How Do You Derive the Hamiltonian for a Particle Under Time-Dependent Force?

[Dopplershift]

Homework Statement

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

Homework 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}

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}

 

[mark726]

Yes, your method looks correct so far. To continue, you can simplify the expression for the Hamiltonian by substituting in the expressions for $p_r$ and $p_\phi$ that you derived. This will give you a final expression for the Hamiltonian in terms of the position and velocity variables, which you can then use to solve for the dynamics of the particle.

To find the total energy, you are correct that you simply add the kinetic and potential energies together. The total energy is a constant of motion and should remain constant as the particle moves under the influence of the attractive force.