Direct Energy, 2018a

314 14.2 Types of Symmetries
be written as a sum of innitesimal transformations in this way. Three
commonly discussed discrete symmetry transformations [187] 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 ∗ , where ∗ denotes complex conjugate.
For example, the wave equation is invariant upon each of these three discrete
symmetries because solutions of the equation remain the same before
and after these symmetry transformations are performed. The transformation
t → ˜t = t + ε, where ε is the continuous parameter which can be
innitesimally small, is an example of a continuous transformation because
it can be separated into a sum of innitesimal symmetries. Both discrete
and continuous symmetries may involve transformations of the independent
variable, the dependent variable, or both variables. In this chapter, we will
study a systematic procedure for identifying continuous symmetries of an
equation, and we will not consider discrete symmetries further.
14.2.2 Regular versus Dynamical
Continuous symmetries can be classied as regular or dynamical. Regular
continuous symmetries involve transformations of the independent variables
and dependent variables. Dynamical symmetries involve transformations
of the independent variables, dependent variables, and the derivatives of
the dependent variables [188]. (Some authors use the term generalized
symmetries instead of dynamical symmetries [164, p. 289].) Only regular
symmetries will be considered. The techniques discussed here generalize to
dynamical symmetries [164], but they are beyond the scope of this text.
14.2.3 Geometrical versus Nongeometrical
Symmetries may also be classied as geometrical or nongeometrical [184]
[185]. Nongeometrical symmetry transformations involve taking a Fourier
transform, performing some transformation of the variables, then taking an
inverse Fourier transform. The resulting transformations are symmetries
if the solution of the equation under consideration are the same before
and after the transformations occur. Nongeometrical symmetries can be
written as functions of an innitesimal parameter but are not continuous.
Nongeometrical symmetries will not be discussed here and are also beyond
the scope of this text.