\section{Notation Glossary}
\label{app:notation}

This appendix provides a glossary of notation used throughout the handbook.

\nomenclature{$y$}{Dependent variable; unknown function}
\nomenclature{$y(t)$}{Dependent variable as a function of time $t$}
\nomenclature{$y(x)$}{Dependent variable as a function of spatial variable $x$}
\nomenclature{$x$}{Independent variable (spatial)}
\nomenclature{$t$}{Independent variable (time)}
\nomenclature{$\lambda$}{Eigenvalue; parameter in characteristic equation}
\nomenclature{$\omega$}{Angular frequency}
\nomenclature{$\omega_0$}{Natural angular frequency}
\nomenclature{$\alpha, \beta, \gamma$}{General constants; damping ratio}
\nomenclature{$\delta$}{Dirac delta function}
\nomenclature{$\mu$}{Separation constant; parameter}
\nomenclature{$\theta$}{Angle; phase shift}
\nomenclature{$\phi$}{Eigenfunction; angle}
\nomenclature{$A, B, C$}{General constants of integration}
\nomenclature{$a, b, c$}{Coefficients in differential equations}
\nomenclature{$n, m, k$}{Integer indices}
\nomenclature{$f(t)$}{Input/forcing function}
\nomenclature{$F(s)$}{Laplace transform of $f(t)$}
\nomenclature{$\mathcal{L}$}{Laplace transform operator}
\nomenclature{$\mathcal{L}^{-1}$}{Inverse Laplace transform}
\nomenclature{$\mathcal{F}$}{Fourier transform operator}
\nomenclature{$u(t)$}{Unit step (Heaviside) function}
\nomenclature{$y_h$}{Homogeneous solution}
\nomenclature{$y_p$}{Particular solution}
\nomenclature{$W(y_1, y_2)$}{Wronskian of $y_1$ and $y_2$}
\nomenclature{$\mathbf{A}$}{Coefficient matrix in systems}
\nomenclature{$\mathbf{x}(t)$}{State vector}
\nomenclature{$I_n$}{Identity matrix of size $n$}
\nomenclature{$e^{\mathbf{A}t}$}{Matrix exponential}
\nomenclature{$\mathbf{u}, \mathbf{v}$}{Vectors in phase plane}
\nomenclature{$u(t-a)$}{Shifted unit step function}
\nomenclature{$*$}{Convolution operator}