|Ω}} to the generating function of [[Bernoulli_number#Generating_function|Bernoulli numbers]] and
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| the [[adjoint endomorphism]] of {{mvar|Ω}},
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| ::<math>\Omega ' = \frac {\operatorname{ad}_\Omega}{\exp(\operatorname{ad}_\Omega)-1} ~ A~,</math>
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| to solve for {{mvar|Ω}} recursively in terms of {{mvar|A}}, "in a continuous analog of the [[Baker–Campbell–Hausdorff formula|CBH expansion]]", as outlined in a subsequent section.
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| 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
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| :<math>\Omega_1(t) =\int_0^t A(t_1)\,dt_1,</math>
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| :<math>\Omega_2(t) =\frac{1}{2}\int_0^t dt_1 \int_0^{t_1} dt_2\ \left[ A(t_1),A(t_2)\right]</math>
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| :<math>\Omega_3(t) =\frac{1}{6} \int_0^t dt_1 \int_0^{t_{1}}d t_2 \int_0^{t_{2}} dt_3 \ (\left[ A(t_1),\left[
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| A(t_2),A(t_3)\right] \right] +\left[ A(t_3),\left[ A(t_2),A(t_{1})\right] \right] )</math>
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| :<math>\Omega_4(t) =\frac{1}{12} \int_0^t dt_1 \int_0^{t_{1}}d t_2 \int_0^{t_{2}} dt_3 \int_0^{t_{3}} dt_4 \ (\left[\left[\left[A_1,A_2\right],
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| A_3\right],A_4\right]</math>
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| :::<math>+\left[A_1,\left[\left[A_2,A_3\right],A_4\right]\right]+
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| \left[A_1,\left[A_2,\left[A_3,A_4\right]\right]\right]+
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| \left[A_2,\left[A_3,\left[A_4,A_1\right]\right]\right]
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| )</math>
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| where [''A'',''B''] ≡ ''AB''−''BA'' is the matrix [[commutator]] of ''A'' and ''B''.
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| These equations may be interpreted as follows: {{math|''Ω''<sub>1</sub>(''t'')}} coincides exactly with the exponent in the scalar ({{mvar|n}} = 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: {{mvar|Ω}} or parts of it are in the [[Lie algebra]] of the [[Lie group]] of the evolution.
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| 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 theory|perturbation]] theories. For instance, in [[classical mechanics]] the [[Symplectic geometry|symplectic]] character of the [[time evolution]] is preserved at every order of approximation. Similarly the [[unitary operator|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).
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| == Convergence of the expansion ==
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| From a mathematical point of view, the convergence problem is the following: given a certain matrix {{math|''A''(''t'')}}, when can the exponent {{math|''Ω''(''t'')}} be obtained as the sum of the Magnus series?
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| A sufficient condition for this series to [[convergent series|converge]] for {{math|''t'' ∈ [0,''T'')}} is
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| :<math> \int_0^T \|A(s)\| ds < \pi</math> | |
| where <math> \| \cdot \|</math> denotes a [[matrix norm]]. This result is generic, in the sense that one may construct specific matrices {{math|''A''(''t'')}} for which the series diverges for any {{math|''t'' > ''T''}}.
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| == Magnus generator ==
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| A recursive procedure to generate all the terms in the Magnus expansion utilizes the matrices {{math| ''S''<sub>''n''</sub><sup>(''k'')</sup>}}, defined recursively through
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| :<math>S_{n}^{(j)} =\sum_{m=1}^{n-j}\left[ \Omega_{m},S_{n-m}^{(j-1)}\right],\qquad\qquad 2\leq j\leq n-1 </math>
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| :<math>S_{n}^{(1)} =\left[ \Omega _{n-1},A \right] ,\qquad S_{n}^{(n-1)}= \mathrm{ad} _{\Omega _{1}}^{n-1} (A) ~,</math>
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| which then furnish
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| :<math>\Omega _{1} =\int_{0}^{t}A (\tau )d\tau </math>
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| :<math>\Omega _{n} =\sum_{j=1}^{n-1}\frac{B_{j}}{j!}\int_{0}^{t}S_{n}^{(j)}(\tau)d\tau ,\qquad\qquad n\geq 2. </math>
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| Here, ad<sub>''k''</sup> is a shorthand for an iterated commutator,
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| :<math>\mathrm{ad}_{\Omega}^0 A = A, \qquad \mathrm{ad}_{\Omega}^{k+1} A = [ \Omega, \mathrm{ad}_{\Omega}^k A ], </math>
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| (see [[adjoint endomorphism]] ), while {{math|''B''<sub>''j''</sub>}} are the [[Bernoulli numbers]].
