3 Scalings and alternative presentations

In this section we shall concentrate on some geometrical remarks concerning the Hamiltonian corresponding to the logarithmic potential

$$% H({\vec{p}},{\vec{x}})={{\vec{p}}^2\over 2}+ {p_0^2\over 2}\ln\left(x^2+y^2/q^2+r_{\rm c}^2\right),$$ (16)

where ${\vec{x}}=(x,y)$, ${\vec{p}}=(p_x,p_y)$. Along the paper, for the numerical examples, the parameters were taken as p²₀=2, $r_{\rm c}=0.1$, h=-0.4059 and 0.696<q<1 (Sects. 2.2 and 4.2). First of all let us mention that the parameters p₀ and $r_{\rm c}$ are not essential. Indeed, scaling to new variables ${\vec{X}}=(X,Y)$, ${\vec{P}}=(P_X,P_Y)$ by ${\vec{x}}=\beta_1{\vec{X}}$, ${\vec{p}}=\beta_2{\vec{P}}$ and using a new time variable $s=\gamma t$ the resulting equations of motion correspond to the Hamiltonian

$$% \widehat{H}({\vec{P}},{\vec{X}})={{\vec{P}}^2\over 2}+ {1\over 2}\ln\left(X^2+Y^2/q^2+1\right)$$ (17)

if $\beta_1=r_{\rm c}$, $\beta_2=p_0$, $\gamma=p_0/r_{\rm c}$. So, the energy moves to $h\rightarrow h/p_0^2-\ln r_{\rm c}\equiv\hat{h}$. This transformation is, in fact, a generalized canonical one. The energy is scaled in such a way that the value of $\widehat{H}$ at the origin is 0. Only the energy and q are necessary to study the full family defined by (17). Alternatively, one can use $r_{\rm c}$ as a parameter keeping h fixed, for instance h=0 as MS89 do (see also Appendix). From now on, in this section, we shall work with (17).

Next we study the problem of representation of the phase space. From (17) we note that for all $\hat{h}\ge 0$, $\widehat{H}({\vec{P}}, {\vec{X}})=\hat{h}$ defines a compact level of energy which can be seen as $\S^3$. Indeed, for small $\hat{h}>0$ one can keep the dominant terms

$$\begin{eqnarray*}{1\over 2}(P_X^2+P_Y^2+X^2+Y^2/q^2)\approx\hat{h}. \end{eqnarray*}$$

Introducing the variables

$$\begin{eqnarray*}x_1={X\over\sqrt{2\hat{h}}},\quad x_2={Y\over q\sqrt{2\hat{h}}... ...3={P_X\over\sqrt{2\hat{h}}},\quad x_4={P_Y\over\sqrt{2\hat{h}}}, \end{eqnarray*}$$

we obtain $\sum_{i=1}^4x_{i}^2=1$.

Now let us pass to general values of $\hat{h}$. Denote by $\rho^2=X^2+Y^2/q^2$ and define $\tilde{\rho}=\sqrt{\ln(1+\rho^2)}$ which is analytic for any $\rho>0$. Then we introduce

$$\begin{eqnarray*}x_1={X\tilde{\rho}/\rho\over\sqrt{2\hat{h}}},\quad x_2= {Y\tilde{\rho}/\rho\over q\sqrt{2\hat{h}}} \end{eqnarray*}$$

and x₃, x₄ as above, so $\sum_{i=1}^4x_{i}^2=1.$

Figure 5: Sketch of the transformation from the xpx to x1x3-plane and to angles $(\varpi ,\lambda )$. The section was drawn for the Hamiltonian (16) with h=-0.4059, q=0.9, $r_{\rm c}=0.1$, p02=2

A simple way to use these variables is as follows. All orbits intersect transversally, for instance, y=0 except the x-axis periodic orbit which is always contained on this plane and appears as the boundary of this surface of section, x₁²+x₃²=1 or $P_X^2+\ln(1+X^2)=2\hat{h}$ - see Fig. 5. One can identify the points in the boundary to a single point. Points (x₁,x₃) such that x₁²+x₃²<1, represent an open disc so that, identifying the boundary to a single point, we obtain a 2D sphere $\S^2$. By definition we send the origin (x₁,x₃)=(0,0) to the south pole (SP) and the boundary x₁²+x₃²=1 to the north pole (NP). As it could be also used the section x=0 and because in this case the boundary corresponds to the y-axis periodic orbit, x₂²+x₄²=1, we are interested in doing the mentioned identification with suitable symmetry properties.

