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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> | |
| 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>
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| 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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Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. If you liked this short article and you would certainly such as to obtain additional facts pertaining to wordpress dropbox backup kindly visit the site. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. By using Word - Press, you can develop very rich, user-friendly and full-functional website.
ve labored so hard to publish and put up on their website. The nominee in each category with the most votes was crowned the 2010 Parents Picks Awards WINNER and has been established as the best product, tip or place in that category. Are you considering getting your website redesigned. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Have you heard about niche marketing and advertising.
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