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Find Hamiltonian given Force

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  1. Feb 27, 2017 #1
    1. The problem statement, all variables and given/known data
    A particle with mass, m, is subject to an attractive force.
    \begin{equation}
    \vec{F}(r,t) = \hat{e}_r \frac{k}{r^2}e^{-\beta t}
    \end{equation}

    Find the Hamitonian of the particle
    2. Relevant equations
    H = T + U
    Where T is the kinetic energy and U is the potential
    \begin{equation}
    H = p_i \dot{q}_i - L
    \end{equation}

    3. The attempt at a solution

    \begin{equation}
    \begin{split}
    F - \nabla (U) \\
    F_r = \frac{\partial U}{\partial r} \\
    U (r) = \int F(r) dr \\
    U(r) = ke^{-\beta t} \int \frac{1}{r^2} dr \\
    U(r) = - \frac{ke^{-\beta t}}{r}
    \end{split}
    \end{equation}
    \begin{equation}
    L = \frac{d}{dt}(\frac{\partial L)}{\partial \dot{r}} - \frac{\partial L}{\partial r} = m \ddot{r} - 2r\dot{\phi}^2 + \frac{\partial U}{\partial r}
    \end{equation}
    \begin{equation}
    T = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\phi}^2)
    \end{equation}
    Given the following:
    \begin{equation}
    \begin{split}
    p_r = \frac{\partial T}{\partial \dot{r}} = m\dot{r} \\
    p_\phi = \frac{\partial T}{\partial \dot{\phi}} = mr^2\dot{\phi}
    \end{split}
    \end{equation}

    Thus,
    \begin{equation}
    \begin{split}
    H = T + U \\
    H = \frac{1}{2m} (p_r^2+\frac{p_\phi}{r^2}) + U(r) \\
    \end{split}
    \end{equation}
    Therefore having
    \begin{equation}
    H = \frac{1}{2m} [(m\dot{r})^2 + m^2r^2\dot{\phi}^2] + (- \frac{ke^{-\beta t}}{r})
    \end{equation}

    as my final answer.


    Is my method correct so far? If so, how do I continue?

    And to find the total energy, I just add T + U, right?

    \begin{equation}
    \frac{1}{2}m(\dot{r}^2 + r^2\dot{\phi}^2) + (- \frac{ke^{-\beta t}}{r})
    \end{equation}
     
  2. jcsd
  3. Mar 5, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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