2 Projective Euler variables and the 4D oscillator

In [Reference Kummer8], it is shown that the Moser regularization exposes the hidden $SO(4)$ symmetry of the Kepler system, and it is intimately related to the $KS$ -regularized Kepler system. Precisely, in [Reference Ferrer and Crespo6, Reference Kummer8] the Kepler system is obtained due to the reduction of the action of a 1D subgroup of $SO(4)$ , which in a more physical context is dubbed as a gauge group. This section provides a set of symplectic coordinates especially well suited for this geometric setting. Moreover, we shall prove that time reparametrizations play a key role in this system, leading straightforwardly to the $KS$ or Moser regularization.

The Projective Euler variables are based on Euler parameters and have been used in several works (see, for example, [Reference Ferrer and Crespo5, Reference Ferrer and Crespo6] and the references therein). This set of variables has proved to be very useful to describe different dynamical aspects of the 4D oscillator. Notably, in [Reference Ferrer and Crespo6], the Projective Euler variables establish the connection with the Kepler system and the spherical rotor straightforwardly. For the benefit of the reader, we briefly recall here the Projective Euler chart, which is defined through the transformation

$$ \begin{align*}\mathcal{PE}: (\rho,\phi,\theta,\psi,P,\Phi,\Theta,\Psi)\rightarrow (\mathbf{ q},\mathbf{ p}),\end{align*} $$

given by

(2.1) $$ \begin{align}q_1=&\sqrt{\rho}\, c_\alpha s_\beta,\quad p_1=\dfrac{2}{ \sqrt{\rho }}\, s_\beta\left(c_\alpha\,P\rho-s_{\alpha}\,{\Theta}\right)+\dfrac{\Phi+\Psi}{\sqrt{\rho }\, c_\alpha} c_\beta,\nonumber\\[1pt] q_2=&\sqrt{\rho}\, s_\alpha s_\gamma,\quad p_2=\dfrac{2}{ \sqrt{\rho }}\, c_\gamma\left(s_\alpha\,P\rho+c_\alpha\,\Theta\right)-\dfrac{\Phi-\Psi}{\sqrt{\rho }\, s_\alpha} s_\gamma,\\[1pt] q_3=&\sqrt{\rho}\, s_\alpha s_\gamma,\quad p_3=\dfrac{2}{ \sqrt{\rho }}\, s_\gamma\left(s_\alpha\,P\rho+c_{\alpha}\,{\Theta}\right)+\dfrac{\Phi-\Psi}{\sqrt{\rho }\, s_\alpha} c_\gamma,\nonumber\\[1pt] q_4=&\sqrt{\rho}\, c_\alpha c_\beta,\quad p_4=\dfrac{2}{ \sqrt{\rho }}\, c_\beta\left(c_\alpha\,P\rho-s_\alpha\,\Theta\right)-\dfrac{\Phi+\Psi}{\sqrt{\rho }\, c_\alpha} s_\beta, \nonumber\end{align}$$

where we have introduced the following notation with the aim of getting compact expressions $c_x=\cos x$ and $s_x=\sin x$ . Moreover, $\alpha =\theta /2$ , $\beta =(\phi +\psi )/2$ , $\gamma =(\phi -\psi )/2$ , and $(\rho ,\phi ,\theta ,\psi )\in \Lambda = R^+\times (0,2\pi )\times (0,\pi )\times (0,2\pi )$ . Excluding the manifolds $\mathcal {M}_1\!\!=\{(q,Q)|q_1\!=\!q_4\!=\!0\}$ and $\mathcal {M}_2=\{(q,Q)|q_2\!=\!q_3\!=\!0\}$ , the Hamiltonian (1.2) in the new variables reads as follows:

(2.2) $$ \begin{align} &\hspace{-0.8cm}\mathcal{H}_\omega= \frac{ \omega}{2}\rho +2\rho P^2 + \frac{1}{\rho}\;\mathcal{Z}(\theta,\_,\_,\Theta,\Phi,\Psi), \end{align} $$

where

(2.3) $$ \begin{align} \mathcal{Z}=2\left(\Theta^2+\frac{\Phi^2+\Psi^2-2\,\Phi\Psi\,\cos\theta}{\sin^2\theta} \right). \end{align} $$

In the above expression, the variables $\phi $ and $\psi $ are cyclic, with $\Phi $ and $\Psi $ as the corresponding first integrals. In addition, the computation of the Poisson brackets shows that the function $\mathcal {Z}$ is an integral for $\mathcal {H}_\omega $ . This function will play a key role in what follows and defines a well-known Hamiltonian dubbed as the spherical rotor, which is obtained by making equal all the principal moments of inertia of the free rigid body system in Euler angles (see [Reference Marsden and Ratiu12]). Mainly, the equations of motion associated with $\mathcal {Z}$ describe the dynamics of a full symmetric rigid body, with uniform distribution of mass and freely rotating without external interactions. That is to say, the flow generated by $\mathcal {Z}$ , when restricted to a sphere of arbitrary radius, corresponds to uniform rotations.

