Equations of motion

Purpose

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

Electric potential energy

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 ))`

Velocity term

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

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.

The energy equation

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.

Further work

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