A particle with mass $m$ and position $\mathbf{r}(t)$ is subject to a force

$$\mathbf{F}=\mathbf{A}(\mathbf{r})+\dot{\mathbf{r}} \times \mathbf{B}(\mathbf{r})$$

(a) Suppose that $\mathbf{A}=-\nabla \phi$. Show that

$$E=\frac{1}{2} m \dot{\mathbf{r}}^{2}+\phi(\mathbf{r})$$

is constant, and interpret this result, explaining why the field $\mathbf{B}$ plays no role.

(b) Suppose, in addition, that $\mathbf{B}=-\nabla \psi$ and that both $\phi$ and $\psi$ depend only on $r=|\mathbf{r}|$. Show that

$$\mathbf{L}=m \mathbf{r} \times \dot{\mathbf{r}}-\psi \mathbf{r}$$

is independent of time if $\psi(r)=\mu / r$, for any constant $\mu$.

(c) Now specialise further to the case $\psi=0$. Explain why the result in (b) implies that the motion of the particle is confined to a plane. Show also that

$$\mathbf{K}=\mathbf{L} \times \dot{\mathbf{r}}-\phi \mathbf{r}$$

is constant provided $\phi(r)$ takes a certain form, to be determined.

[ Recall that $\mathbf{r} \cdot \dot{\mathbf{r}}=r \dot{r}$ and that if $f$ depends only on $r=|\mathbf{r}|$ then $\left.\boldsymbol{\nabla} f=f^{\prime}(r) \hat{\mathbf{r}} .\right]$