[![image credit Symmetry Magazine](https://www.steemimg.com/images/2016/08/12/symmetry-magazine8e03e.png)](http://www.symmetrymagazine.org/) _[image credit](http://www.symmetrymagazine.org/)_ Dr. Bryan here. Your friendly Steem physicist. I think math is art. Physics too (or why else would they have rewarded me with a Master of Arts in Physics?). So, I like to draw equations on the whiteboard. Here's one about the _locally conserved [Noether](https://en.wikipedia.org/wiki/Emmy_Noether) [Conserved Current](https://en.wikipedia.org/wiki/Conserved_current)_ (you've probably never heard of this equation, but its [among the most important in theoretical physics](https://en.wikipedia.org/wiki/Noether%27s_theorem)): [![noether-current-equationd0107.jpg](https://www.steemimg.com/images/2016/08/11/noether-current-equationd0107.jpg)](https://steemit.com/steemit/@nonlinearone/power-up-steemit-make-the-power-up-pledge-poweruppledge) [image credit: me](https://www.steemimg.com/image/Z6ADU) I also like to think of myself as a little teeny bit like Prometheus - I wanna steal the fire from the scientific gods an give it to everybody. ![prometheus-Free_stock_photo__Prometheus_Fountain_-_Free_Image_on_Pixabay_-_84427350c87.png](https://www.steemimg.com/images/2016/08/12/prometheus-Free_stock_photo__Prometheus_Fountain_-_Free_Image_on_Pixabay_-_84427350c87.png) [image credit - creative commons](https://pixabay.com/en/prometheus-fountain-844273/) So, in addition to showing the equation in all of its glory (instead of dumbing it all down like everyone else seems to do), I also decided to add annotations to every part of the equation so you can get a glimpse behind the curtain and see what those symbols mean. I'll also post some links to Wikipedia so you can geek out hardcore (read on to find that info). ## Noether's (first) Theorem What's the equation about? It's one of the important equations from Noether's Theorem - basically that symmetries lead to conservation laws. > Noether's (first)[1] theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918.[2] The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. Source: https://en.wikipedia.org/wiki/Noether%27s_theorem https://www.youtube.com/watch?v=CxlHLqJ9I0A [video credit goes to the creator of that video - youtube link](https://www.youtube.com/watch?v=CxlHLqJ9I0A) #### What are those weird u things? Those are the greek letter u. They mean that the thing they are near is a four-vector (three space coordinates plus a time coordinate). ## Annotations ### 1 - Conserved Noether Current > The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity. In modern (since ca. 1980[4]) terminology, the conserved quantity is called the Noether charge, while the flow carrying that charge is called the Noether current. The Noether current is defined up to a solenoidal (divergenceless) vector field. Source: https://en.wikipedia.org/wiki/Noether%27s_theorem#Informal_statement_of_the_theorem ### 2 - Momentum Density How the Lagrangian Density changes with respect to the space-time variation of the field phi. ![qft-Free_illustration__Computer__Quantum__Science__Space_-_Free_Image_on_Pixabay_-_869567c0fd5.png](https://www.steemimg.com/images/2016/08/12/qft-Free_illustration__Computer__Quantum__Science__Space_-_Free_Image_on_Pixabay_-_869567c0fd5.png) [image credit: creative commons](https://pixabay.com/en/computer-quantum-science-space-869567/) ### 3 - Momentum Density Equation The partial derivative of the Lagrangian Density with respect to the space-time change of the field phi. ### 4 - Four-Divergence > In differential geometry, the four-gradient ... (or 4-gradient) is the four-vector analogue of the gradient ... from Gibbs-Heaviside vector calculus. https://en.wikipedia.org/wiki/Four-gradient ![Free_illustration__Symmetry__Fractal__Backdrop_-_Free_Image_on_Pixabay_-_1444126c5afa.png](https://www.steemimg.com/images/2016/08/12/Free_illustration__Symmetry__Fractal__Backdrop_-_Free_Image_on_Pixabay_-_1444126c5afa.png) [image credit creative commons](https://pixabay.com/en/symmetry-fractal-backdrop-geometry-1444126/) ### 5 - Infinitesimal change under a symmetry transformation > Under a symmetry transformation, various quantities will change. One efficient way to describe this is to say that a field phi will change under a symmetry transformation by an amount ... lambda [upside down y looking thing]. Source: Quantum Field Theory for the Gifted Amateur Section 10.1 ### 6 - The Field > Quantum field theory frequently makes use of the Lagrangian formalism from classical field theory. This formalism is analogous to the Lagrangian formalism used in classical mechanics to solve for the motion of a particle under the influence of a field. In classical field theory, one writes down a Lagrangian density (curly L) involving a field, φ(x,t), and possibly its first derivatives (∂φ/∂t and ∇φ), and then applies a field-theoretic form of the Euler–Lagrange equation. Writing coordinates (t, x) = (x0, x1, x2, x3) = xμ, this form of the Euler–Lagrange equation is[10] > https://en.wikipedia.org/wiki/Quantum_field_theory#Lagrangian_formalism [![cup-Free_stock_photo__Photo_Montage__Cup__Sky__Clouds_-_Free_Image_on_Pixabay_-_488177e2f00.png](https://www.steemimg.com/images/2016/08/12/cup-Free_stock_photo__Photo_Montage__Cup__Sky__Clouds_-_Free_Image_on_Pixabay_-_488177e2f00.png)](https://www.steemimg.com/image/cMScZ) [image credit creative commons](https://pixabay.com/en/photo-montage-cup-sky-clouds-sun-488177/) My previous post of the Euler-Lagrange equation (mentioned above) https://steemit.com/science/@nonlinearone/the-euler-lagrange-equation-quantum-field-theory ### 7 - Symmetry transformation infinitesimal change equation The Dphi term is shorthand for the partial derivative of phi with respect to lambda evaluated at lambda equals zero. ## Still confused? OK, there is only so much I can do! But ... maybe now that you've seen behind the curtain you can poke around and find out more. There is an amazing wealth of information on Wikipedia! #### Google Talk on Emmy Noether and the Fabric of Reality https://www.youtube.com/watch?v=1_MpQG2xXVo [video credit - the owner of that video on YouTube](https://www.youtube.com/watch?v=1_MpQG2xXVo) _p.s. Emmy Noether is the greatest mathematician/scientist you've never heard of. She will be the subject of a future post. Spoiler alert: she is amazing - how many other Jewish female mathematician/physicists do you think there were in Germany at around 1916?