Noether's Theorem, Symmetries & Lorentz/Poincare Group Self-Study - help?

[ZenMaster]

Hello folks,

I'm interested in getting a much deeper understanding of symmetries and how they pretty much define the universe; e.g. translation symmetry in time = Conservation of Energy?? according to Wikipedia. I'm *extremely* interested in how symmetries lead to universal laws.

My level of education is up to 3.5 years of Mathematics/Statistics in university.. I had to drop due to funding issues. I wasn't a spectacular student but I would have graduated. I learned some set theory, but by now my memories foggy.

Can anyone recommend resources to study from? Online preferred for convenience, but I'm definitely willing to shell out for decent textbooks (preferred university level, since casual "Chapters-ish" books gloss over the real explanations).

Thanks for any suggestions you may have! I'll take any and every one. :nana:

Dan

[Cartesiantheater]

I'd start with linear algebra, if you have become rusty on that. Particularly linear transformations.

Then start noticing things that are invariant under such transformations.


Here's a classic one:

Given two moving bodies in opposite directions along the x-axis, starting at the origin (Newtonian physics here), the total kinetic energy between them will be

(1/2)M (v1² + v2²)

But that value is exactly the same even if you interchange the two velocities.




As it pertains to Lorentz, the space-time interval defines a symmetry. This was discussed in the following thread:

http://forums.armageddononline.org/minkowski-space-and-t24773.html





I don't know of any one text that focuses only on symmetry/invariance, but I'm sure there is one. The single one I am most familiar with is the invariance of the spacetime interval in special relativity.

[equestrian]

Greetings ZenMaster

IMO, set theory won't get you very far, and studying symmetries, time, and the universe are all so theoretical that it won't help a bit or fill in your last half year.

T'were me I'd get a good physics book, and start at page one and learn the principles along with the math that is required. The level of math will be obvious as you progress through the book.

Doing it that way - the course outline (in the table of contents) and math requirements (solving the problems at the end of each chapter) will be all laid out for you, and the end result will get you where you wanna go.

As far as the universe goes - I'd avoid all conventional theory and study Frank D Stacey's, Physics of the Earth.

P.S. t'were me - I'd not skip a chapter anywhere!

james
/

[Cartesiantheater]

I think he's shooting for theoretical here.


Noether's theorem just says that when you have a (differentiable) symmetry that means you have something that is conserved. I don't know how to prove that, but it's pretty easy to show some particular examples. Like in two bodies orbiting each other (or one body orbiting another because...), there is obvious symmetry. So that would mean something is conserved. What is conserved? Using the Euler-Lagrange equation you end up with that mass times distance squared times the second time derivative of the angle = 0 ==> i.e. ANGULAR MOMENTUM is conserved.

(use use the Euler-Lagrange equation on the Lagrangian with respect to both the position vector and the angle- that is you'll have two laws of motion by the time you're done, and ONE of them implies conservation of angular momentum)

[ZenMaster]

Thanks for the replies :)

@ James
Well .. I'm not really looking to finish off my education (for this project). I also want to stick with set theory despite its more pure mathematical / theoretical nature, because it plays a key role in Special Relativity, and Special Relativity is key to the project. That said, I think I'll take a look at that Physics of the Earth book just for kicks.

@ Cartesiantheater
For the Lorentz equations, your derivation of them on this site is really good. I'll look back over it. I've been doing some light hunting on the web and that "Langrangian" keeps coming up. I hate to ask questions before I've done my own homework first so I won't yet start asking further about the Lagrangian and Euler-Lagrange equations :)

Thank you both for the advice, I think I'll start here.

Daniel

[Cartesiantheater]

No problem. If you want to do set theory, let me say that it really goes hand in hand with linear algebra (well, it helps it a great deal). From set theory, you'll then go to relations, and then functions, and of course these tie in with linear algebra, particularly linear transformations.


