Direct Energy, 2018a

14 LIE ANALYSIS 313
independent variables such as the Lagrangian L = L(t, y, dy ). For such
dt
quantities, we will have to distinguish between total and partial derivatives
carefully.
The analysis here is in no way mathematically rigorous. Furthermore,
the examples in this chapter are not original. References to the literature
are included below.
These techniques generalize to more complicated equations. They apply
to equations with multiple independent and multiple dependent variables,
and they apply when these variables are complex [164]. Also, these techniques
apply to partial dierential equations as well as ordinary dierential
equations, and they even apply to systems of equations [164]. See references
[164] for howto generalize the methods introduced in this chapter
to the other situations.
14.2 Types of Symmetries
14.2.1 Discrete versus Continuous
This chapter is concerned with identifying symmetries of equations. We
say that an equation contains a symmetry if the solution to the equation
is the same both before and after a symmetry transformation is applied.
The wave equation is given by
d 2 y
dt + 2 ω2 0y =0 (14.2)
where ω 0 is a constant. When t represents time, ω 0 has units of frequency.
The wave equation is invariant upon the discrete symmetry
y → ỹ = −y. (14.3)
This transformation is a symmetry because when all y's in the equation
are transformed, the resulting equation contains the same solutions as the
original equation.
d 2 ỹ
dt + 2 ω2 0ỹ =0 (14.4)
d 2 (−y)
+ ω
dt
0(−y) 2 =0 (14.5)
2
d 2 y
dt + 2 ω2 0y =0 (14.6)
Symmetries can be classied as either continuous or discrete. Continuous
symmetries can be expressed as a sum of innitesimally small symmetries
related by a continuous parameter. A discrete symmetry cannot