Direct Energy, 2018a

14 LIE ANALYSIS 315
14.3 Continuous Symmetries and Innitesimal Generators
14.3.1 Denition of Innitesimal Generator
Symmetry transformations can be described as transformations of the independent
and dependent variables. Continuous symmetry transformations
can be described as transformations of these variables which depend on a,
possibly innitesimal, parameter ε.
t → ˜t = F (ε)t (14.7)
y → ỹ = F (ε)y (14.8)
The operator F (ε) describes the transformation. It is a function of the
innitesimal parameter ε, and it may also depend on t and y. Furthermore,
it is an operator meaning that it may involve derivative operations.
We are considering only continuous symmetries, so we can study the
behavior in the limit as ε → 0. The operator F (ε) can be written as a
Taylor series in the small parameter ε.
F (ε) =1+εU + 1 2! ε2 U 2 + ... (14.9)
The term U in the expansion above is called the innitesimal generator. It
may be separated into two components.
U = ξ∂ t + η∂ y (14.10)
The function ξ describes innitesimal variation in the independent variable.
The function η describes innitesimal variation in the dependent variable,
and it was introduced in Sec. 11.4. Both ξ and η may depend on both the
independent variable and the dependent variable.
ξ = ξ(t, y) (14.11)
η = η(t, y) (14.12)
In the limit of ε → 0, we can ignore terms of order ε 2 or higher.
F (ε) ≈ 1+εU (14.13)
An innitesimal generator describes a continuous symmetry transformation.
If we know an innitesimal generator for some continuous symmetry,
we can nd the corresponding transformation
t → e εU t and y → e εU y. (14.14)