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14.2: Types of Symmetries

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  • 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

    \[\frac{d^2y}{dt^2} + \omega_0^2y = 0\]

    where \(\omega_0\) is a constant. When \(t\) represents time, \(\omega_0\) has units of frequency. The wave equation is invariant upon the discrete symmetry

    \[y \rightarrow \tilde y = -y.\]

    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.

    \[\begin{align} \frac{d^2 \tilde y}{dt^2} + \omega_0^2 \tilde y &= 0 \\[4pt] \frac{d^2 (-y)}{dt^2} + \omega_0^2(-y) &= 0 \\[4pt] \frac{d^2y}{dt^2} + \omega_0^2y &= 0 \end{align}\]

    Symmetries can be classified as either continuous or discrete. Continuous symmetries can be expressed as a sum of infinitesimally small symmetries related by a continuous parameter. A discrete symmetry cannot be written as a sum of infinitesimal transformations in this way. Three commonly discussed discrete symmetry transformations [187] are:

    • Time reversal \(t \rightarrow \tilde t =(-1)^{\mathfrak{n}} t\), for integer \(\mathfrak{n}\)
    • Parity \(y \rightarrow \tilde y =(-1)^{\mathfrak{n}} y\), for integer \(\mathfrak{n}\)
    • Charge conjugation \(y \rightarrow \tilde y = y^*\), 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 \(t \rightarrow \tilde t = t + \varepsilon\), where \(\varepsilon\) is the continuous parameter which can be infinitesimally small, is an example of a continuous transformation because it can be separated into a sum of infinitesimal 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 classified 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 classified 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 infinitesimal parameter but are not continuous. Nongeometrical symmetries will not be discussed here and are also beyond the scope of this text.