content_image/350855.jpg content_image/350856.jpg content_image/350854.jpg content_image/350857.jpg content_image/350853.jpg content_image/350852.jpg content_image/350851.jpg
where r$_{min}$ and r$_{max}$ are the real positive solutions of the equation 

We find after some calculation 

where sin α = Jc κ 2 ( 3 κ 2 2 E ) 3 $^{2}$, and 0 ≤ α ≤ π $_{2}$. 

We now return to Eq. (2.8). An immediate consequence of integration of Eq. (2.8) is the the relation r 2 

In the particular case of r$_{0}$ = r$_{min}$ ̸ = 0, and r ( t ) > r$_{min}$ , t > 0, the integral from the right-hand side of (2.16) can be expressed by means of the elliptic integral of the third kind Π( φ, n, k ) (see Ref. [19], 3.137, Eq. 3), namely we have 

<img src='content_image/350855.jpg'>

<img src='content_image/350856.jpg'>

<img src='content_image/350854.jpg'>

<img src='content_image/350857.jpg'>

<img src='content_image/350853.jpg'>

where $^{r}$− is the negative root of the polynomial from the left-hand side of Eq. (2.13) satisfying 

Clearly, Eq. (2.17) defines r as an implicit function of ϕ . 

<img src='content_image/350852.jpg'>

Furthermore, for r$_{0}$ = r$_{max}$ ̸ = 0, and r ( t ) < r$_{max}$ , t > 0, the implicit equation for the trajectory can be obtained from (2.16) with the help of the elliptic functions of the third kind Π( φ, n, k ) and first kind F ( φ, k ) (see [19], 3.137, Eq. 4). It follows that 

<img src='content_image/350851.jpg'>

