14.7: Problems

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14.1. Three commonly discussed discrete symmetry transformations 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 \(y \rightarrow \tilde{y} = y^*\)

Verify that the wave equation, \(\ddot{y} + \omega_0^2y = 0\), is invariant upon each of these discrete transformations.

14.2. Repeat the problem above for the equation \(\ddot{y} + y^{-3} = 0\).

14.3. The Thomas Fermi equation is given by \(\ddot{y} = y^{3/2}t^{-1/2}\).

(a) Verify that it is not invariant upon the discrete symmetry transformation of time reversal,

\(t \rightarrow \tilde{t} = (−1)^{\mathfrak{n}} t\) for integer \(\mathfrak{n}\).

(b) Verify that it is not invariant upon the discrete symmetry transformation of parity,

\(y \rightarrow \tilde{y} = (−1)^{\mathfrak{n}} y\) for integer \(\mathfrak{n}\).

\(t \rightarrow \tilde{t} = (−1)^{\mathfrak{n}} t \quad \text{and} \quad y \rightarrow \tilde{y} = (−1)^{\mathfrak{n}} y\).

14.4. Find the prolongation of the infinitesimal generator

\[U = \xi \partial_t + \eta \partial_y \nonumber \]

acting on the Lagrangian

\[\mathcal{L} = \frac{1}{2}\dot{y}^2 + \frac{1}{3}ty^{2}. \nonumber \]

Write your answer in terms of \(\xi\) and \(\eta\) but not \(\eta^t\) or \(\eta^{tt}\).

14.5. Find the infinitesimal generators for the equation, \(\ddot{y} + y^{-3} = 0\). (This problem is discussed in [190].)

Answer:

\[U_1 = \partial_t \nonumber \]

\[U_2 = 2t\partial_t + y\partial_y \nonumber \]

\[U_3 = t^2\partial_t + ty\partial_y \nonumber \]

14.6. The equation \(\ddot{y} + y^{-3} = 0\) has the three infinitesimal generators listed in the problem above. These infinitesimal generators form a group. The commutator was defined in Section 14.3.3, and the commutator of any pair of these infinitesimal generators can be calculated by

\[[U_a, U_b] = U_aU_b - U_bU_a. \nonumber \]

Using the equation above, show that the commutator for each of the three pairs of infinitesimal generators results in another element of the group.

14.7. Derive the infinitesimal generators for the wave equation, \(\ddot{y} + \omega_0^2y = 0\). (This problem is discussed in [191].)

Answer:

\[U_1 = \partial_t \nonumber \]

\[U_2 = y\partial_y \nonumber \]

\[U_3 = \sin (\omega_0t)\partial_y \nonumber \]

\[U_4 = \cos (\omega_0t)\partial_y \nonumber \]

\[U_5 = \sin (2\omega_0t)\partial_t + \omega_0y \cos (2\omega_0t) \partial_y \nonumber \]

\[U_6 = \cos (2\omega_0t) \partial_t − \omega_0y \sin (2\omega_0t) \partial_y \nonumber \]

\[U_7 = y \cos (\omega_0t) \partial_t − \omega_0y 2 \sin (\omega_0t) \partial_y \nonumber \]

\[U_8 = y \sin (\omega_0t) \partial_t + \omega_0y 2 \cos (\omega_0t) \partial_y \nonumber \]

14.8. The wave equation \(\ddot{y} + \omega_0^2y = 0\) has the eight infinitesimal generators listed in the problem above. The corresponding Lagrangian is

\[\mathcal{L} = \frac{1}{2}\dot{y}^2 + \frac{1}{2}\omega_0^2y^{2}. \nonumber \]

Find the invariants corresponding to the following infinitesimal generators.

(a) \(U_1 = \partial_t \)

(b) \(U_3 = \sin (\omega_0t)\partial_y \)

14.9. In Problem 11.8, we encountered the equation given by \(\ddot{y} = g \sin y\) for constant \(g\).

(a) Show that \(U = \partial_t \) is an infinitesimal generator of this equation.

(b) Show that \(U = y\partial_y \) is not an infinitesimal generator of this equation.

14.10. The Lagrangian

\[\mathcal{L} = \frac{1}{2}\dot{y}^2 + \frac{1}{2}y^{-2} \nonumber \]

corresponds to the equation of motion \(\ddot{y} + y^{-3} = 0\). This equation of motion has three infinitesimal generators:

\[U_1 = \partial_t \nonumber \]

\[U_2 = 2t\partial_t + y\partial_y \nonumber \]

\[U_3 = t^2\partial_t + ty\partial_y \nonumber \]

Use Noether's theorem to find the invariants that correspond to each of these infinitesimal generators. (We encountered this Lagrangian in problem 11.3.)