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Chemistry – Noether’s Theorem

In the history of science, many women contributed to the field. Whether it is modern day where women are treated as equals when being involved in anything. Or in the past, where women were treated unfairly and not allowed to do much, like go to school. The was the case for the brilliant Emmy Noether, with her many accomplishments. Her greatest accomplishment was Noether’s theorem, which proved that if the system has the right continuous symmetry the conservation laws correspond to that system. A big change when looking at concepts like the law of conservation.

Emmy Noether was born March 23, 1882 in Erlangen, Germany. She studied mathematics at the university of Erlangen, but unfortunately women at the time could only attend informally with the permission of the instructor. So she would “attend” to classes at the University of Göttingen in 1903-1904. Good news for Noether women could be full students in Erlangen, so she went back in 1904, received a Ph. D in 1907 from her thesis on algebraic invariants. In 1915, she was invited by Hilbert and Klein to help them understand the math behind none other than Albert Einstein and the recent published theory of general relativity. Even with the disproval from the university of a woman teaching at a university level, Hilbert and Klein were able to persuade Noether to stay.  July 23, 1918 was the day where Noether’s theorem was made, and changed modern physics for the better. (add more about her life if needed to)

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The law of conservation of angular momentum states that when external torque on the system is zero, its angular momentum remains constant. By having rotational symmetry around the axis, the momentum in the direction the object is travelling in is conserved. Rotational symmetry means that viewing the system at a different point will not change the system. For example, think of a satellite orbiting around the planet. Let say one starts orbiting form the continent Asia, and the other starts at the continent North America. If they are both the same distance away from the earth. They both have the same gravitational potential energy, and the energy stays the same. Since they also travel at the same speed, the angular momentum is conserved. This is very similar to the law of conservation of momentum.

Noether’s theorem also relates greatly to the law of the conservation of momentum. For something to have a translational symmetry, it would still have the same momentum whether it was measured at a different spot. For example, if measuring the momentum of a bowling ball rolling on a smooth flat surface. If no external forces are acting on the ball is assumed, then the bowling ball will keep moving (until it goes to knock the pins down) and the mass and velocity will be conserved. The change of speed would be due to space itself, but Noether’s theorem ties to the conservation of momentum. So, even if the bowling ball was bowled on a different ally (in the same are with the same smooth surface), it will still move the same. The physics is not changed from linear movements within a uniform space. The same could be said about the conservation of energy

The conservation of energy states that energy cannot be created or destroyed, only transferred from one object to another.  In this case (enter source) Noether showed that the principle of energy conservation follows from symmetry under time translations. What that means is the energy of the system will be the same whether it is done at a different time. There is a symmetry of time, which is called time translation. For an example, a road with hills, and two cars hit each other. The momentum is not conserved, it could be lost or gained. The direction of the gravitational field changes with respect to the road. So, with symmetric translation, the momentum will be different if the collision happens on different points on the road. Although, the gravitational field will not change at a different point of time. So, whether the same crash happens at 8:00 a.m., midnight, or even 5 years from now. That gravitational field stays the same. This is time translation symmetry where energy is conserved. This is the theorem that helps prove the first law of thermodynamics.

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