In general, if your problem displays any kind of symmetry* you can exploit that to simplify things. I think most people are capable of doing this intuitively when the symmetry is obvious. The Buckingham pi theorem is a great example of a systematic way to find and exploit a symmetry that isn’t so obvious.
* By “symmetry” I really mean “invariance under a group of transformations”.
This is a great point. Other than fairly easy geometric and time symmetries, do you have any advice or know of any resources which might be helpful towards finding these symmetries?
Here’s what I do know: Sometimes you can recognize these symmetries by analyzing a model differential equation. Here’s a book on the subject that I haven’t read, but might read in the future. My PhD advisor tells me I already know one reliable way to find these symmetries (e.g., like how to find the change of variables used here), so reading this would be a poor use of time in his view. This approach also requires knowing a fair bit more about a phenomena than just which variables it depends on.
The book you linked is the sort of thing I had in mind. The historical motivation for Lie groups was to develop a systematic way to use symmetry to attack differential equations.
This is a great point. Other than fairly easy geometric and time symmetries, do you have any advice or know of any resources which might be helpful towards finding these symmetries?
Are you familiar with Noether’s Theorem? It comes up in some explanations of Buckingham pi, but the point is mostly “if you already know that something is symmetric, then something is conserved.”
The most similar thing I can think of, in terms of “resources for finding symmetries,” might be related to finding Lyapunov stability functions. It seems there’s not too much in the way of automated function-finding for arbitrary systems; I’ve seen at least one automated approach for systems with polynomial dynamics, though.
Not familiar with Noether’s theorem. Seems useful for constructing models, and perhaps determining if something else beyond mass, momentum, and energy is conserved. Is the converse true as well, i.e., does conservation imply that symmetries exist?
I’m also afraid I know nearly nothing about non-linear stability, so I’m not sure what you’re referring to, but it sounds interesting. I’ll have to read the Wikipedia page. I’d be interested if you know any other good resources for learning this.
Is the converse true as well, i.e., does conservation imply that symmetries exist?
I think this is what Lie groups are all about, but that’s a bit deeper in group theory than I’m comfortable speaking on.
I’d be interested if you know any other good resources for learning this.
I learned it the long way by taking classes, and don’t recall being particularly impressed by any textbooks. (I can lend you the ones I used.) I remember thinking that reading through Akella’s lecture notes was about as good as taking the course, and so if you have the time to devote to it you might be able to get those from him by asking nicely.
Conservation gives a local symmetry but there may not be a global symmetry.
For instance, you can imagine a physical system with no forces at all, so everything is conserved. But there are still some parameters that define the location of the particles. Then the physical system is locally very symmetric, but it may still have some symmetric global structure where the particles are constrained to lie on a surface of nontrivial topology.
Noether’s theorem has nothing to do with Buckingham’s theorem. Buckingham’s theorem is quite general (and vacuous), while Noether’s theorem is only about hamiltonian/lagrangian mechanics.
Added: Actually, Buckingham and Noether do have something in common: they both taught at Bryn Mawr.
Noether’s theorem has nothing to do with Buckingham’s theorem.
Both of them are relevant to the project of exploiting symmetry, and deal with solidifying a mostly understood situation. (You can’t apply Buckingham’s theorem unless you know all the relevant pieces.) The more practical piece that I had in mind is that someone eager to apply Noether’s theorem will need to look for symmetries; they may have found techniques for hunting for symmetries that will be useful in general. It might be worth looking into material that teaches it, not because it itself is directly useful, but because the community that knows it may know other useful things.
It’s a quite bit more general than Lagrangian mechanics. You can extend it to any functional that takes functions between two manifolds to complex numbers.
In general, if your problem displays any kind of symmetry* you can exploit that to simplify things. I think most people are capable of doing this intuitively when the symmetry is obvious. The Buckingham pi theorem is a great example of a systematic way to find and exploit a symmetry that isn’t so obvious.
* By “symmetry” I really mean “invariance under a group of transformations”.
This is a great point. Other than fairly easy geometric and time symmetries, do you have any advice or know of any resources which might be helpful towards finding these symmetries?
Here’s what I do know: Sometimes you can recognize these symmetries by analyzing a model differential equation. Here’s a book on the subject that I haven’t read, but might read in the future. My PhD advisor tells me I already know one reliable way to find these symmetries (e.g., like how to find the change of variables used here), so reading this would be a poor use of time in his view. This approach also requires knowing a fair bit more about a phenomena than just which variables it depends on.
The book you linked is the sort of thing I had in mind. The historical motivation for Lie groups was to develop a systematic way to use symmetry to attack differential equations.
Are you familiar with Noether’s Theorem? It comes up in some explanations of Buckingham pi, but the point is mostly “if you already know that something is symmetric, then something is conserved.”
The most similar thing I can think of, in terms of “resources for finding symmetries,” might be related to finding Lyapunov stability functions. It seems there’s not too much in the way of automated function-finding for arbitrary systems; I’ve seen at least one automated approach for systems with polynomial dynamics, though.
Not familiar with Noether’s theorem. Seems useful for constructing models, and perhaps determining if something else beyond mass, momentum, and energy is conserved. Is the converse true as well, i.e., does conservation imply that symmetries exist?
I’m also afraid I know nearly nothing about non-linear stability, so I’m not sure what you’re referring to, but it sounds interesting. I’ll have to read the Wikipedia page. I’d be interested if you know any other good resources for learning this.
I think this is what Lie groups are all about, but that’s a bit deeper in group theory than I’m comfortable speaking on.
I learned it the long way by taking classes, and don’t recall being particularly impressed by any textbooks. (I can lend you the ones I used.) I remember thinking that reading through Akella’s lecture notes was about as good as taking the course, and so if you have the time to devote to it you might be able to get those from him by asking nicely.
Conservation gives a local symmetry but there may not be a global symmetry.
For instance, you can imagine a physical system with no forces at all, so everything is conserved. But there are still some parameters that define the location of the particles. Then the physical system is locally very symmetric, but it may still have some symmetric global structure where the particles are constrained to lie on a surface of nontrivial topology.
Noether’s theorem has nothing to do with Buckingham’s theorem. Buckingham’s theorem is quite general (and vacuous), while Noether’s theorem is only about hamiltonian/lagrangian mechanics.
Added: Actually, Buckingham and Noether do have something in common: they both taught at Bryn Mawr.
Both of them are relevant to the project of exploiting symmetry, and deal with solidifying a mostly understood situation. (You can’t apply Buckingham’s theorem unless you know all the relevant pieces.) The more practical piece that I had in mind is that someone eager to apply Noether’s theorem will need to look for symmetries; they may have found techniques for hunting for symmetries that will be useful in general. It might be worth looking into material that teaches it, not because it itself is directly useful, but because the community that knows it may know other useful things.
It’s a quite bit more general than Lagrangian mechanics. You can extend it to any functional that takes functions between two manifolds to complex numbers.
In what sense do you mean Buckingham’s theorem is vacuous?