A particle of mass $m$ experiences a repulsive central force of magnitude $k / r^{2}$, where $r=|\mathbf{r}|$ is its distance from the origin. Write down the Hamiltonian of the system.

The Laplace-Runge-Lenz vector for this system is defined by

$$\mathbf{A}=\mathbf{p} \times \mathbf{L}+m k \hat{\mathbf{r}}$$

where $\mathbf{L}=\mathbf{r} \times \mathbf{p}$ is the angular momentum and $\hat{\mathbf{r}}=\mathbf{r} / r$ is the radial unit vector. Show that

$$\{\mathbf{L}, H\}=\{\mathbf{A}, H\}=\mathbf{0},$$

where $\{\cdot, \cdot\}$ is the Poisson bracket. What are the integrals of motion of the system? Show that the polar equation of the orbit can be written as

$$r=\frac{\lambda}{e \cos \theta-1},$$

where $\lambda$ and $e$ are non-negative constants.