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Emmy Noether was a brilliant mathematician, who was described by Einstein as "the most significant creative mathematical genius thus far produced since the higher education of women began". In fact, she may have been one of the most important mathematicians of all time when it comes to changing physics forever. She discovered a theorem that links together seemingly unrelated concepts that are very fundamental to our understanding of physics.

Noether's theorem, (technically Noether's first theorem) states that there is an inherent link between certain kinds of symmetry within the universe, and conservation laws (such as conservation of momentum, energy, and angular momentum). If one exists, then so must the other.

In this video, we start by understanding what we mean by a symmetry. Specifically, this refers to unchanging behaviours of any system that we study even when a specific variable is changed. For example, if we move a ball to a different position in space, its behaviour does not suddenly change. This is "translational symmetry". We also look at "temporal symmetry" (symmetry over time) and "rotational symmetry" (symmetry over angular displacement). Basically, a system's behaviours do not inherently change, and these symmetries exist, because the laws of physics stay the same regardless of position, time, or angle!

Noether's theorem states that if such a symmetry exists, then there HAS to be a conservation law that corresponds to it. Conservation of momentum comes about because of translational symmetry. Conservation of energy comes about because of temporal symmetry. And conservation of angular momentum comes about because of rotational symmetry. So this possibly gives us a reason as to WHY these conservation laws exist in the first place. But how do we know that symmetries must have an associated conservation law?

To understand this, we take a look at the Euler-Lagrange equation. This allows us to use some basic Lagrangian mechanics to understand how a system changes over time. One special case of the Euler-Lagrange equation can be shown to be equivalent to Newton's Second Law of Motion - the force on the system being equal to its rate of change of momentum. So if the force exerted on the overall system is zero, then its momentum is constant (or conserved)! This also applies to other types of symmetry and conservation law through the use of the Euler-Lagrange equations.

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Timestamps:
0:00 - Emmy Noether
0:40 - Noether's Theorem: Symmetries
3:33 - Check out Wren to Calculate Your Carbon Footprint!
5:10 - Symmetries and Conservation Laws
6:28 - Lagrangian Mechanics
7:28 - The Euler-Lagrange Equation

(My video on Lagrangian Mechanics and the Euler-Lagrange Equation:    • Why Lagrangian Mechanics is BETTER th...  )

#physics #scientist #mathematics

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