Direct Energy, 2018a

322 14.4 Derivation of the Innitesimal Generators
14.4 Derivation of the Innitesimal Generators
14.4.1 Procedure to Find Innitesimal Generators
We are studying dierential equations, which can be written as
F (t, y, ẏ, ...) =0 (14.40)
for some function F . We are looking for continuous symmetries that can
be applied to this equation such that the original equation and the transformed
equation have the same solutions. The symmetries are denoted by
innitesimal generators
U = ξ∂ t + η∂ y (14.41)
that describe how the independent variable t and dependent variable y
transform. Upon a symmetry transformation, the independent variable
and dependent variable transform, but so do the derivatives of the dependent
variable, ẏ, ÿ, ... The prolongation of an innitesimal generator is a
generalization of the innitesimal generator that describes the transformation
of the independent variable, the dependent variable, and derivatives
of the dependent variable [164, p. 94].
The nth prolongation of a generator U is dened as
pr (n) U = ξ∂ t + η∂ y + η t ∂ẏ + η tt ∂ÿ + η ttt ∂ ... y + ..., (14.42)
and it has terms involving η tn . The functions η t and η tt are dened [164],
η t = η t (t, y, ẏ) = d (η − ξẏ)+ξÿ (14.43)
dt
η tt = η tt (t, y, ẏ) = d2
(η − ξẏ)+ξ... y (14.44)
dt2 The quantities η ttt , η tttt , and so on can be dened similarly, but they will
not be needed for the examples below. The prolongation of the innitesimal
generator is an operator that describes the transformation of t, y, ẏ, ÿ, and
so on up to the nth derivative. Some authors [189] use the term tangential
mapping instead of prolongation.
The procedure to nd all possible continuous symmetries of an equation
is based on the idea that the solutions of an equation remain unchanged
upon a symmetry operation. For a given transformation to be a symmetry
operation, not only must all the solutions remain unchanged, but so must
all derivatives of the solutions. Thus, for a dierential equation of the form
F (t, y, ẏ, ...) =0, all symmetries U obey the symmetry condition