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ayan849
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Being not an expert, my question might sound naive to students of mahematics. My question is how on Earth a Lie group helps to solve an ode. Can anyone explain me in simple terms?
ayan849 said:My question is how on Earth a Lie group helps to solve an ode. Can anyone explain me in simple terms?
Stephen Tashi said:I'd be glad to discuss it here and that might motivate me to study it more. I think you need at least a facility with the calculus of several variables in order to understand the material. After introducing continouous gropus, Emanuel discusses "The Method Characteristics", so understanding the geometric interpretation of that technique is probably necessary.
How big a non-expert are you? How serious are you about findng an answer?
Stephen Tashi said:Do you already know something about Lie goups?
And I should have written "The Method Of Characteristics".
lavinia said:I have read most Chevalley's book.
The general form for a first-order ODE is:
[itex] f(x,y,y') = 0 \ [/itex] (5.1)
"We assume this ODE is invariant under a [one parameter] group whose symbol is [itex] U f [/itex].
We start with a function [itex] f(x,y) [/itex]. The condition under which it is invariant with respect to the the group is determined next. By this we mean that
[itex] f_1 = f(x_1, y_1) = f(x,y) [/itex] (4.1)
[tex] f(x_1,y_1) = f(x,y) + \alpha U f + \frac{\alpha^2}{2} U^2 f + ... [/tex]
we observe that a necessary a sufficient condition for [itex] f [/itex] to be an invariant function of the group, is for
[itex] Uf = 0 [/itex] (4.2)
Sina said:So really what you care about is the Lie group of transformations on the phase space which take solution curves to solution curves.
Another type of invariance is now discusses. Our goal will be to find the conditions wherein one curve of a one-parameter family of curves transforms into itself or into another curve of the family.
Let
[itex] f = f(x,y) = c [/itex] (4.8)
be a one-parameter family of curves. We select one of these curves, which is written as
[itex] f_1 = f(x_1,y_1) = c. [/itex]
Under the Transformation (2.3) this becomes
[itex] f_1 = f(\phi(x,y,\alpha),\psi(x,y,\alpha)) = \omega(x,y,\alpha)[/itex] (4.9)
or
[itex] \omega(x,y,\alpha) = c_1 [/itex].
By letting [itex] \alpha [/itex] vary with [itex] c_1 [/itex] fixed this relation becomes the other members of the family of curves.
This is obviously a late response, but I'll take a swing at answering this given that there's another thread on Lie groups and ODEs. What I have to say fits this somewhat old thread much better than the new one.ayan849 said:Can anybody care to give a geometrical interpretation? I can't understand a bit of what is going on here :(
A Lie group is a mathematical concept that describes a continuous group of transformations. This is useful in solving ODEs because it allows for the use of symmetry and invariance principles, which can simplify and sometimes even solve the equations.
Yes, a Lie group can be used for any type of ODE, including linear, non-linear, and partial differential equations. The application of Lie group theory will depend on the specific form and properties of the ODE.
Symmetry plays a crucial role in the application of Lie group theory to ODEs. By identifying symmetries in the equation, we can reduce the number of independent variables and simplify the equation. This can lead to a more manageable problem and in some cases, even an analytical solution.
In most cases, a Lie group will not provide a general solution to an ODE. However, it can provide valuable insights and techniques for solving specific cases and finding solutions that may not be apparent with traditional methods.
One limitation of using Lie groups is that they are only applicable to ODEs that possess certain symmetries. Additionally, the calculations involved in applying Lie group theory can become complex and require advanced mathematical skills. Therefore, it may not always be the most practical approach for solving ODEs.