ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Noether's Theorem: Symmetry in Nature

Updated on June 1, 2012

Noether's Theorem (also called The Noether Theorem) is a theorem discovered by mathematician Emmy Noether. It is extremely useful in many areas of theoretical physics, and is frequently used in classical and quantum field theory to predict the conservation laws that must be obeyed in all processes, such as the conservation of energy.

What is Noether's Theorem?

Expressed in words, Noether's Theorem says:

"For every continuous symmetry, there are corresponding quantities whose values are conserved."

This butterfly has reflection symmetry.
This butterfly has reflection symmetry.
This star has rotational symmetry order 5.
This star has rotational symmetry order 5.

What is Symmetry?

The easiest way to understand symmetry is with an example. Consider the pattern on a butterfly's wings. The pattern is completely symmetrical - the butterfly's left wing is the exact mirror image of the right wing. If you flipped an image of the butterfly over in a mirror, swapping left with right, it would look exactly the same as the original. This means that the butterfly has a reflection symmetry.

That's one type of symmetry, and is the kind that most people refer to when they say something is symmetrical. But there are also other kinds. Consider the star shown in the picture. It has rotational symmetry, because you can rotate it by 72 degrees and it will look exactly the same as it did before the rotation. Because you can do this 5 times until you get back to where you started, we say the star has rotational symmetry order 5.

The universe also has rotational symmetry. What that means is that if you do an experiment, and then you rotate all your apparatus and do the same experiment again, you will always get the same result. It's like the image of the star looking the same after it has been rotated by 72 degrees, but with one difference: the universe doesn't care how many degrees you rotate by, the result will be the same regardless. Rotating the universe is more like rotating a circle than a star - no matter how many degrees you rotate by, everything looks the same. The universe has continuous rotational symmetry.

From Noether's theorem, we know that there is a conservation law associated with this symmetry. It turns out that the continuous rotational symmetry of the universe leads to the law that angular momentum is always conserved. Put simply, this means that if you have a bicycle wheel spinning freely, with no frictional forces acting on it, the wheel will continue to spin at exactly the same speed forever. It's angular momentum will be conserved.

Read ngriffin360's hub on conservation of angular momentum.

The universe also has time symmetry, which means that the outcome of an experiment is not affected by when the experiment is performed - i.e. if you do an experiment today, and do the same experiment next Tuesday, being careful to keep all the variables the same, you should expect the same result. This symmetry leads to the conservation of energy.

There is also translation symmetry - if you do an experiment in one place, and then relocate your laboratory ten miles south of where you are now, then again you should get the same results (ignoring local effects due to variations in the Earth's gravitational field etc). This symmetry leads to the conservation of momentum.

The mathematics needed to prove Noether's Theorem and to show how each symmetry leads to its conservation law can be found in any of the following maths books. The proof isn't very long, but you need to know calculus to follow it.

Emmy Noether's Wonderful Theorem - for UK Amazon users

topquark works as a researcher in theoretical particle physics and blogs about science at The Particle Pen.

working

This website uses cookies

As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://corp.maven.io/privacy-policy

Show Details
Necessary
HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
LoginThis is necessary to sign in to the HubPages Service.
Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
AkismetThis is used to detect comment spam. (Privacy Policy)
HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
Features
Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
MavenThis supports the Maven widget and search functionality. (Privacy Policy)
Marketing
Google AdSenseThis is an ad network. (Privacy Policy)
Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
Index ExchangeThis is an ad network. (Privacy Policy)
SovrnThis is an ad network. (Privacy Policy)
Facebook AdsThis is an ad network. (Privacy Policy)
Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
AppNexusThis is an ad network. (Privacy Policy)
OpenxThis is an ad network. (Privacy Policy)
Rubicon ProjectThis is an ad network. (Privacy Policy)
TripleLiftThis is an ad network. (Privacy Policy)
Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
Statistics
Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
ClickscoThis is a data management platform studying reader behavior (Privacy Policy)