Direct Energy, 2018a

334 14.5 Invariants
Next use Eq. 14.118 to nd the invariant.
Υ=η ∂L
∂ẏ
=ẏ (14.144)
Qualitatively, ẏ represents the slope ofthe line, so this invariant tells us
that the slope ofthe solutions to the line equation must be constant.
Another continuous symmetry ofthis equation is described by the in-
nitesimal generator U = t∂ y with ξ =0and η = t. We can solve for the
prolongation ofthe generator acting on the Lagrangian.
( )
d
pr (n) UL = η t ẏ =ẏ
dt (t − 0) + 0 =ẏ
We can nd G using Eq.14.117, and we can nd the invariant using Eq.14.118.
dG
dt =ẏ + 1 2ẏ2 · 0=ẏ (14.145)
G = y (14.146)
Υ=y − tẏ (14.147)
Qualitatively, this invariant represents the y-intercept ofthe line, so this
invariant tells us that the y-intercept ofthe solution to the line equation
must be constant.
14.5.5 Pendulum Equation Invariants Example
Consider the equation describing a pendulum, studied in Problem 11.8.
The energy conversion process is described by the Lagrangian
which corresponds to the equation ofmotion
L = 1 2 mẏ2 − mg cos y (14.148)
ÿ = g sin y. (14.149)
In these equations m represents the mass, and g represents the gravitational
constants. Both m and g are assumed constant here. This equation of
motion has only one continuous symmetry described by the innitesimal
generator U = ∂ t with ξ =1and η =0. We can use Noether's theorem to
nd the corresponding invariant.