Aside from linear algebra, brush up on multi-variable and vector calculus (wouldn't hurt to review polar, cylindrical and spherical coordinates). After you are good there, study the Calculus of Variations (which is where the Euler-Lagrange equation comes into play, and with that you can get the laws of motion of the object(s) you are studying, and by looking at those you can see if there are conserved quantities).



But if special relativity is the MAIN physics portion, here is my advice:

Linear algebra! You need a good grasp of vectors and matrices if you plan on studying special relativity in a more mathematical light. Because special relativity in a more sophisticated mathematics relies heavily on what are called four vectors, and obviously linear transformations. Which then is developed into tensors (if you want to learn General Relativity, that's needed).


Here is an online book I am currently going through. When you're ready, it gives the more mathematically sophisticated treatment of special relativity (in the sense that ties it to general relativity).

http://pages.pomona.edu/~tmoore/grw/


Outstanding, IMHO.

[ZenMaster]

Yup, I don't think there's any way around it. I'm going to have to brush up on ALL of my old math skills! X-)

Ah well, the practice will be good. I'm thinking I might go full bore and see about Quantum Physics.. although that might be a complete impossibility at my skill level. :-.

[dedanoe]

if a theory tends to propagate symmetry then its equations, plots, and tables must also be symmetrical as it is the case in the lever law only: F_1 x D_1 =||= 2_D x 2_F and the lever law is not as simple as it sounds. just follow this one: SO.TURN.ON EVERYTHING IS FALS.E &&& A.SELF IS EVERYTHING ON.TURN.SO look for the similarities between the lever law and yin yang symbol. in fact if you want to learn real physics you better go to AiKiDo lessons.

before learning the generally accepted physics demand explanation for what TIME is cause it is the first variable that comes into play which from the start has no strict definition. the big bang theory is about space-time popping up from nothing as it is the case with the entire physics and its first lecture kinetics. all you need are wide definitions and do not let others take the right from you to freely attempt and build your own theory. we are obligated to continually reevaluate everything. authorities cannot conserve physics at any cost as if it was some bloody tradition.

[ZenMaster]

.. and do not let others take the right from you to freely attempt and build your own theory. we are obligated to continually reevaluate everything. authorities cannot conserve physics at any cost as if it was some bloody tradition.

You say we are obligated to reevaluate everything, but also go on to assert that time must be the first variable. With ya on the first point, but I disagree with you on the nature of "time". I do not take Einstein's union of space with time on faith, but have studied Special Relativity and I can't deny the logic regardless of who came up with.

That's the great thing about science. Dogma may seep into it, as in any other societal structure, but science is one of the few societal structures that not only allows, but *encourages* you to try and destroy the dogma (with thoughtfulness).

[dedanoe]

every physical system can be described with several magnitudes, say in our case those magnitudes are x, y, z, w, s, t. then the ordered array of all the magnitudes describing the system is event E = (x, y, z, w, s, t). then the laws that are some equations regulate which events are legal, say in our case those laws are F1(E1) = 0 and F2(E2) = 0 and F3(E3) = 0. out of this system, three equations are soluble via the remaining three x = H1(w, s, t) and y = H2(w, s, t) and z = H3(w, s, t). the array (x, y, z) is trajectory of legal events. and finally, time is the ordered array of parameters T = (w, s, t). as you can see time is system specific and time can be multidimensional.

[ZenMaster]

every physical system can be described with several magnitudes, ...

time is the ordered array of parameters T = (w, s, t). as you can see time is system specific and time can be multidimensional.

Well, you're describing variables, or simply unknowns. The word "time" itself does not generalize, and is instead the fundamental S.I. unit, the "clock" that allows change. As far as I'm aware, the idea of a system of multidimensional time is so far away from mainstream, it's pure crackpottery.

As for time being the "first" variable .. although this too is well outside mainstream physics, who's to say time is not an emergent property of quantum physics? It's not impossible.

[dedanoe]

write down the law of lever as Weight1 x Distance1 = Weight2 x Distance2 then show us how your time flow puts this lever in motion?

[ZenMaster]

I never said it did. It seems to me that's what you were saying ..


before learning the generally accepted physics demand explanation for what TIME is cause it is the first variable that comes into play which from the start has no strict definition.

