Direct Energy, 2018a

14 LIE ANALYSIS 339
14.9. In Problem 11.8, we encountered the equation given by ÿ = g sin y
for constant g.
(a) Show that U = ∂ t is an innitesimal generator of this equation.
(b) Show that U = y∂ y is not an innitesimal generator of this
equation.
14.10. The Lagrangian
L = 1 2ẏ2 + 1 2 y−2
corresponds to the equation of motion ÿ + y −3 =0. This equation
of motion has three innitesimal generators:
U 1 = ∂ t
U 2 =2t∂ t + y∂ y
U 3 = t 2 ∂ t + ty∂ y
Use Noether's theorem to nd the invariants that correspond to each
of these innitesimal generators. (We encountered this Lagrangian in
problem 11.3.)