as such, you can turn one such solution into another solution by replacing x->x+a for any value of $a$. since $a$ can be any real number, there's an infinite number of such transformations and they're all parametrized by the set of allowed $a$

so that set of transformations on solutions actually forms a group

Okay

So, solutions form a group.

One way to appreciate their use with diff eq's is to think of Lie symmetries as special continuous coordinate transformations which transform one differential equation into an equivalent (but hopefully simpler) differential equation which can more easily be solved

no, not the solutions themselves: the transformations that turn one solution into another

@bolbteppa That’s the thing, Waah!

it's a group of transformations, not a group of solutions

@Semiclassical Yeah of course, group is a set of operations not the functions

Do we get introduced to Lie Symmetry post graduation or in UG?

Or never

depends on what coursework you do. in physics courses you may not see it at all

i don't know about math

In mathematics course?

@Semiclassical Are you a self-taught physicist-mathematician?

by and large, no: i did physics/math in undergrad and physics grad school

Renowned Mathematician And Physicist Freeman Dyson Has Died At Age 96

that said, becoming "self-taught" is inevitable at a certain point. doing research means becoming an expert on your own terms using your own resources

Yikes

oh, that's a bummer

@bolbteppa When we got introduced to live symmetry?

In a course on Lie theory or in a 'group theory in physics' course, it depends on the department

the trouble is that Lie symmetry analysis is a bit of a specialized tool

the question of "can I solve this ODE exactly" is not as relevant as it once was

given the role of numerical methods

RIP to Freeman Dyson

that's not to say it's irrelevant, but it's off the beaten track for most mathematicians. they'd certainly be aware of it, but I'm not sure how much they'd specifically kno about it

@Semiclassical Lie symmetry contains which lie the liar one or I’m lying on couch ?

it contains the lie that you pronounce Lie like that :P

(it's pronounced like LEE, because that's how you say Sophus Lie's last name)

What? How? LIE is pronounced as L A EE

Nope. Lie was Sophus Lie's last name, and it's pronounced LEE.

he was norweigian, for a measure of context.

Okay!

@Knight explain these notes to us on this topic

I like that the first page in the first chapter tells you how to pronounce Lie correctly :P

also, that first page does contain a hint of what makes Lie symmetry "geometric": you're studying symmetries on the space of solution curves

Why have people chosen to adapt the green-human-like with big eyes creature to be a representative of aliens?

for instance, the example I gave above for y'=f(y) can be summed up as: Suppose i have a solution curve for that ode. then if I slide it to the right or left, then that's another solution curve. so the set of solution curves is invariant under horizontal translations

by contrast, if you had y'=f(x) then it'd be invariant under vertical translations

with more complicated odes, it's not at all obvious what sorts of invariance is possible. but if you can find it, then your life becomes a lot easier

When telescopes listen for radio waves far into the galaxy, how do they target exactly that place? I mean, aren't there other RF waves interfering. Also, when you aim the telescope at a planet, how do you know that the RF signal is actually from there and is not something else that has bounced of and so on...?

@bolbteppa Those are excellent text for my level. It has explained symmetry very well by saying “symmetry means changing something so that it is unchanged” (although it doesn’t write that explicitly but I got it that way)

@SolomonSlow Hello ! How are you?

Symmetries are the bundle isomorphisms that leave the action invariant

oh crap

remember how in mario 3 there was a p meter that went up as you go faster

i just realized p means momentum

maybe

@SirCumference mariowiki.com/Power_Meter Stands for power, but that's still physics.

dang it

@SirCumference Well, $P=Fv=mav=ap$, so it is relatable

Ignoring cases we also end up with $a=1$

I am a poor introductory physics student now

we'll use some deformed natural units so the function $a$ is set to 1

"deformed natural" is oxymoronic

i wonder if you actually can do that with natural units

set non-constants to 1

As long as you aren't talking to experts you can do anything you want

fair enough lol