What are the equations of motion that describe a charged particle moving around an oppositely charged pole?

The electric potential energy is:

`U = k_e q int_(-a)^a (Q d xi) / (2 a) 1/L_xi`

` = (k_e Q q)/(2a) int_(-a)^a (d xi)/sqrt(r^2 + (xi - z)^2`

` = (k_e Q q)/(2a) [ ln ( xi - z + sqrt(r^2 + (xi - z)^2) ) ]_(-a)^a`

` = (k_e Q q)/(2a) ln((sqrt(r^2 + (z - a)^2) - (z-a)) / (sqrt(r^2 + (z+a)^2)-(z+a)))`

To simplify further analysis it is convenient to convert to these prolate spheriodal coordinates `(mu,nu,varphi)`:

`x = a sinh mu sin nu cos varphi`

`y = a sinh mu sin nu sin varphi`

`z = a cosh mu cos nu`

Also note that:

`r = a sinh mu sin nu`

For simplicity also define `k` as:

`k = (k_e q Q)/(2a)`

Substituting these values in gives:

`U(mu,nu) = k ln(( a cosh mu - a cos nu - (a cosh mu cos nu-a)) / (a cosh mu + a cos nu - (a cosh mu cos nu+a)))`

`U(mu) = k ln( cosh mu + 1 ) - k ln( cosh mu - 1 ))`

For the analysis we require a term for kinetic energy, which requires velocity.
Let `u = sinh mu sin nu`. `dot x` and `dot y` are:

`dot x = -a dot varphi u sin varphi + a dot u cos varphi`

`dot y = a dot varphi u cos varphi + a dot u sin varphi`

From this it can be seen:

`dot x^2 + dot y^2 = a^2 dot varphi^2 u^2 + a^2 dot u^2`

Now:

`dot u = dot nu sinh mu cos nu + dot mu cosh mu sin nu`

`dot z = -a dot nu cosh mu sin nu + a dot mu sinh mu cos nu`

From these we can see:

`a^2 dot u^2 + dot z^2
= a^2 (dot mu^2 + dot nu^2) (sinh^2 mu cos^2 nu + cosh^2 mu sin^2 nu)`

` = a^2 (dot mu^2 + dot nu^2) (sinh^2 mu + sin^2 nu)`

Now we get:

`v^2 = dot x^2 + dot y^2 + dot z^2 =
a^2 dot varphi^2 sinh^2 mu sin^2 nu
+ a^2 (dot mu^2 + dot nu^2) (sinh^2 mu + sin^2 nu)`

The Lagrangian is:

`cc L(mu,dot mu,nu,dot nu,dot varphi)
= KE - PE`

`= 1/2 m v^2 - U(mu)`

`= 1/2m [ a^2 dot varphi^2 sinh^2 mu sin^2 nu
+ a^2 (dot mu^2 + dot nu^2) (sinh^2 mu + sin^2 nu) ] - U(mu)`

The conjugate momentum to `varphi` is the constant:

`p_varphi = (del cc L)/(del dot varphi) = m a^2 dot varphi sinh^2 mu sin^2 nu`

Note that `m a^2 dot varphi sinh^2 mu sin^2 nu = m r^2 dot phi` which is the angular
momentum, so angular momentum in the system is conserved.

In order to derive the equations of motion it is possible to create a Routhian equation, from this
generate Euler-Lagrange equations, and then with some difficulty integrate them. Curiously all this activity
leads one around in a circle back to `E = KE + PE` in a form which we can construct without going to
any of this effort.

`E =
(m a^2)/2 (dot mu^2 + dot nu^2)(sinh^2 mu + sin^2 nu)
+ p_varphi^2 / (2 m a^2 sinh^2 mu sin^2 nu)
- 2k ln[tanh(mu/2)]`

In the potential energy portion of the equation `ln [tanh(mu/2)] = -1/2 ln((cosh(mu)+1)/(cosh(mu)-1))` which gives
the potential energy as calculated above. The kinetic energy portion of the equation has been rearranged
to remove `dot varphi`.

`dot mu^2 + dot nu^2` is a interesting part of this equation - it is a major component of the velocity and because the total
energy is constant it is dependant on just two of the position coordinates `mu` and `nu`. It is a convenient
quantity to check in simulations. There is also a shorter form of the energy equation:

`E =
(m a^2)/2 (dot mu^2 + dot nu^2)(sinh^2 mu + sin^2 nu)
+ 1/2 p_varphi dot phi
- 2k ln[tanh(mu/2)]`

The brevity comes at the expense of adding an extra variable.

It's possible/probable that the energy equation in one of the forms shown above together with the conjugate moment equation

` p_varphi = m a^2 dot varphi sinh^2 mu sin^2 nu`

form the best description of the equations of motion of the system.

Ideally I would have parametric descriptions of the motion, but this may not be achievable.