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Symmetry and Conservation in Physics: The Famous Work of Emmy Noether

The following is a paper written by me for the class MATH 23b: Introduction to Proofs at Brandeis University and is posted for course credit.

Noether’s Theorem was proven, as the name suggests, by the mathematician Emmy Noether  in 1918. While Noether was a mathematician, her famous theorem pertains to the field of physics, and is often regarded as one of the most important concepts in the field. Noether’s Theorem shows the connection between conservation laws and continuous symmetries of systems. It states that if there is a continuous symmetry in the Lagrangian of a system, then there is an associated conservation law. Before one can make sense of this, a few terms need to be discussed.

First is the definition of symmetry. Symmetry is a transformation which keeps the object it is being applied to “the same” in some sense of the word. For a symmetry to be continuous, it means that the scale of the transformation does not change the fact that the object is symmetric under that transformation. This is best understood through an example. Consider a circle, which has both reflective and rotational symmetry. If you cut the circle in half and flip it about that line some integer number of times it appears the same as before. This reflection is not a continuous symmetry however, because it can only be reflected an integer number of times and nothing in between. Now consider the rotational symmetry of a circle. If you rotate a circle about its center by some angle, it appears the same, regardless of the size of the angle by which you rotate it. This is what we mean by continuous symmetries.

The physics to which Noether’s Theorem pertains is called Lagrangian Mechanics which is build upon “the principle of least action.” This principle simply states that when considering the motion of an object, that object will minimize the quantity called the action3. The action is defined to be the time integral of the Lagrangian, which is some function of the generalized coordinates of the system considered and the derivatives of those coordinates. That’s a mouthful, but it comes down to this, the Lagrangian is: L=L(q,q’) for a set of generalized coordinates denoted by q where the prime denotes the derivative, and the action is: S[q(t)]=integral(L(q,q’)dt) from t to t+dt.  To minimize the action, we take its derivative and set that equal to zero, the outcome of which is the “Euler-Lagrange Equation:” dL/dq-(d/dx)(dLdq’)=0.

In Noether’s Theorem, the symmetries considered are continuous symmetries of the Lagrangian, meaning that for some coordinate q, L(q+q,dq’+dq’)=L(q,q’) for an arbitrary change in q, where q can be any of the coordinates of the system.  To minimize the action, we take its derivative and set that equal to zero, the outcome of which is the “Euler-Lagrange Equation:” dL/dq-(d/dx)(dLdq’)=0. If the Lagrangian is symmetric with respect to a coordinate q then we know that dL/dq=0 since the Lagrangian does not change with changes in the coordinate. This means that for a system with a Lagrangian symmetric in q, the Euler-Lagrange equation reads (d/dx)(dL/dq’)=0, and thus we say that the quantity dL/dq’=constant and thus is conserved for all time.  Some common symmetries in the Lagrangian are spatial translational, spatial rotational, and time translational symmetries, and these symmetries correspond to the conserved quantities; linear momentum, angular momentum, and energy respectively. For example when considering a particle with mass m in unaccelerated motion in one dimension, the classical Lagrangian is  L=(½)m(x’)^2 where this is the kinetic energy of the particle and the derivative of the position of this particle x’ is the velocity. From this the Euler-Lagrange Equation gives (d/dx)(mx’)=0 since the Lagrangian is symmetric in the x direction and thus mx’=constant. Where this conserved quantity mx’ is equal to the linear momentum of the particle. In other words, the linear momentum of the particle is conserved.

Noether’s Theorem shows a fundamental connection between mathematical symmetries in a system and the most important physical quantities. This connection is reflected in the common wording of Noether’s Theorem, “Where there is symmetry in a physical system, there is a conserved quantity associated with it.” This gives those working in physics an idea of where conservation comes from and how to find conservation laws, which allow for a much greater understanding of physical systems. This is why Noether’s Theorem came to be known as one of the most amazing concepts in physics.

References

1 Emmy Noether, Encyclopedia Britannica;https://www.britannica.com/biography/Emmy-Noether

2 The Noether Theorem, Columbia University; http://phys.columbia.edu/~nicolis/NewFiles/Noether_theorem.pdf

3 The Principle of Least Action, The Feynman Lectures, California Institute of Technology;             http://www.feynmanlectures.caltech.edu/II_19.html

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This seems like a good time to casually mention that

The Associated Press

ProPublica

and The Intercept

all have services for anyone interested in sharing information through, uh, non-traditional means.

