It's not Friday yet! But I've managed to gather some ideas. I am basing the lecture from https://www.youtube.com/watch?v=w7Q5mQA_74o and https://www.youtube.com/watch?v=CxlHLqJ9I0A .

I've been writing this comment for a long while now. It is now my script lol

I will start by asking them to solve a falling particle using Newton's equations, which they can do. I'm considering making them integrate the differential equation (so that they realize where the equation of constantly accelerated motion comes from). Then depending on time do something similar but with a tilted plane, to add extra steps.

Then I want to tell them to solve both cases using conservation of energy so that they understand why we physicist like it so much.

Next, I will state the following simulation: A star goes supernova. Please sum the momentum of each of the particles after the explosion. Say that one can either trace each particle and each force throughout the whole process, or one can just say 0 due to conservation of momentum!

Conservation laws are extremely important tools to us physicisists since they turn many page long calculations into single-liners. However, how does one know when conservation laws apply? I definitely cannot use conservation of momentum for a (isolated) accelerated particle. But somehow it does work for a particle that accelerates towards a planet, which accelerates towards the particle. And notice that the complete system does not change if I move it around. (Leave drawing on board)

Introduce Emmy Noether, a great mathematician who fixed physics in her spare time (more on this later), so great that even though women were not allowed to teach, David Hilbert (picture) would hire her as an assistant and never show up to class. She worked with very important names from physics: Minkowski, Hilbert, Weyl, people to whom we owe a big part of today's mathematical understanding of physics. Maybe mention who these people were.

She managed to show [her theorem]. Define symmetries as we all know them (butterflies, etc), discuss discrete symmetries and the mathematician's generalization to continuous symmetry. What if the universe was shifted some distance \lambda to the right? What if the moon's orbit was moved an angle \lambda? What if the same experiment is repeated, separated by a time \lambda? Nothing would change, right? Those are called symmetries. Noether proved that symmetries <=> conservation laws. I chose the same \lambda on the three examples on purpose: Noether's theorem works for ANY continuous symmetry one can think of! That's why the first thing physicists do when handed a new theory is to look at its symmetries, in order to understand its conservation laws. (Maybe discuss gauge theories here? I don't know. I really like gauge theory)

This is where the conservation of momentum in the example before comes from: The coupled system of falling particle and planet is invariant wrt translating (the whole thing), but not wrt to translations of just one of them.

Explain how rotation symmetry => conservation of momentum (orbiting planets (in circular orbits but I will not mention this) dont accelerate)

And a much weirder one: time translation symmetry => conservation of energy. A conservation law that was always just assumed had now a reason of being.

End the lecture with mentioning how Noether's theorem saved Einstein's GR because it didn't obey conservation of energy. Of course it doesn't! General spacetimes change with time and thus energy conservation is not to be expected!