2. Time reversal symmetry:
˜qi(t) = qi(−t)
Again: qisatisfying Newton’s 2nd law implies ˜qialso satisfies Newton’s 2nd law.
3. Spatial rotation symmetry:
˜qi(t) = Rqi(t)
where R:R3→R3is a rotation. Again: qisatisfying Newton’s 2nd law implies ˜qialso does.
Then JB breaks the table because he’s not strong enough to move the Earth.
4. Spatial translation symmetry:
˜qi(t) = qi(t) + k, k ∈R3
Again, same thing. 3) and 4) are isometries of R3, that is, functions T:R3→R3such that
||T x −T y || =||x−y||,∀x, y ∈R3.
Theorem 1 Every isometry T:R3→]R3is the composite of:
1. a rotation
2. a translation
and possibly
3. parity (total spacial inversion):
x7→ −x, (x∈R3)
Let’s show that if T:R3→R3is an isometry and
˜qi(t) = T qi(t)
then qisatisfies Newton’s 2nd law implies ˜qidoes.
m¨
˜qi(t) = md2
dt2T qi(t)
=md
dt ddtT qi(t)
Now use the theorem:
T x =Sx +k x ∈R3
where k∈R3and Sis an orthogonal linear transformation (i.e. 3 ×3 matrix with S S†= 1, or a
linear transformation with ||Sx|| =||x||,∀x∈R3).
Then d
dt T qi(t) = S˙qi(t)
and d2
dt2T qi(t) = S¨qi(t)
2
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