Direct Energy, 2018a

14 LIE ANALYSIS 337
14.7 Problems
14.1. Three commonly discussed discrete symmetry transformations are:
Time reversal t → ˜t =(−1) n t for integer n
Parity y → ỹ =(−1) n y for integer n
Charge conjugation y → ỹ = y ∗
Verify that the wave equation, ÿ + ω 2 0y =0, is invariant upon each of
these discrete transformations.
14.2. Repeat the problem above for the equation ÿ + y −3 =0.
14.3. The Thomas Fermi equation is given by ÿ = y 3/2 t −1/2 .
(a) Verify that it is not invariant upon the discrete symmetry transformation
of time reversal,
t → ˜t =(−1) n t for integer n.
(b) Verify that it is not invariant upon the discrete symmetry transformation
of parity,
y → ỹ =(−1) n y for integer n.
(c) Verify that it is invariant upon the discrete symmetry transformation
t → ˜t =(−1) n t and y → ỹ =(−1) n y.
14.4. Find the prolongation of the innitesimal generator
acting on the Lagrangian
U = ξ∂ t + η∂ y
L = 1 2ẏ2 + 1 3 ty2 .
Write your answer in terms of ξ and η but not η t or η tt .
14.5. Find the innitesimal generators for the equation, ÿ + y −3 =0. (This
problem is discussed in [190].)
Answer:
U 1 = ∂ t
U 2 =2t∂ t + y∂ y
U 3 = t 2 ∂ t + ty∂ y