Re: Re: smart change of variables?

*To*: mathgroup at smc.vnet.net*Subject*: [mg88486] Re: [mg87719] Re: [mg87695] smart change of variables?*From*: janda at th.if.uj.edu.pl*Date*: Tue, 6 May 2008 06:43:18 -0400 (EDT)*References*: <200804151051.GAA27960@smc.vnet.net>

Dear Mathematica Experts, I would like to change variables of a coupled system of ordinary differential equations F1(A,B,A',B',A'',B'',A''',A'''')=0 F2(A,B,A',B',A'',B'',A''',A'''')=0 A''',A'''' -third,fourth ord. deriv. of A. F1 and F2 are polynomials in their variables (first order in A'''',A''',B''; second order in A'',B',B; fourth order in A). We can substitute F2 by F3=F3(A,B,A',B',A'',B'') a second order polynomial. I am trying to find decoupling variables to the system. I expect that Groebner Basis of the polynomials can help in detecting such variables, however the derivatives make the problem a bit harder. Can anyone suggest a function or a package in Mathematica, which could help in determining good variables ? Thank you in advance, Sincerely Artur Janda > Barrow wrote: >> Dear all, >> >> I have a arithmetic problem. >> I have an expression expr = (p1 + 2*p2)*(k1 + 2*k2) >> where p1 + p2 = k1 + k2 >> I wanna make the following change of variables, >> s = (p1 + p2)^2 >> t = (p1 - k1)^2 >> u = (p1 - k2)^2 >> >> Is it possible to tell Mathematica to express expr >> in terms of s, t, and u automatically? >> >> Thanks so much. >> any ideas would be appreciated. >> Sincerely Barrow > > More or less. You can create a Groebner basis out of the defining > polynomials, such that variables p1, p2, k1, and k2 are ordered higher > than s, t, and u. Then generalized division (aka polynomial reduction) > of expr will do what it can to make such a replacement. > > polys = {s-(p1+p2)^2,t-(p1-k1)^2,u-(p1-k2)^2,p1+p2-(k1+k2)}; > vars = {p1,p2,k1,k2,s,t,u}; > gb = GroebnerBasis[polys, vars]; > > In[19]:= InputForm[PolynomialReduce[(p1+2*p2)*(k1+2*k2), gb, vars][[2]]] > Out[19]//InputForm= 3*k2*p2 + (3*s)/2 + t/2 - u/2 > > In general the replacement will be dependent on variable and monomial > ordering. But I'm fairly certain that you will be stuck with some part > not replaced no matter waht orders are used, since this particular > variable/monomial ordering does not manage to reduce k2*p2. > > Daniel Lichtblau > Wolfram Research > >