Please feel free to share any others.

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pretty much

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A Century-Old Buenos Aires Theater Converted Into a 21,000-Square-Foot Bookstore

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What would a manned mission across the solar system look like? Take a trip to 2040 with us and find out!

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NASA’s Mars propaganda posters make us all want to go to Mars.

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NDT is The Man!

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also, really like your blog, one of my alts is swimming-in-shattered-starsDOTtumblrDOTcom/about I met Kent Rominger at CSU where he gave a talk. One of his professors was married to my Lit instructor. Have a great day....

It’s quite coincidental that you mention Kent Rominger. I’m the Social Media/PR Director for the film ‘I want to be an Astronaut’ of which I connected with the director of the film, David Ruck, who took on the film as his final thesis project for his MFA at American University. 

Essentially, I saw the movie trailer his team posted for their Kickstarter campaign, I was hooked on the concept, contacted his team, and after some minimal cross-promotional involvement, I was invited down to DC to speak on a panel of people who were in the film, then it was screened for a small audience.

I quickly formed a nice relationship with David which developed into me being his PR/Communications person, and 3 months later, we received a message from astronaut Rick Mastracchio - then living aboard the International Space Station - who expressed his interest in viewing the entire film after - like me - saw the trailer circulating.

In collaboration with Rick, we moved (slowly) through NASA HQ via phone conferences/emails and eventually, in March 2014, the film was uplinked to the ISS for the first ever private screening of a documentary film about human spaceflight - in space! The story was picked up by CNN, USA Today and others, which was pretty fantastic. I ended up forging the beginnings of a relationship with CEO Chris Carberry and Marketing Director M Wade Holler of the non-profit space advocacy organization Explore Mars, who I coordinated the first ever “on Earth” screening for an audience to tip off the 2014 Humans 2 Mars Summit, which takes place annually in April/May at George Washington University of DC.

The film was not only premiered there (in Lisner Auditorium), we hosted a post-screening panel discussion featuring David J. Ruck (director/filmmaker), former aerospace technician Greg Cecil (who worked specifically on the thermal tiles for Space Shuttle Atlantis, later to be laid off with the 7,000 others during the retiring of the STS program), and Dan Hendrickson (former director of space policy at the AIAA; now, director of business development for Astrobotic), but legendary science journalist Miles O’Brien of PBS NewsHour and FRONTLINE moderated the event. It was an extraordinary privilege to say the least, as there were some heavy hitters within the space industry present that evening, such as John Logsdon (space historian), Ken Ham (astronaut and one of the top tier execs for Bigelow Aerospace).

The following three days at the H2M Summit were incredibly fulfilling, enlightening, and presented a unique networking opportunity that we (David and I) took full advantage of, which is how we ended up intersecting with “Rommel”, who not only befriended us, but ended up becoming a great supporter of ours. Through his relationships with Aerojet Rocketdyne, Boeing, Lockheed Martin, and with him being the CEO of then ATK (now Orbital ATK after the merger of Orbital Sciences Corp. and ATK), further screenings were coordinated that summer through Rominger who brought us to the 30th Space Symposium in Colorado Springs, CO; EAA AirVenture in Oshkosh, WI; and the official ATK-sponsored IMAX theater at the Clark Planetarium in Salt Lake City, UT…other venues we were eventually featured at notwithstanding.

Kent is a solid guy and became such an incremental supporter of our mission, so to speak. He was a key figure in my journey to understand the behind the scenes - exciting and not so exciting - of the space industry, which has molded my perspective on the then and now of where we’ve been, where we’re going, and the where/what/when/how and why of it all. 

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Scientific Theory vs. Law

Chat with a friend about an established scientific theory, and she might reply, “Well, that’s just a theory.” But a conversation about an established scientific law rarely ends with “Well, that’s just a law.” Why is that? What is the difference between a theory and a law… and is one “better”? 

Scientific laws and theories have different jobs to do. A scientific law predicts the results of certain initial conditions. It might predict your unborn child’s possible hair colors, or how far a baseball travels when launched at a certain angle.

In contrast, a theory tries to provide the most logical explanation about why things happen as they do. A theory might invoke dominant and recessive genes to explain how brown-haired parents ended up with a red-headed child, or use gravity to shed light on the parabolic trajectory of a baseball.

