Re: Non-conventional Dimenion Analysis??
- From: tessel@xxxxxx
- Date: Thu, 30 Jun 2005 04:53:29 +0000 (UTC)
On Wed, 29 Jun 2005, Gert-Jan wrote:
I was wondering the following according Dimension Analysis.
For what it's worth, I think this is usually called "dimensional analysis".
If you have found the dimensionless groups (DG) from Buckingham PI theorema, then consequently the relationship between the different groups should be found from observations.
Before I say anything else, here is a book I really like which discusses the Buckingham Pi theorem in the context of Lie's theory of the symmetry of differential equations:
Brian J. Cantwell, Introduction to Symmetry Analysis, Cambridge University Press, 2002
I think that by "dimensionless group" you might mean what Cantwell would probably call a "rational invariant" of some "unidimensional dilational subgroup" of the "point symmetry group" of the differential equation governing a mathematical model of some physical scenario.
I wonder whether it is forbidden to raise a dimensionless group to the power of another dimensionless group or not???
I think you might sense (correctly) that not all invariants need be rational, and that not all subgroups need be dilational. If so, Lie's "symmetry analysis" of differential equations should be just what you want. This is a far-reaching generalization of the Pi theorem. (Actually, IIRC, it predates the work of Buckingham, who didn't know about the relevance of Lie theory.) In addition to offering hope of finding exact solutions, it also is very useful in conjunction with other analytic techniques even in cases where finding a relevant exact solution is too hard.
Thus something like DG1=DG2**alfa*DG2**DG3.
Er, if the book above doesn't help, can you explain your example in more detail?
At the moment I thick that the functional form is not necesarily prescribed (e.g. multiplicative or additive) by the theory. What do you thick?
approach to studying a differential equation, we use the invariants of a subgroup* of the symmetry group to form new dependent and independent variables, then rewrite the original differential equation (or system of DEs) in terms of these new variables. So in this sense, the functional relationship between the new variable is determined by the relationship between the old ones. But if the original equation was a PDE, the new equation should have one fewer independent variable. If it was an ODE, the new equation should have its order decremented by one. So in this sense, following Lie, we can simplify the problem of solving the original equation.From Lie's point of view, in the simplest possible "symmetry ansatz"
[*Often a one dimensional subgroup, in fact, often a one dimensional -dilational- subgroup, which is the only kind considered by Buckingham.]
On the face of it, Buckingham's approach does -not- require the user to know (or guess) a governing DE, but in practice, most who have tried to follow his scheme seem to find that it rarely works except when employed by an experienced scientist with deep insight into the relevant physics.
Lie's approach starts with a DE but is more general, and could also be said to be more truly "algorithmic" than Buckingham's approach. Indeed, elaborations of Lie's approach have been implemented, e.g. in Maple, although the user still needs to know how to use the routines and how to interpret the results. As an added bonus, the theory of Lie groups and Lie algebras grew out of Lie's attempt to do for differential equations what Galois had done for algebraic equations. It is only fairly recently that computer algebra has advanced to the point where his theory of symmetry analysis has become practical. Nowadays, whenever you call "dsolve" in some computer algebra system, such as Maple or Mathematica, you are most likely using Lie theory plus "differential algebra". I think that is very cool!
HTH,
"T. Essel"
.
- References:
- Non-conventional Dimenion Analysis??
- From: Gert-Jan
- Non-conventional Dimenion Analysis??
- Prev by Date: Re: A question of discrete space-time, part 3
- Next by Date: Re: Gravitational waves in strong field limit.
- Previous by thread: Non-conventional Dimenion Analysis??
- Next by thread: References to a pure path integral approach to QFT
- Index(es):
Relevant Pages
|
|
