Direct Energy, 2018a

318 14.3 Continuous Symmetries and Innitesimal Generators
Figure 14.2: The solid line shows a solution to the wave equation. The
dotted and dashed lines show solutions found using the symmetry transformation
t → t and y → y(1 + ε) which has innitesimal generator U = y∂ y .
remove the restraint today, then we know the path taken by the mass when
we repeat the experiment tomorrow , and we know this idea from symmetry
analysis without having to re-analyze the system.
All linear equations, including the wave equation, contain a continuous
symmetry transformation described by the innitesimal generator
U = y∂ y (14.26)
which corresponds to ξ =0and η = y. Again, we can nd the corresponding
nite transformation using Eq. 14.14.
t → ( e εy∂ y ) t =
(
1+εy∂ y + 1 )
2! (εy∂ y) 2 + ... t = t (14.27)
and
y → ( e εy∂ y ) y =
(
1+εy∂ y + 1 )
2! (εy∂ y) 2 + ... y = y(1 + ε). (14.28)
To summarize this transformation,
t → t (14.29)