Direct Energy, 2018a

338 14.7 Problems
14.6. The equation ÿ + y −3 =0has the three innitesimal generators listed
in the problem above. These innitesimal generators form a group.
The commutator was dened in Section 14.3.3, and the commutator
of any pair of these innitesimal generators can be calculated by
[U a ,U b ]=U a U b − U b U a .
Using the equation above, show that the commutator for each of the
three pairs of innitesimal generators results in another element of
the group.
14.7. Derive the innitesimal generators for the wave equation, ÿ+ω 2 0y =0.
(This problem is discussed in [191].)
Answer:
U 1 = ∂ t
U 2 = y∂ y
U 3 =sin(ω 0 t) ∂ y
U 4 =cos(ω 0 t) ∂ y
U 5 = sin(2ω 0 t)∂ t + ω 0 y cos(2ω 0 t)∂ y
U 6 =cos(2ω 0 t)∂ t − ω 0 y sin(2ω 0 t)∂ y
U 7 = y cos (ω 0 t) ∂ t − ω 0 y 2 sin (ω 0 t) ∂ y
U 8 = y sin (ω 0 t) ∂ t + ω 0 y 2 cos (ω 0 t) ∂ y
14.8. The wave equation ÿ + ω 2 0y =0has the eight innitesimal generators
listed in the problem above. The corresponding Lagrangian is
L = 1 2ẏ2 − 1 2 ω2 0y 2 .
Find the invariants corresponding to the following innitesimal generators.
(a) U 1 = ∂ t
(b) U 3 =sin(ω 0 t) ∂ y
(c) U 5 = sin(2ω 0 t)∂ t + ω 0 y cos(2ω 0 t)∂ y