Lie integration

The Lie integration (according to Sophus Lie ) is a method for the numerical integration of differential equations . In contrast to the conventional methods, the equations here can be solved by differentiation instead of integration .

Basics

Lie operator

The Lie operator D is a linear differential operator: Let be a domain and (here and ) be of the form ${\displaystyle G\subset \mathbb {C} ^{n}}$ ${\displaystyle D:C^{\infty }(G,\mathbb {C} )\longrightarrow C^{\infty }(G,\mathbb {C} )}$ ${\displaystyle C^{\infty }(G,\mathbb {C} )=\{f:G\longrightarrow \mathbb {C} \,|\,f{\text{ holomorph}}\,\}}$ ${\displaystyle z=(z_{1},\ldots ,z_{n})\in G\subset \mathbb {C} ^{n}}$

${\displaystyle D=\theta _{1}{\frac {\partial }{\partial z_{1}}}+\theta _{2}{\frac {\partial }{\partial z_{2}}}+\ldots +\theta _{n}{\frac {\partial }{\partial z_{n}}}.}$

The functions are holomorphic (i.e. they can be expanded into a converging power series ). ${\displaystyle \theta _{i}(z):G\longrightarrow \mathbb {C} }$

Lie rows

The Lie operator can be applied to a function (which is holomorphic in the same region as ): ${\displaystyle f(z)}$ ${\displaystyle \theta _{i}(z)}$

${\displaystyle D(f)=\theta _{1}{\frac {\partial f}{\partial z_{1}}}+\theta _{2}{\frac {\partial f}{\partial z_{2}}}+\ldots +\theta _{n}{\frac {\partial f}{\partial z_{n}}}.}$

The Lie series L is now defined as follows:

${\displaystyle L(z,t)=\sum _{\nu =0}^{\infty }{\frac {t^{\nu }}{\nu !}}D^{\nu }f(z)=f(z)+tDf(z)+{\frac {t^{2}}{2!}}D^{2}f(z)+\ldots ,}$

where the double application of the Lie operator means on, and so on. Since the Taylor series of the exponential function by ${\displaystyle D^{2}}$ ${\displaystyle f(z)}$

${\displaystyle e^{x}=\left(1+{\frac {x}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\ldots \right)}$

is given, the Lie series can be written symbolically in the following form:

${\displaystyle L\left(z,t\right)=e^{tD}f\left(z\right)}$.

Exchange rate

For the Lie series a commutation theorem applies : Let it be  a holomorphic function and the  power series of developed in  converges at the point  with . Then applies ${\displaystyle F:G\subset \mathbb {C} ^{n}\rightarrow \mathbb {C} ^{n}}$ ${\displaystyle z\in G}$ ${\displaystyle F}$ ${\displaystyle Z\in G}$ ${\displaystyle Z=\sum _{\nu =0}^{\infty }{\frac {t^{\nu }}{\nu !}}D^{\nu }z}$

${\displaystyle F\left(Z\right)=\sum _{\nu =0}^{\infty }{\frac {t^{\nu }}{\nu !}}D^{\nu }F(z)}$,

whatever in the form

${\displaystyle F\left(e^{tD}z\right)=e^{tD}F(z)}$

can be written. The last representation motivates the designation interchangeability clause: You can interchange the order of application of  and  . ${\displaystyle e^{tD}}$ ${\displaystyle F}$

The method

The solution of a differential equation by Lie integration works as follows. A system of first-order differential equations is given:

${\displaystyle {\frac {\mathrm {d} z_{i}}{\mathrm {d} t}}=\theta _{i}(z)}$, ${\displaystyle i=0\ldots n}$

Then the solutions of the equations can be described by a Lie series:

${\displaystyle z_{i}=e^{tD}\xi _{i}}$

where here are the initial conditions . To prove this, the following is first derived from the time: ${\displaystyle \xi _{i}}$ ${\displaystyle z_{i}(t=0)}$ ${\displaystyle z_{i}}$

${\displaystyle {\frac {\mathrm {d} z_{i}}{\mathrm {d} t}}=De^{tD}\xi _{i}}$.

The exchange theorem then results and follows from the definition of the Lie operator ${\displaystyle De^{tD}\xi _{i}=e^{tD}D\xi _{i}}$

${\displaystyle D\xi _{i}=\theta _{i}\left(\xi _{i}\right)}$

and thus the proof of the statement:

${\displaystyle {\frac {\mathrm {d} z_{i}}{\mathrm {d} t}}=e^{tD}\theta _{i}\left(\xi _{i}\right)=\theta _{i}\left(e^{tD}\xi _{i}\right)=\theta _{i}\left(z_{i}\right)}$.

example

As a demonstration of the process, the equation of motion of the harmonic oscillator is solved using Lie integration. The movement of the oscillator can be described by a differential equation of the second order:

${\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+\alpha ^{2}x=0}$.

