By: Gunnar Mein

Newton's world

If you got exposed to any physics in school, you have probably heard - and then forgotten - about Newton's laws:

A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force.

When a body is acted upon by a net force, the body's acceleration multiplied by its mass is equal to the net force.

If two bodies exert forces on each other, these forces have the same magnitude but opposite directions.

While these are succinct, they are very often not the starting point for a physics discussion. Physicists like to attack problems from "conservation laws". You can derive three conservation laws from Newton's laws:

The total energy (kinetic and potential) of a system is conserved (i.e. it doesn't change over time)

The total momentum of a system is also conserved.

The total angular momentum (the "rotational" cousin of momentum) is also conserved.

It's always nice to have such simple differential equations be true, but where do these laws come from? What is it about our universe that makes these so? The equations give us little answer. They are simple, but not satisfying. This is where Emmy Noether's work comes in.

Introducing Emmy Noether

Born in 1882, Amalie Emmy Noether was a German mathematician who made great contributions to the eld of abstract algebra. In her time, she was denied full professorship because she was a woman, and later expelled from teaching altogether because she was a Jew in rising Nazi Germany. American physicists Leon M. Lederman and Christopher T. Hill argue in their book "Symmetry and the Beautiful Universe" that Noether's theorem is "certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem". Expelled from Germany, she found refuge at Bryn Mawr College and taught at the Institute for Advanced Study in Princeton.

Noether's First Theorem

To really understand her work, we would need to use calculus and learn about a way of describing physics called the "Lagrangian formulation of mechanics", which would take us a little too far. But all that aside, "Noether's ﬁrst theorem" states that: "If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time."

In physics, a symmetry means that we can make a change to a system (a "transformation" of e.g. coordinates) without changing the physics of it. An example of such a symmetry is under "translation in time" - if we do our experiment tomorrow instead of today, we expect the outcomes to be the same. Physics does not change over time. Similarly, symmetry under "translation in space" would mean that we expect the experiment to work the same whether we do it here or next door, or half-way around the world, or in another galaxy. Physics does not change with location. The last one of the symmetries that I want to point out is the one under rotation: I can rotate my experiment by any number of degrees, and I still expect the physics of it to be the same.

Translational and rotational symmetery

These "symmetries" seem trivial, but appreciate for a moment how crucial they are. If we could not assume that physics is the same here as in the rest of our galaxy, then what point would there be in speculating about how the galaxy works? Trying to describe it would be hopeless. Similarly, if physics could be different tomorrow, then why bother even writing it down? And so we hold on to these symmetries. We have no reason to doubt them, and we won't.

The great things about Noether's theorem is that it both encourages us by saying that there will be a conservation law coming from every such symmetry, and it also shows us the way - the law will follow from the correct formulation of the symmetry in just a few lines of (advanced) math. And thus:

Symmetry under space translation gives us conservation of momentum

Symmetry under rotation gives us conservation of angular momentum

Symmetry under time translation gives us conservation of total energy

To a physicist, this is very satisfying. The conservation laws follow from some really basic, really important principles of our universe.

There is one more little thing that comes from this (a "corollary" as the mathematicians say). Every now and then, about every 5-10 years, someone will claim to have invented a "momentum-less drive", a drive for a spaceship that violates conservation of momentum. On the surface, that's a very desirable thing because it would mean that spacecraft could move without carrying "propellant", a fancy word for "stuff we throw out the back in order to accelerate forward." But if we look a little bit at some elementary logic, we remember that "A implies B" (e.g. symmetry implies conservation) also means that "not B implies not A". Which in our case would mean "no conservation implies no symmetry". For the speciﬁc example of violating conservation of momentum, we need to conclude that if momentum is not conserved, then physics is not the same from place to place. And as we discussed before, that is certainly not something we are willing to concede. And that's why when someone once again comes up with such a drive, I - and all fans of Emmy Noether - will continue to bet against it.

Death and legacy

Emmy Noether died in 1935 at age 52, of complications after a seemingly successful surgery. One contemporary physicist wrote this on the occasion of her death:

The Late Emmy Noether

To the Editor of the New York Times: The efforts of most human beings are consumed in the struggle for their daily bread but most of those who are, either through fortune or some special gift, relieved of this struggle are largely absorbed in further improving their worldly lot. Beneath the effort directed toward the accumulation of worldly goods lies all too frequently the illusion that this is the most substantial and desirable end to be achieved; but there is, fortunately, a minority composed of those who recognize early in their lives that the most beautiful and satisfying experiences open to humankind are not derived from the outside, but are bound up with the development of the individual’s own feeling, thinking and acting. The genuine artists, investigators and thinkers have always been persons of this kind. However inconspicuously the life of these individuals runs its course, none the less the fruits of their endeavors are the most valuable contributions which one generation can make to its successors.

Within the past few days a distinguished mathematician, Professor Emmy Noether, formerly connected with the University of Göttingen and for the past two years at Bryn Mawr College, died in her ﬁfty-third year. In the judgment of the most competent living mathematicians, Fräulein Noether was the most signiﬁcant creative mathematical genius thus far produced since higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and uniﬁed form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulae are discovered necessary for the deeper penetration into the laws of nature.

Born in a Jewish family distinguished for the love of learning, Emmy Noether, who, in spite of the efforts of the great Göttingen mathematician, Hilbert, never reached the academic standing due her in her own country, none the less surrounded herself with a group of students and investigators at Göttingen, who have already become distinguished as teachers and investigators. Her unselﬁsh, signiﬁcant work over a period of many years was rewarded by he new rulers of Germany with a dismissal, which cost her the means of maintaining her simple life and the opportunity to carry on her mathematical studies. Farsighted friends of science in this country were fortunately able to make such arrangements at Bryn Mawr College and at Princeton that she found in America up to the day of her death not only colleagues who esteemed her friendship but grateful pupils who enthusiasm made her last years the happiest and perhaps the most fruitful of her entire career.

Albert Einstein

Princeton University, May 1, 1935

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