_ [![Noetherdd336.jpg](https://www.steemimg.com/images/2016/08/12/Noetherdd336.jpg)](https://www.steemimg.com/image/cMQj8) [image credit wikipedia](https://en.wikipedia.org/wiki/File:Noether.jpg) #### Recommended reading Quantum Field Theory for the Gifted Amateur. Chapter 10: Symmetry. Section 10.2 Noether's theorem. [![Dr. Bryan Profile Pic](https://www.steemimg.com/images/2016/08/06/drwhopic7e23f_jpg__408408_05cf4.th.png)]( https://steemit.com/introduceyourself/@nonlinearone/hello-world-physics-ph-d-with-a-love-for-particle-physics-code-hiking-and-blockchains-here) https://steemit.com/introduceyourself/@nonlinearone/hello-world-physics-ph-d-with-a-love-for-particle-physics-code-hiking-and-blockchains-here
post_id | 578,785 |
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author | nonlinearone |
permlink | math-is-art-a-rendition-of-the-most-important-equation-you-ve-never-heard-of-noether-conserved-current |
category | mathematics |
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I'm not a math/numbers guy at all. But this was interesting. Unsurprisingly, for me, the most interesting part is that Noether is a female Jewish mathematician in turn of the 20th Century Germany. I really look forward to the post about her story. Followed :)
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Thanks @gregoryschneider - she was pretty amazing and thanks for the follow. Followed back :)
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Followed from a fellow PhD.
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Great and thanks! What is your Ph.D. in?
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Biochemistry
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Wow, talk about information packed! I guess i'll stick with growing toamtoes, it's all greek to me!
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Lol, well some of it _is_ greek, so good observation. Tomatoes are fun too :)
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Love Math & Numbers! @nonlinearone Thanks for sharing. Look forward to Emmy Noether! :D
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Thanks @virtualgrowth!
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Science is truly wonderful, scientists make the world a better place.
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In answer to your question regarding female Jewish mathematicians around her time ... none.
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permlink | re-nonlinearone-math-is-art-a-rendition-of-the-most-important-equation-you-ve-never-heard-of-noether-conserved-current-20160813t002729329z |
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You've now inspired me to write a post on Noetherian Rings.
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Yay!
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Also ... is this theorem about how conservation of energy exists under a symmetric relation on the variables used in the equation, i.e. `x_1*x_2*x_3 + x_2*x3 + x_2*x_1 + x_1*x_3` can be changed (under a simple symmetric group action of (1,2,3)) to the same equation?
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Honestly I have no idea but it sounds interesting :)
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All of the symmetry can be described by group theory, right? Is there really anything more complicated than that going on, even though you're looking at vector fields? Emmy Noether was awesome! She was supposedly really uncouth when eating in public, lol.
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permlink | re-nonlinearone-math-is-art-a-rendition-of-the-most-important-equation-you-ve-never-heard-of-noether-conserved-current-20160813t003320876z |
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I dont know if its more complicated but I wouldnt say group theory is all thats going on. The derivation involves the calculus of variations and the principle of least action.
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Thanks a lot for this nice post! Noether theorem is one of the most important theorem in fundamental physics (together with spin-statistics theorem, in my opinion). Greetings from another physicists laying around on steemit, by the way ^^
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Lemouth thanks! Please hit me up on my email and say hi - Steem physicists need to keep in touch- see my #contactme post :)
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I am usually available on the chat, but mostly away from keyboard at the moment (vacation time). Hope to see you there maybe in about 10 days! Cheers, B>
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Corrections: "Those are the greek letter u" should read "Those are the greek letter mu". https://en.wikipedia.org/wiki/Mu_(letter) See also: https://en.wikipedia.org/wiki/Four-vector#Four-vector_algebra
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Thanks for the upvote @ned :)
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That first image is 2MB & way larger dimensions than it needs to be. I'd like to ask that you take a little more care with that kind of thing. Try scaling down and saving as a JPG before using an image in a blog post. ;)
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