Problem 1
Three commonly discussed discrete symmetry transformations are: Time reversal \(t \rightarrow \tilde{t}=(-1)^{\mathrm{n}} t\) for integer \(\mathfrak{n}\) Parity \(y \rightarrow \tilde{y}=(-1)^{\mathrm{n}} y\) for integer \(\mathfrak{n}\) Charge conjugation \(y \rightarrow \tilde{y}=y^{*}\) Verify that the wave equation, \(\ddot{y}+\omega_{0}^{2} y=0\), is invariant upon each of these discrete transformations.
Problem 10
The Lagrangian $$\mathcal{L}=\frac{1}{2} \dot{y}^{2}+\frac{1}{2} y^{-2}$$ corresponds to the equation of motion \(\ddot{y}+y^{-3}=0\). This equation of motion has three infinitesimal generators: $$\begin{array}{c}U_{1}=\partial_{t} \\\U_{2}=2 t \partial_{t}+y \partial_{y} \\\ U_{3}=t^{2} \partial_{t}+t y \partial_{y}\end{array}$$ Use Noether's theorem to find the invariants that correspond to each of these infinitesimal generators.