Given, for instance, x₁ and x₃ on the section through y=0, we define the angles $0\le\varpi\le\pi/2$ and $0\le\lambda<2\pi$ as (see Fig. 5)

$$% \sin\varpi=\sqrt{x_1^2+x_3^2},\quad \tan\lambda={x_3\over x_1},$$ (18)

such that, on $\S^2$, $2\varpi$ is the polar angle measured from the SP while $\lambda$ is the usual azimuthal angle. Then introducing Cartesian coordinates $(\xi,\eta,\zeta)$

$$% \xi=\sin 2\varpi\cos\lambda,\quad \eta=\sin 2\varpi\sin\lambda,\quad \zeta=-\cos 2\varpi$$ (19)

we get the desired representation. This transformation is similar to the use of a Hopf fibration of $\S^3$ (see Stiefel & Scheifele 1971).

Hence the "natural'' space to represent the full dynamics is $\S^2$. Periodic orbits along the axes appear at the poles and the "pendulum-like oscillations'' for loops fit on this sphere. One can also understand this representation by looking back to Fig. 1 or Fig. 4. Points with $\theta= -\pi/2$ and $\pi/2$ are identified. In this way we obtain a cylinder. Then points on the upper curve (1:1 direct periodic orbit) are identified to a single point, as well as points on the lower curve (1:1 retrograde periodic orbit). The cylinder becomes $\S^2$ and then it is turned to have (0,0) and $(0,\pi/2)$ at the SP and NP respectively. But note that in this case the pendulum oscillations, as described in Sects. 2.1 and 2.2, correspond to boxes.

For completeness we list the transformation in case the formulation (16) is desired. Let then, for instance, (x,p_x) be the values in the y=0 surface of section with, say, p_y>0, unless we are at the boundary. Compute first $X={x/r_{\rm c}}$, P_X=p_x/p₀, $\hat{h}=h/p_0^2-\ln r_{\rm c}$ and then

$$% x_1={\rm sign}\,(X){\sqrt{\ln(1+X^2)}\over\sqrt{2\hat{h}}},\quad x_3={P_X\over\sqrt{2\hat{h}}}\cdot$$ (20)

Having x₁ and x₃, (18) and (19) give us $(\xi,\eta,\zeta)$. A similar transformation follows for (y,p_y) in the x=0 section.

Figure 6 displays, for q=0.75, $r_{\rm c}=0.1$ and h=-0.4059 (similar to Fig. 1 (right) but for different variables) the cross section x=0. The NP, that for this section corresponds to the unstable y-axis orbit, is seen as an hyperbolic point inside a gross stochastic layer. The SP, that corresponds to the x-axis orbit, also appears as an hyperbolic point, because for this value of q the latter orbit is unstable. Note the 2-periodic orbit that bifurcates from the x-axis orbit that gives rise to the banana orbits. The other elliptic points on the northern hemisphere correspond to the 1:1 periodic orbit (direct and retrograde). Therefore "pendulum-like'' oscillations correspond to loops while rotations to boxes. In other words, for boxes, $\lambda$ ranges from 0 to $2\pi$ while for loops, $\lambda$ is confined to some subinterval of $[0,2\pi)$. For a better visualization, orbits in the "hidden'' part of the sphere (SP and almost all the southern hemisphere) were drawn with much less points.

Figure 6: Surface of section x=0 on the sphere $\xi ^2+\eta ^2+\zeta ^2=1$, for the Hamiltonian (16) with the same parameters than Fig. 5 but q=0.75. The "hidden'' part was drawn with less points