Direct Energy, 2018a

14 LIE ANALYSIS 317
Figure 14.1: The solid line shows a solution to the wave equation. The
dotted and dashed lines show solutions found using the symmetry transformation
t → t + ε and y → y which has innitesimal generator U = ∂ t .
the equation without having to go through the work ofsolving the equation
again. The wave equation, Eq. 14.19, has solutions ofthe form
y(t) =c 0 cos (ω 0 t)+c 1 sin (ω 0 t) (14.24)
where boundary conditions determine the constants c 0 and c 1 . The symmetry
described by the innitesimal generator U = ∂ t tells us that
y(t) =c 0 cos (ω 0 (t + ε)) + c 1 sin (ω 0 (t + ε)) (14.25)
must also be a solution. Using Eq. 14.24, we have found a family of
related solutions because Eq. 14.25 is a solution for all nite or innitesimal
constants ε. Figure 14.1 illustrates this idea. The known solution is shown
as a solid line. The dotted and dashed lines illustrate related solutions,
for dierent constant ε values. We encountered the wave equation in the
mass spring example ofSection 11.5 and the capacitor inductor example
ofSection 11.6, so symmetry analysis provides information about both of
these energy conversion processes. It tells us that ifwe run the energy
conversion process and nd one physical path y(t), then for appropriate
boundary conditions, y(t + ε) is also a physical path. This symmetry is
present in all time invariant systems. Qualitatively for the mass spring
example, it tells us that ifwe know the path taken by the mass when we