Graduate Example of how a rotation matrix preserves symmetry of PDE

[JTC]

Good Day

I have been having a hellish time connection Lie Algebra, Lie Groups, Differential Geometry, etc.
But I am making a lot of progress.

There is, however, one issue that continues to elude me.

I often read how Lie developed Lie Groups to study symmetries of PDE's

May I ask if someone could exemplify this with a very simple, concrete example?

For example, I understand orthogonal matrices (Lie Groups) and how their basis is skew symmetric matrices as generators (Lie Algebras) and I can connect this with the need to study Differential geometry.

But where can I find (or could someone provide) a simple, example of how rotation matrices preserve symmetries of PDS (and, also, explain what a synmmetry of a PDE is)

 

[fresh_42]

This is a very good question and I'll try to find / create an example. Noether's paper about invariants of differential expressions (Göttingen, 1918) is quite clear and surprisingly easy to read compared to the modern versions of her famous theorem. I would quote it, but it's not in English.

 

[anorlunda]

But where can I find (or could someone provide) a simple, example of how rotation matrices preserve symmetries of PDS (and, also, explain what a synmmetry of a PDE is)

It may be a more elementary level than a textbook, but this Wikipedia article provides several derivations of Noether's Theorem in different contexts plus examples of its use. https://en.wikipedia.org/wiki/Noether's_theorem

 

[JTC]

It may be a more elementary level than a textbook, but this Wikipedia article provides several derivations of Noether's Theorem in different contexts plus examples of its use. https://en.wikipedia.org/wiki/Noether's_theorem

Hi, I am "aware" of Noether's work but I feel it is too advanced for me and am hoping for a simpler example. One must exist. I just feel much happier when things are in context of simple examples.

 

[JTC]

This is a very good question and I'll try to find / create an example. Noether's paper about invariants of differential expressions (Göttingen, 1918) is quite clear and surprisingly easy to read compared to the modern versions of her famous theorem. I would quote it, but it's not in English.

If you do type up a response, please, if you can, keep it simple. It would help me a lot.

 

[Orodruin]

There are many examples of how symmetries can be used in relation to PDEs. One of the more encountered ones is to use the symmetries in order to reduce the possible forms of a solution. To be explicit, consider the search for the Green's function of Poisson's equation in $n$ dimensions, i.e.,
$$
\nabla^2 G(\vec x) = \delta^{(n)}(\vec x)
$$
This differential equation is invariant under rotations about the origin, implying that the Green's function $G(\vec x)$ can be written only as a function of $r = \lvert \vec x\rvert$. Once we know this, we can significantly simplify the differential equation and eventually solve it as an ODE in $r$ (with the solution depending on the number of dimensions we consider).

Another application of symmetries to PDEs is to use the symmetries of a Sturm-Liouville problem, including the boundary conditions, to find out what types of eigenfunctions are admissible as solutions, the number of degenerate eigenvalues, etc, using representation theory.

Edit: In particular, it is a rather nice exercise to look at how the functions $e^{in\varphi}$ form the irreps of the group $SO(2)$ as represented by functions on the unit circle, or how the spherical harmonics form the irreps of the group $SO(3)$ as represented by functions on the unit sphere.

 

[fresh_42]

Here's another nice treatise including an easy example.
http://www.mathpages.com/home/kmath564/kmath564.htm