SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Spontaneous Symmetry Breaking - Classical Physics
A small bead of mass m threaded on a rotating circle of radius r;
constant frequency ω. We use the Lagrangian formalism:
1 L = 1 2 m˙⃗r 2 −V (⃗r ) = 1 2 mr 2 ˙ϑ 2 + 1 2 mω2 r 2 sin 2 ϑ−mgr(1−cos ϑ)
df .
2 L = T ϑ − U ϑ ↔ U ϑ = mgr[ (1 − cos ϑ) − 1
} {{ }
2 (mω2 /g) sin 2 ϑ ]
} {{ }
Potential Centrifugal
Bead on a Circle
z
ω
π _ υ
y
r
m
x
f = −mg
U=mgr(1−cos υ)
The lowest energy solutions ↔ ˙ϑ = 0 ↔ dU
dϑ = 0
Result: sin ϑ s = 0 and cos ϑ o = 1/β; β df . = rω 2 /g
Conclusion. Two types of lowest-energy solutions:
1. ϑ s = 0 - with the axial symmetry of the problem
2. ϑ o ≠ 0 - breaking the original axial symmetry !!
Jerzy DUDEK
SYMMETRIES in PHYSICS