First, this equation is converted into a system of two first-order differential equations:

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=y=\theta _{1}(x,y)}$,

${\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} t}}=-\alpha ^{2}x=\theta _{2}(x,y)}$.

The initial conditions are referred to as and . The Lie operator thus has the following form: ${\displaystyle x(t=0)=\xi }$ ${\displaystyle y(t=0)=\eta }$

${\displaystyle D=\theta _{1}{\frac {\partial }{\partial \xi }}+\theta _{2}{\frac {\partial }{\partial \eta }}=\eta {\frac {\partial }{\partial \xi }}-\alpha ^{2}\xi {\frac {\partial }{\partial \eta }}}$.

The solutions of the differential equations are now given by the Lie series:

${\displaystyle x=e^{\tau D}\xi ,}$

${\displaystyle y=e^{\tau D}\eta }$

where here represents the time step of integration. In order to show the solution explicitly, the Lie series is shown in its developed form: ${\displaystyle \tau }$ ${\displaystyle t-t_{0}}$

${\displaystyle x=e^{\tau D}\xi =\left(1+\tau D+{\frac {\tau ^{2}}{2!}}D^{2}+{\frac {\tau ^{3}}{3!}}D^{3}+\ldots \right)\xi }$

The individual terms in the series are now calculated:

${\displaystyle D\xi =\eta =\theta _{1}}$

${\displaystyle D^{2}\xi =D\eta =-\alpha ^{2}\xi =\theta _{2}}$

${\displaystyle D^{3}\xi =-\alpha ^{2}D\xi =-\alpha ^{2}\eta }$

${\displaystyle D^{4}\xi =-\alpha ^{2}D\eta =\alpha ^{4}\xi }$

${\displaystyle D^{5}\xi =-\alpha ^{4}D\xi =\alpha ^{4}\eta \,}$

${\displaystyle D^{6}\xi =-\alpha ^{4}D\eta =-\alpha ^{6}\xi }$

In general it can be shown that in this case:

${\displaystyle D^{2n}{\xi }=(-1)^{n}\alpha ^{2n}\xi }$

${\displaystyle D^{2n+1}{\xi }=(-1)^{n}\alpha ^{2n}\eta }$

Now the individual terms can be inserted into the Lie series:

${\displaystyle x=\xi +\tau \eta -{\frac {\tau ^{2}}{2!}}\alpha ^{2}\xi -{\frac {\tau ^{3}}{3!}}\alpha ^{2}\eta +{\frac {\tau ^{4}}{4!}}\alpha ^{4}\xi \ldots }$

After a factorization of and there is finally ${\displaystyle \xi }$ ${\displaystyle \eta }$

${\displaystyle x=\xi \left(1-{\frac {\tau ^{2}}{2!}}\alpha ^{2}+{\frac {\tau ^{4}}{4!}}\alpha ^{4}-{\frac {\tau ^{6}}{6!}}\alpha ^{6}+\ldots \right)+{\frac {\eta }{\alpha }}\left(\tau \alpha -{\frac {\tau ^{3}}{3!}}\alpha ^{3}+{\frac {\tau ^{5}}{5!}}\alpha ^{5}-{\frac {\tau ^{7}}{7!}}\alpha ^{7}+\ldots \right)}$

The two rows in brackets are the power series of the cosine and sine functions. The solution of the equation of motion of the harmonic oscillator now follows:

${\displaystyle x(t)=\xi \cos \alpha \tau +{\frac {\eta }{\alpha }}\sin \alpha \tau }$.

Notes on the Lie integration

• Since the solutions for Lie integration are given as a power series in the independent variable (here ), it is very easy to specify an integration algorithm with step size control . ${\displaystyle \tau }$

• The method is very exact for the numerical solution of differential equations. By selecting the time step and the number of terms of the Lie series that are calculated for the solution, the accuracy can be controlled: the more Lie terms that are calculated, the larger the time step can be (and vice versa).

• For many differential equations, the terms of the Lie series can be calculated recursively. This makes the integration process very fast.

• To solve a differential equation using Lie integration, you only need to know about the derivatives of the equations. However, these can always be determined up to any high order. In addition, in contrast to integration using computer algebra systems (such as Mathematica or Maple ) , the differentiation of equations can be completely automated.

For the reasons mentioned above, the Lie integration is used especially in celestial mechanics for the numerical integration of the planetary motion, since speed and accuracy are of great importance here (see Lie integrator ).

literature

• Wolfgang Gröbner: The Lie series and their applications. German Publishing House of Science, Berlin 1960, DNB 451675177 .
• Rudolf Dvorak, Florian Freistetter, J. Kurths: Chaos and Stability in Planetary Systems. Springer, 2005, ISBN 3-540-28208-4 .
• N. Asghari et al: Stability of terrestrial planets in the habitable zone of Gl 777 A, HD 72659, Gl 614, 47 Uma and HD 4208. In: Astronomy & Astrophysics . 426/2004, pp. 353-365 ISSN  0004-6361 .