Did I misinterpret you?

And why is the lever law so important to you?

[Cartesiantheater]

I never said it did. It seems to me that's what you were saying ..



Did I misinterpret you?

And why is the lever law so important to you?

He has an entirely different theory of physics that rests on ideas about levers. He more or less dismisses most every branch of physics. His theory is far more mathematical than it is physics, since it has no experimental basis and is formed from core ideas rather than from core measurements. I.e. his theory is more deductive than inductive.

Sometimes you will see this in the history of science, but usually it is the other way around. Usually it gets you nowhere, but once in a while it gets you were no one else can (Einstein is an example of both: he went nowhere with his Grand Unification ideas, but went where few else did with special relativity [in that he derived from first principles what everyone else figured from measurements or prevailing theory {special relativity was not uniquely Einstein's. He just looked at it a little differently}]).

okay that was longer than I intended.

[Cartesiantheater]

Hey ZenMaster, if you ever come back, in debating about the conservation of energy (a poster who doesn't believe in it... ), I copied from my classical mechanics textbook a passage on how the conservation of energy is derived from a symmetry of time. I thought I'd paste it here for you if you're interested in the type of math you will have to understand to be able to start to get into Noether's Theorem, and how symmetries relate to conservation laws, etc.






7.9 Conservation Theorems Revisited


Conservation of Energy

We saw in our previous arguments that time is homogeneous within an inertial reference frame. Therefore, the Lagrangian that describes a closed system (i.e., a system not interacting with anything outside the system) cannot depend explicitly on time, that is

(7.124)




∂L/∂t = 0



so that the total derivative of the Lagrangian becomes

(7.125)




dL/dt = ∑j ∂L/∂qj q'j + ∑j ∂L/∂q'j q''j



where the usual term, ∂L/∂t, does not appear now. But Lagrange's equations are

(7.126)


∂L/∂qj = d/dt ∂L/∂q'j


Using Equation 7.126 to substitute ∂L/∂q in Equation 7.125, we have


dL/dt = ∑j q'j d/dt ∂L/∂q'j + ∑j ∂L/∂q'j q''j

or



dL/dt - ∑j d/dt (q'j ∂L/∂q'j) = 0


so that

(7.127)


d/dt (L - ∑j q'j ∂L/∂q'j) = 0

The quantity in the parentheses is therefore constant in time; denote this constant by -H

(7.128)


L - ∑j q'j ∂L/∂q'j = -H = constant

If the potential energy U does not depend explicitly on the velocities x'α,i, or the time t, then U = U(xα,i). The relations connecting the rectangular coordinates and the generalized coordinates are of the form xα,i = xα,i(qj) or qj = qj(xα,i), where we exclude the possibility of an explicit time dependence in the transformation equations. Therefore, U = U(qj), and ∂U/∂q'j = 0. Thus,




∂L/∂q'j = ∂(T - U)/∂q'j = ∂T/∂q'j


Equation 7.128 can be written as

(7.129)



(T - U ) - ∑jq'j ∂T/∂q'j = -H


and, using Equation 7.122, we have



(T - U) - 2T = -H


or

(7.130)



T + U = E = H = constant


The total energy E is a constant of the motion for this case.

The function H, called the Hamiltonian of the system, may be defined as in Equation 7.128 (but see Section 7.10). It is important to note that the Hamiltonian H is equal to the total energy E only if the following conditions are met:

1.


The equations of the transformation connecting the rectangular and generalized coordinates (Equation 7.116) must be independent of the time, thus ensuring that the kinetic energy is a homogeneous quadratic function of the q'j

2.


The potential energy must be velocity independent, thus allowing hte elimination of the terms ∂U/∂q'j from the equation for H





From: Classical Dynamics of Particles and Systems, Fifth Edition, Thornton, Marion. pg 260-261



EDIT- I also thought I'd post this:


http://www.sjsu.edu/faculty/watkins/noetherth.htm


This is an article discussing Noether's theorem.