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15.E: Advanced Hamiltonian Mechanics (Exercises)

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    14245
  • 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.
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