PHYS 6124 videos :
week 10
Discrete symmetries : an intro to finite groups
What is a symmetry of laws of motion? The families of symmetry-related full state space cycles are replaced by fewer and often much shorter "relative" cycles, and the notion of a prime periodic orbit is replaced by the notion of a "relative" periodic orbit, the shortest segment that tiles the cycle under the action of the group.
October 20
clip 1 - discrete symmetry, an example
3-disk pinball
(version of Jan 4, 2015)
October 20
clip 2 - what is a group?
what is a group?
(version of Jan 4, 2015)
October 20
clip 3 - coordinate transformations
active, passive
(version of Jan 4, 2015)
October 20
clip 4 - following Mefisto
symmetry defined three (3) times
(version of Jan 4, 2015)
October 20
Clip 5 - subgroups
subgroups, classes, group orbits, reduced state space
(version of Jan 4, 2015)
October 22
Clip 6 - characters
this requires character
(version of Apr 8, 2015)
October 22
Clip 7 - characters
hard work builds character
(version of Apr 8, 2015)
October 22
Clip 8 - symmetry group C3
the symmetry group of a propeller
(version of Apr 8, 2015)
October 22
Clip 9 - irreps of C3
irreps of C3
(version of Apr 8, 2015)
October 22
Clip 10 - the symmetry group D3
rotations and reflections in the plane
(version of Apr 8, 2015)
October 22
Clip 11 - irreps
irreducible representations
(version of Apr 8, 2015)
October 22
Clip 12 - characters
Frobenius character formula
(version of Apr 8, 2015)
October 22
Clip 13 - characters
character orthogonality relations
(version of Jan 4, 2015)
October 22
Clip 14 - the summary :
it is all about class and character
(version of Jan 4, 2015)
Discrete symmetry :
reduction of dynamics to a fundamental domain
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"It does not say anyplace in the Bible that if equations of motion
have a symmetry, solutions should have it too."
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While everyone can visualize the fundamental domain for a 3-disk billiard, the simpler problem - symmetry reduction of 1d dynamics that is equivariant under a reflection, the most common symmetry in applications - seems to baffle everyone. So here is a step-by-step walk through to this simplest of all symmetry reductions.
October 20
1-dimensional example
a 1-dimensional system with a symmetry
(version of Jan 4, 2015)
October 22
fundamental domain
fundamental domain
(version of Jan 4, 2015)
October 22
Tiling
Tiling of state space by a finite group
(version of Jan 4, 2015)
October 20
Fundamental tile
Make the "fundamental tile" your hood
(version of Jan 4, 2015)
October 20
Symmetry-reduction
Symmetry-reduced dynamics
(version of Jan 4, 2015)
October 20
Regular representation
Regular representation of permuting tiles
(version of Apr 8, 2015)
October 20
Clip 14d - discussion
class and character
(version of Oct 22, 2020)
October 20
Group theory
voodoo
(version of Jan 4, 2015)
October 20
Tell no Lie
to plumbers
(version of Jan 4, 2015)