Classical Mechanics

2.1. Symmetry and groups
Mathematically, it is possible to construct a space with any desired symmetry
using standard techniques. We begin with a simple case, reserving more involved
examples for later chapters. To begin, we rst dene a mathematical object
capable of representing symmetry. We may think of a symmetry as a collection
of transformations that leave some essential properties of a system unchanged.
Such a collection, G, of transformations must have certain properties:
1. We may always dene an identity transformation, e, which leaves the system
unchanged: ∃ e ∈ G.
2. For every transformation, g, taking the system from description A to another
equivalent description A , there must be another transformation, de-
′
1
′
noted g , that reverses this, taking A to A. The combined effect of the two
1 1
transformations is therefore g g = e. We may write: ∀g ∈ G, ∃g ∈ G
1
g g = e.
3. Any two transformations must give a result which is also achievable by a
transformation. That is, ∀g , g ∈ G, ∃g ∈ G g g = g .
1 2 3 1 2 3
4. Applying three transformations in a given order has the same effect if we
replace either consecutive pair by their combined result. Thus, we have
associativity: ∀g , g , g ∈ G, g ( g g ) = ( g g ) g .
1 2 3 1 2 3 1 2 3
These are the dening properties of a mathematical group. Precisely, a group
is a set, S, of objects together with a binary operation satisfying properties 14.
We provide some simple examples.
The binary, or Boolean, group, B, consists of the pair B = {{ 1, 1} , }
where is ordinary multiplication.
The multiplication table is therefore
1 1
1 1 1
1 1 1
Naturally, 1 is the identity, while each element is its own inverse. Closure is
evident by looking at the table, while associativity is checked by tabulating all
triple products:
1 (1 ( 1)) = 1 = (1 1) ( 1)
1 ( 1 ( 1)) = 1 = (1 ( 1)) ( 1)
etc.
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