Discovering Group Theory, Day 17

Realizing that space-time symmetries provide more intuitive introduction to the abstract notion of a symmetry used in physics,  on day 17 we switched gears by discussing Richard Feynman’s lecture “Symmetry in Physical Law,” which students watched outside the class.

Feynman introduces the notion of a symmetry of a physical law, which we disected in the context of Coulomb’s law.

  • We defined a coordinate system and discussed the difference between passive and active transformations.
  • We worked out the translational and rotational spatial symmetries in 2D and showed that translations form a group.
  • We also introduced translations in time and showed how they leave the kinematic equations unchanged.

As a transition to the discussion of moving observers, we watched the first half of a wonderful video from the 60’s titled Frames of Reference. 

 

 

Discovering Group Theory, Day 15

On Day 15, we continued working with complex numbers, but now in exponential or polar form. After an introduction to complex numbers exponential form and its relation to the rectangular form, students worked on exercises covering:

  • Complex numbers in the polar form and their relation to the rectangular form
  • Complex conjugation and modulus
  • Complex numbers as points in a complex plane

About half of the class managed to get to the proof that the complex numbers form a group under multiplication, which was easier to do in the polar form.

We ended the class by watching a TED talk by Murray Gell-Mann,  “Beauty, truth, and …, physics?”.