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| Finally, when this recursion is worked out explicitly, it is possible to express {{math|''Ω''<sub>''n''</sub>(''t'')}} as a linear combination of ''n''-fold integrals of ''n''−1 nested commutators involving {{mvar|n}} matrices {{mvar|A}},
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| :<math> \Omega_n(t) = \sum_{j=1}^{n-1} \frac{B_j}{j!} \,
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| \sum_{
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| k_1 + \cdots + k_j = n-1 \atop
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| k_1 \ge 1, \ldots, k_j \ge 1}
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| \, \int_0^t \,
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| \mathrm{ad}_{\Omega_{k_1}(\tau )} \, \mathrm{ad}_{\Omega_{k_2}(\tau )} \cdots
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| \, \mathrm{ad}_{\Omega_{k_j}(\tau )} A(\tau ) \, d\tau \qquad n \ge 2,</math>
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| an expression which becomes increasingly intricate with {{mvar|n}}.
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| == Applications ==
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| Since the 1960s, the Magnus expansion has been successfully applied as a perturbative tool in numerous areas of physics and chemistry, from [[atomic physics|atomic]] and [[molecular physics]] to [[nuclear magnetic resonance]] and [[quantum electrodynamics]]. It has been also used since 1998 as a tool to construct practical algorithms for the numerical integration of matrix linear differential equations. As they inherit from the Magnus expansion the
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| preservation of qualitative traits of the problem, the corresponding schemes are prototypical examples of [[geometric integrator|geometric numerical integrators]].
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| == See also ==
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| * [[Baker–Campbell–Hausdorff formula]]
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| * [[Fer expansion]]
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| == References ==
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| * {{cite journal | doi = 10.1002/cpa.3160070404 | title = On the exponential solution of differential equations for a linear operator | year = 1954 | author = W. Magnus | journal = Comm. Pure and Appl. Math. | volume = VII | issue = 4 | pages = 649–673}}
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| * {{cite journal | doi = 10.1088/0305-4470/31/1/023 | title = Magnus and Fer expansions for matrix differential equations: The convergence problem | year = 1998 | author = S. Blanes, F. Casas, J.A. Oteo, J. Ros | journal = J. Phys. A: Math. Gen. | volume = 31 | issue = 1 | pages = 259–268}}
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| * {{cite journal | doi = 10.1098/rsta.1999.0362 | title = On the solution of linear differential equations in Lie groups | year = 1999 | author = A. Iserles, S.P. Nørsett | journal = Phil. Trans. R. Soc. Lond. A | volume = 357 | issue = 1754 | pages = 983–1019}}
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| * {{cite journal | doi = 10.1016/j.physrep.2008.11.001 | title = The Magnus expansion and some of its applications| year = 2009 | author = S. Blanes, F. Casas, J.A. Oteo, J. Ros | journal = Phys. Rep. | volume = 470 | issue = 5-6 | pages = 151–238}}
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| [[Category:Ordinary differential equations]]
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| [[Category:Lie algebras]]
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| [[Category:Recursion schemes]]
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| [[Category:Mathematical physics]]
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e - Shop Word - Press is a excellent cart for your on the web shopping organization. Good luck on continue learning how to make a wordpress website. This CMS has great flexibility to adapt various extensions and add-ons. If you need a special plugin for your website , there are thousands of plugins that can be used to meet those needs. The top 4 reasons to use Business Word - Press Themes for a business website are:.
Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. The higher your blog ranks on search engines, the more likely people will find your online marketing site. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. Many times the camera is following Mia, taking in her point of view in almost every frame.
Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. Setting Up Your Business Online Using Free Wordpress Websites. Enough automated blog posts plus a system keeps you and your clients happy. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way.
The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. But the Joomla was created as the CMS over years of hard work. Next you'll go by way of to your simple Word - Press site. If you have any concerns concerning where and ways to use backup plugin, you can contact us at our page. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. If your site does well you can get paid professional designer to create a unique Word - Press theme.
As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Visit our website to learn more about how you can benefit. Just download it from the website and start using the same. In addition, Word - Press design integration is also possible. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities.