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# 15.E: Advanced Hamiltonian Mechanics (Exercises)

1. Poisson brackets are a powerful means of elucidating when observables are constant of motion and whether two observables can be simultaneously measured with unlimited precision. Consider a spherically symmetric Hamiltonian $H = \frac{1}{2m} \left( p^2_r + \frac{p^{2}_{\theta}}{r^2} + \frac{p^2_{\phi}}{r^2 \sin^2 \theta} \right) + U(r) \nonumber$ for a mass $$m$$ where $$U(r$$ is a central potential. Use the Poisson bracket plus the time dependence to determine the following:

1. Does $$p_{\phi}$$ commute with $$H$$ and is it a constant of motion?
2. Does $$p^2_{\theta} + \frac{p^2_{\phi}}{ \sin^2 \theta }$$ commute with $$H$$ and is it a constant of motion?
3. Does $$p_r$$ commute with $$H$$ and is it a constant of motion?
4. Does $$p_{\phi}$$ commute with $$p_{\theta}$$ and what does the result imply?

2. Consider the Poisson brackets for angular momentum $$L$$

1. Show $$\{L_i, r_j \} = \epsilon_{ijk}r_k$$, where the Levi-Cevita tensor is, $\epsilon_{ijk} = \begin{cases} +1 & \mbox{if } ijk \mbox{ are cyclically permuted}\\ −1 & \mbox{if } ijk \mbox{ are anti-cyclically permuted} \\ 0 & \mbox{if } i = j \mbox{ or } i = k \mbox{ or } j = k \end{cases} \nonumber$
2. Show $$\{L_i, p_j \} = \epsilon_{ijk}p_{k}$$.
3. Show $$\{L_i, L_j \} = \epsilon_{ijk}L_k$$. The following identity may be useful: $$\epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km} − \delta_{jm}\delta_{kl }$$.
4. Show $$\{L_i, L^2 \} = 0$$.

3. Consider the Hamiltonian of a two-dimensional harmonic oscillator, $H = \frac{\mathbf{p}^2 }{2m} + \frac{1 }{2 }m ( \omega^2_1r^2_1 + \omega^2_2r^2_2 ) \nonumber$ What condition is satisfied if $$L^2$$ a conserved quantity?

4. Consider the motion of a particle of mass $$m$$ in an isotropic harmonic oscillator potential $$U = \frac{1}{ 2} kr^2$$ and take the orbital plane to be the $$x − y$$ plane. The Hamiltonian is then $H \equiv S_0 = \frac{1}{2m}(p^2_x + p^2_y) +\frac{1}{2}k(x^2 + y^2) \nonumber$

Introduce the three quantities

$S_1 = \frac{1}{2m}(p^2_x − p^2_y) +\frac{1}{2}k(x^2 − y^2) \nonumber$

$S_2 = \frac{1}{ m} p_{x}p_{y} + kxy \nonumber$

$S_3 = \omega (xp_{y} − yp_{x}) \nonumber$

with $$\omega = \sqrt{\frac{k}{m}}$$. Use Poisson brackets to solve the following:

1. Show that $$\{S_0, S_i\}=0$$ for $$i = 1, 2, 3$$ proving that $$(S_1, S_2, S_3)$$ are constants of motion.
2. Show that $\{S_1, S_2\}=2\omega S_3 \nonumber$ $\{S_2, S_3\}=2\omega S_1 \nonumber$ $\{S_3, S_1\}=2\omega S_2 \nonumber$

so that $$(2\omega )^{ −1 } (S_1, S_2, S_3)$$ have the same Poisson bracket relations as the components of a 3-dimensional angular momentum.

$S^2_0 = S^2_1 + S^2_2 + S^2_3 \nonumber$

5. Assume that the transformation equations between the two sets of coordinates $$(q, p)$$ and $$(Q, P)$$ are

$Q = \ln (1 + q^{\frac{1}{2}} \cos p) \nonumber$

$P = 2(1 + q^{\frac{1}{2}} \cos p)q^{\frac{1}{2}} \sin p) \nonumber$

1. Assuming that $$q, p$$ are canonical variables, i.e. $$[q, p]=1$$, show directly from the above transformation equations that $$Q, P$$ are canonical variables.
2. Show that the generating function that generates this transformation between the two sets of canonical variables is $F_3 = −[e^Q − 1]^2 \tan p \nonumber$

6. Consider a bound two-body system comprising a mass $$m$$ in an orbit at a distance $$r$$ from a mass $$M$$. The attractive central force binding the two-body system is

$\mathbf{F} = \frac{k}{r^2}\mathbf{\hat{r}} \nonumber$

where $$k$$ is negative. Use Poisson brackets to prove that the eccentricity vector $$A = p\times L+\mu k\hat{r}$$ is a conserved quantity.

7. Consider the case of a single mass m where the Hamiltonian $$H =\frac{1}{2}p^2$$.

1. Use the generating function $$S(q, P, t)$$ to solve the Hamilton-Jacobi equation with the canonical transformation $$q = q(Q, P)$$ and $$p = p(Q, P)$$ and determine the equations relating the $$(q, p)$$ variables to the transformed coordinate and momentum $$(Q, P)$$.
2. If there is a perturbing Hamiltonian $$\Delta H =\frac{1}{2}q^2$$, then $$P$$ will not be constant. Express the transformed Hamiltonian $$H$$ (using the transformation given above in terms of $$P$$, $$Q$$, and $$t$$). Solve for $$Q(t)$$ and $$P(t)$$ and show that the perturbed solution $$q[Q(t), P(t)]$$, $$p[Q(t), P(t)]$$ is simple harmonic.