Magnus expansion

Encyclopedia

In mathematics

and physics

, the

(1907–1990), provides an exponential representation of the solution of a first order linear homogeneous 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.

associated with the linear ordinary differential equation

for the unknown

When

This is still valid for

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, the

**Magnus expansion**, named after Wilhelm MagnusWilhelm Magnus

Wilhelm Magnus was a mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations....

(1907–1990), provides an exponential representation of the solution of a first order linear homogeneous differential equation

Equation

An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

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*n*×*n*coefficient matrix*A*(*t*) we want to solve the initial value problemInitial value problem

In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...

associated with the linear ordinary differential equation

for the unknown

*n*-dimensional vector function*Y*(*t*).When

*n*= 1, the solution readsThis is still valid for

*n*> 1 if the matrix*A*(*t*) satisfies for any pair of values of*t*,*t*_{1 and t2. In particular, this is the case if the matrix is constant. In the general case, however, the expression above is no longer the solution of the problem. The approach proposed by Magnus to solve the matrix initial value problem is to express the solution by means of the exponential of a certain n × n matrix function , which is subsequently constructed as a seriesSeries (mathematics)A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.... expansion, where, for the sake of simplicity, it is customary to write down for and to take t0 = 0. 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 where is the matrix commutatorCommutatorIn mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:... of A and B. These equations may be interpreted as follows: coincides exactly with the exponent in the scalar (n = 1) case, but this equation cannot give the whole solution. If one insists in having an exponential representation the exponent has to be corrected. The rest of the Magnus series provides that correction. In applications one can rarely sum exactly the Magnus series and 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 perturbationPerturbation theoryPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem... theories. For instance, in classical mechanicsClassical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces... the symplectic character of the time evolutionTime evolutionTime evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies... is preserved at every order of approximation. Similarly the unitaryUnitaryUnitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary... character of the time evolution operator in quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic... is also preserved (in contrast to the Dyson seriesDyson seriesIn scattering theory, the Dyson series, formulated by British-born American physicist Freeman Dyson, is a perturbative series, and each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics at the second order the difference from...). Convergence of the expansion From a mathematical point of view, the convergence problem is the following: given a certain matrix , when can the exponent be obtained as the sum of the Magnus series? A sufficient condition for this series to converge for is where denotes a matrix norm. This result is generic, in the sense that one may consider specific matrices for which the series diverges for any . Magnus generator It is possible to design a recursive procedure to generate all the terms in the Magnus expansion. Specifically, with the matrices defined recursively through one has Here is a shorthand for an iterated commutator, and are the Bernoulli numbers. When this recursion is worked out explicitly, it is possible to express as a linear combination of -fold integrals of nested commutators containing matrices , an expression that becomes increasingly intricate with . Applications Since the 1960s, the Magnus expansion has been successfully applied as a perturbative tool in numerous areas of physics and chemistry, from atomicAtomic physicsAtomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and... and molecular physicsMolecular physicsMolecular physics is the study of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics. Its most important experimental techniques are the various types of spectroscopy... to nuclear magnetic resonanceNuclear magnetic resonanceNuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation... and quantum electrodynamicsQuantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved.... 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 preservation of qualitative traits of the problem, the corresponding schemes are prototypical examples of geometric numerical integratorsGeometric integratorIn the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation.-Pendulum example:.... The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL. }