In simplest terms, a law predicts what happens while a theory proposes why. A theory will never grow up into a law, though the development of one often triggers progress on the other. 

We weren’t handed a universal instruction manual. Instead, we continually propose, challenge, revise, or even replace our scientific ideas as a work in progress. Laws usually resist change since they wouldn’t have been adopted if they didn’t fit the data, though we occasionally revise laws in the face of new unexpected information. A theory’s acceptance, however, is often gladiatorial. Multiple theories may compete to supply the best explanation of a new scientific discovery. Upon further research, scientists tend to favor the theory that can explain most of the data, though there may still be gaps in our understanding. 

Even incorrect theories have their value. Discredited alchemy was the birthplace of modern chemistry, and medicine made great strides long before we understood the roles of bacteria and viruses. That said, better theories often lead to exciting new discoveries that were unimaginable under the old way of thinking. Nor should we assume all of our current scientific theories will stand the test of time. A single unexpected result is enough to challenge the status quo. However, vulnerability to some potentially better explanation doesn’t weaken a current scientific theory. Instead, it shields science from becoming unchallenged dogma. 

A good scientific law is a finely-tuned machine, accomplishing its task brilliantly but ignorant of why it works as well as it does. A good scientific theory is a bruised, but unbowed, fighter who risks defeat if unable to overpower or adapt to the next challenger. Though different, science needs both laws and theories to understand the whole picture. So next time someone comments that it’s just a theory, challenge them to go nine rounds with the champ and see if they can do any better.

From the TED-Ed Lesson What’s the difference between a scientific law and theory? - Matt Anticole

Animation by Zedem Media

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Humans to Mars by 2028? Check Out Lockheed Martin’s Red Planet ‘Base Camp’ Idea

Lockheed Martin has launched its campaign to establish a “Mars Base Camp” — a vision for sending humans to Mars by 2028.

In its Mars Base Camp concept video, the aerospace compant Lockheed Martin lays out a plan that would transport astronauts from Earth to a Mars-orbiting science laboratory, where they could perform real-time scientific exploration, analyze Martian rock and soil samples,and confirm the ideal place to land humans on the surface.

Lockheed Martin foresees launching this orbiting science station in 2028, setting the stage for a human landing mission in the 2030s. ~ Space.com

Image: A detailed look at Lockheed Martin’s envisioned “Mars Base Camp” outpost, which the company aims to launch in 2028.                   Credit: Lockheed Martin

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On-vehicle video of UP Aerospace’s SL-10 rocket launch, capsule deployment and re-entry.

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paramecium bursaria

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Diagrama do traje espacial da missão Apolo.

Apollo spacesuit diagram.

Credit: Paul Calle/NASA

#nasa #apolloprogram #moon #space #astronaut #astrogram #observatoriog1 #vintage

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How to Do Business with NASA

It’s Small Business Week! To celebrate, we’re breaking down the process and explaining how YOUR small business can work with us. Here are 10 steps:

Prior to working with us, identify which of your products or services best fit within our industry. It’s also important to know the Federal Supply Class or Service Codes (FSC/SVC) for your products or services. Prepare a capability brief in both printed and electronic versions with an emphasis on Government work.

In order to register your business with us, there are three systems you’ll need to use. The Data Universal Numbering System (DUNS), the System for Award Management (SAM) and the NASA Vendor Data Base (NVDB). After you’ve survived all those acronyms, your business is registered!

Here at NASA we have centers around the country that each procure different types of business. Where does your product or service fit in? The best thing to do is visit THIS site and find out more about each center. You can also take a look at our Acquisition Forecast to find out about expected contract opportunities.

You can find current procurement opportunities in your product or service area by checking the Federal Business Opportunities website. This site also helps you identify our requirements and even send you e-mail notifications of released requirements.

Contracting procedures can be tedious, it’s always a good idea to familiarize yourself with the Federal Acquisition Regulations (FAR), as well as our supplement to those regulations. Which can be found HERE.

Did you know that many of our purchases are orders on the Federal Supply Schedule contracts? They are, which means you can contact the U.S. General Services Administration (GSA) for information on how to obtain a contract.

There are some very beneficial resources available to you throughout this process. You can request training and counseling on marketing, financial and contracting issues at minimal or no cost from Procurement Technical Assistance Centers (PTACs).

You also have the option to consult with the SBA’s Procurement Center Representatives (PCRs) and the SBA Business Development Centers. The SBA provides each of our centers with a liaison.

