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I am self-studying a physics textbook on waves. While discussing solutions to linear homogeneous ODEs, the author talked about the exponential as "irreducible" solutions and on a footnote, said that they were, formally, "irreducible representations of the translation group". I am understanding the material that is presented in the book, but I think I would like to explore the relevant area of mathematics alongside. Which branch of mathematics is this and what are the introductory references?


Another persistent theme in the book is solving differential equations (oscillating systems, but a general approach would also be good for me) using arguments such as linearity and symmetry. What are good books on differential equations that approach the subject in a similar fashion and are accessible.

I have studied calculus and ODEs before, but it was very "application oriented" and most of the material was presented as "methods" (i.e you have a linear 1st order ODE, you use integrating factor, etc). I know with a lot of contemplation I can connect some dots myself, but it will be helpful to have some guiding material.

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First question: en.wikipedia.org/wiki/Representation_theory. Your third question doesn't seem to be an actual question. –  anon Aug 18 '11 at 8:08
    
I've studied symmetry from 'Symmetry and Integration Methods for Differential Equations' written by George W. Bluman. It is quite descriptive while somehow being misleading to certain extent. My recommendation is to take a look into 'Applications of Lie Groups to Differential Equations', which is quite a standard reference as far as I know. –  newbie Aug 18 '11 at 8:23
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With regard to your last question, it is clearly stated in Amazon's description of Arnold's book:'A fresh modern approach to the geometric qualitative theory of ordinary differential equations'.So in order to read that book, you may need some knowledge of differential geometry,etc. Briefly introducing how advanced math education you have received would be helpful. –  newbie Aug 18 '11 at 8:36
    
@anon what are the prerequisites and what would you recommend as a suitable introductory text –  kuch nahi Aug 18 '11 at 17:11
    
Prerequisites would be linear and abstract algebra at a high enough level. Beyond knowing what representations and characters are, I don't actually know representation theory, so I can't recommend any texts. Though I believe googling for PDFs should provide a wealth of free, understandable introductory material that should satisfy basic curiosity. –  anon Aug 18 '11 at 17:19
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newbie mentioned Peter Olver's "Applications of Lie Groups to Differential Equations", which is a good book. An alternative is Sattinger and Weaver's "Lie Groups and Algebras, with Applications to Physics, Geometry, and Mechanics". I didn't know anything about Lie groups before I read those two books, so it's certainly possible to learn them that way.

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