Jet Prolongation Formulas for Lie Group Symmetries

[Twigg]

I can never derive the prolongation formulas correctly when I want to prove the Lie group symmetries of PDEs. (If I'm lucky I get the transformed tangent bundle coordinate right and botch the rest.) I've gone through a number of textbooks and such in the past, but I haven't found any clear, non-horrendous method for reproducing the prolongation formulas for a symmetry generator on the go. Anyone have any hints/content/advice?

 

[strangerep]

Which textbooks have you tried? I tried Olver (with only partial success) and then Stephani (with a little more success, though Olver goes further). I never found one that was "clear".

Stephani is aimed more at physicists than mathematicians, so it depends what you're looking for.

 

[George Jones]

Does Hydon cover this?

https://www.amazon.com/dp/0521497868/?tag=pfamazon01-20

Maybe Hydon isn't advanced and/or rigorous enough.

 

[Twigg]

Sorry for taking so long getting back to you guys, had my hands full. I started a couple years ago out of Olver and got little from it. After reading Olver and looking at some other stuff I tried as an exercise to derive the Lorentz group from Maxwell's Equations using the symmetry method outlined in chapter 3 or so of Olver (if I remember correctly), the method where you prove infinitesimal invariance using a generator of a one parameter symmetry group. It became a complicated mess that was worse than what I started with, a large intensively coupled system of PDEs describing the symmetry group is what I remember getting. In light of that what I'm really looking for is a guide to using the symmetry method effectively. if there isn't any good resource, do you guys maybe have a worked example of something like I tried? Thanks again!

 

[strangerep]

do you guys maybe have a worked example of something like I tried?

I only have (relatively simple) worked examples for a couple of ODEs (free particle, and Kepler), following methods outlined in Stephani. Even those took many pages to work out fully.

It's much, much worse if you want to find dynamical symmetries, rather than merely point symmetries.

 

[Twigg]

Alrighty, glad to know it's not just me abusing a good method then. Olver made it look a lot more convenient that it seems to me. I have one last idea to try before I start thinking about simpler alternatives. I'm going to see if I can't write a program to do the heavy algebraic lifting. I'm thinking of starting with a MATLAB function that computes the prolongation formulas symbolically, and another function that uses those prolongation formulas to get the symmetry conditions from a given PDE system input. I'll post it in this thread if I come up with anything. Thanks all!

 

[strangerep]

I'm going to see if I can't write a program to do the heavy algebraic lifting. I'm thinking of starting with a MATLAB function that computes the prolongation formulas symbolically, and another function that uses those prolongation formulas to get the symmetry conditions from a given PDE system input.

I recall mention of such software in either Olver, or Stephani, (or both). But, iirc, they only produce the PDEs that you have to solve to find the symmetries -- which is not really the hard part of the problem, or so I found.

I also found it far too easy to apply the prolongation formulas incorrectly -- which is why I made the effort to work through Stephani's Kepler example in detail.

 

[Twigg]

I recall mention of such software in either Olver, or Stephani, (or both). But, iirc, they only produce the PDEs that you have to solve to find the symmetries -- which is not really the hard part of the problem, or so I found.

That's pretty much all I had in mind. I figure the PDEs won't be terribly helpful. The first time I tried doing it by hand for Maxwell's equations, I remember having ten 0th order jet coordinates (4 for space and time + 6 for the E and B fields) plus twenty four 1st order jet coordinates, making for thirty four generator components. I don't remember how many PDEs I would have gotten from that, but I remember it was more than a few.

I also found it far too easy to apply the prolongation formulas incorrectly -- which is why I made the effort to work through Stephani's Kepler example in detail.

I'll take a look at it, thanks for the tip!