There is also an option to get free and confidential mentoring by former CEOs through SCORE.

Direct contracting is not the only route for small businesses. Consider subcontracting opportunities, and get information through the SBA’s SUB-Net or Subcontracting Opportunities Directory. Solicitations or notices are posted by prime contractors. Our list of prime vendors is located on our Marshall Space Flight Center’s website.

Explore other small business programs, such as our Mentor-Protégé Program, the Small Business Innovation Research Program and the Historically Black Colleges and Universities and Minority-Serving Institutions Program. Information on these and other programs is available on our Office of Small Business Programs website.

After you have identified your customers, researched their requirements and familiarized yourself with our procurement regulations and strategies, it’s time to market your product or service. Present your capabilities directly to the NASA Centers that buy your products or services. Realize that, as with yours, their time is valuable. If the match is a good one, you can provide them with a cost-effective, quality solution to their requirements. Good luck!

Here are a Few Small Businesses We’re Already Working With…

Dynetics Technical Services, Inc., of Huntsville, AL works with us on enterprise information technology services so that we have the right tools to reach for new heights. This company was also named Agency Small Business Prime Contractor of the Year.

Arcata Associates, Inc., of Las Vegas, NV manages operations and maintenance for our Dryden Aeronautical Test Range in Edwards, CA. Their work ensures that we can continue our critical work in aviation research and development. This company was even named Agency Small Business Subcontractor of the Year.

Make sure to follow us on Tumblr for your regular dose of space: http://nasa.tumblr.com

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Just 40 light years from Earth, three planets might host life forms adapted to infrared worlds

Is there life beyond our solar system? If there is, our best bet for finding it may lie in three nearby, Earth-like exoplanets.                                

For the first time, an international team of astronomers from MIT, the University of Liège in Belgium, and elsewhere have detected three planets orbiting an ultracool dwarf star, just 40 light years from Earth. The sizes and temperatures of these worlds are comparable to those of Earth and Venus, and are the best targets found so far for the search for life outside the solar system. The results are published today in the journal Nature.

The scientists discovered the planets using TRAPPIST (TRAnsiting Planets and PlanetesImals Small Telescope), a 60-centimeter telescope operated by the University of Liège, based in Chile. TRAPPIST is designed to focus on 60 nearby dwarf stars—very small, cool stars that are so faint they are invisible to optical telescopes. Belgian scientists designed TRAPPIST to monitor dwarf stars at infrared wavelengths and search for planets around them.

The team focused the telescope on the ultracool dwarf star, 2MASS J23062928-0502285, now known as TRAPPIST-1, a Jupiter-sized star that is one-eighth the size of our sun and significantly cooler. Over several months starting in September 2015, the scientists observed the star’s infrared signal fade slightly at regular intervals, suggesting that several objects were passing in front of the star.

With further observations, the team confirmed the objects were indeed planets, with similar sizes to Earth and Venus. The two innermost planets orbit the star in 1.5 and 2.4 days, though they receive only four and two times the amount of radiation, respectively, as the Earth receives from the sun. The third planet may orbit the star in anywhere from four to 73 days, and may receive even less radiation than Earth. Given their size and proximity to their ultracool star, all three planets may have regions with temperatures well below 400 kelvins, within a range that is suitable for sustaining liquid water and life.

Because the system is just 40 light years from Earth, co-author Julien de Wit, a postdoc in the Department of Earth, Atmospheric, and Planetary Sciences, says scientists will soon be able to study the planets’ atmospheric compositions, as well as assess their habitability and whether life actually exists within this planetary system.

“These planets are so close, and their star so small, we can study their atmosphere and composition, and further down the road, which is within our generation, assess if they are actually inhabited,” de Wit says. “All of these things are achievable, and within reach now. This is a jackpot for the field.”

Read More

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Searching for a job requires the perfect résumé or CV, and there are countless online guides dedicated to CV-crafting that promise to get you to the second round of the hiring process. But if there’s one thing you’d never do, it’s list your failures on there, right?

Well, that’s exactly what one esteemed Princeton University professor has done this week, and based on the viral response, maybe it’s something we should all be a little more open to.

As Ana Swanson reports for The Washington Post, professor of psychology and public affairs Johannes Haushofer wrote his ‘CV of Failures’ a few years ago for his students, but last week decided to put the document online in the hopes of making a broader audience realise that we all experience failure on the way to success.

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