2.1 The role of time reparametrizations

Now, we explore two reparametrizations of the independent variable for the 4D oscillator system. Each of them will reveal a particular 3D subdynamics of the original system. The following theorem was first given in [Reference Ferrer and Crespo6 ].

Theorem 2.1 Considering the time reparametrization $d\tau =(4\rho )^{-1}\,ds$ on system (1.2), and fixing the energy level $\mathcal {H}_\omega = h$ , the flow is described by the Hamiltonian $\tilde {\mathcal {H}}_\omega =\frac {1}{4\rho }(\mathcal {H}_\omega - h)$ on the manifold $\tilde {\mathcal {H}}_\omega =0$ , which in the Projective Euler chart is given by

(2.4) $$ \begin{align} \tilde{\mathcal{H}}_\omega= \tilde{\mathcal{K}}_\mu + \frac{\Psi^2-2\Phi\Psi\cos \theta}{2\rho^2\,\sin^2\theta}, \end{align} $$

where

(2.5) $$ \begin{align} \tilde{\mathcal{K}}_\mu=\frac{1}{2}\left(P^2 + \frac{\Theta^2}{\rho^2}+\frac{\Phi^2}{\rho^2\,\sin^2\theta} \right)- \frac{\mu}{\rho}, \end{align} $$

and $\mu = h/4$ .

Proof It is a straightforward computation using the Projective Euler chart (see [Reference Ferrer and Crespo6, Theorem 1]).   ▪

Remark 2.2 Notice that by restricting to $\Phi =0$ or $\Psi =0$ , we are led to the 3D Kepler system in spherical coordinates (see [Reference Goldstein, Poole and Safko7]). Indeed, the expressions for these momenta in Cartesian coordinates are given by

(2.6) $$ \begin{align} \Phi(q,p)=\frac{1}{2}(p_1 q_4-p_4 q_1 - p_2 q_3 + p_3 q_2),\nonumber\\ \Psi(q,p)=\frac{1}{2}(p_1 q_4-p_4 q_1+ p_2 q_3 - p_3 q_2 ). \end{align} $$

Namely, each of them is one of the possible choices for the bilinear relation constraint imposed for the $KS$ transformation. Moreover, in [Reference Ferrer and Crespo6], we also showed that the $KS$ map connects both systems in rectangular coordinates, just by inverting the spherical coordinates transformation

$$ \begin{align*}\mathcal{S}:(\rho,\phi,\theta,R,\Phi,\Theta)\rightarrow(x_1,x_2,x_3,y_1,y_2,y_3),\end{align*} $$

(2.7) $$ \begin{align} x_1&=\rho\,s_\theta\,c_\phi,&y_1&={1}/{2\,\rho\,s_\theta } \left[c_\phi \left(\Theta s_{2\theta}+2 P \rho s^2_\theta \right)-2 \Phi s_\phi\right],& \\ x_2&=\rho\,s_\theta\,s_\phi,&y_2&={1}/{2\,\rho\,s_\theta } \left[s_\phi \left(\Theta s_{2 \theta }+2 P \rho s^2_\theta \right)+2 \Phi c_\phi\right],& \nonumber \\ x_3&=\rho\,c_\theta,& y_3&=P c_ \theta-{\Theta s_\theta}/{\rho },& \nonumber \end{align} $$

with $(\rho ,\phi ,\theta )\in \Lambda _4= R^+\times (0,2\pi )\times (0,\pi )$ . In other words, the composition of the previous transformations $\mathcal {S}\circ P_4\circ \mathcal {PE}^{-1}$

$$ \begin{align*}(q,p)\rightarrow (\rho,\phi,\theta,\psi,P,\Phi,\Theta,\Psi)\rightarrow (\rho,\phi,\theta,P,\Phi,\Theta)\rightarrow(x_1,x_2,x_3,y_1,y_2,y_3),\end{align*} $$

where $P_4$ is the projection over $(\rho ,\phi ,\theta ,P,\Phi ,\Theta )$ , is given by

(2.8) $$ \begin{align} &x_1=2(q_2q_4-q_1q_3), \; & y_1 ={1}/{2\rho}\,( p_4 q_2 + p_2 q_4- p_1 q_3-p_3 q_1), \nonumber \\ &x_2=2(q_1q_2+q_3q_4), \; & y_2 ={1}/{2\rho}\,( p_2 q_1 + p_1 q_2 + p_4 q_3 + p_3 q_4), \\ &x_3= q_1^2 -q_2^2-q_3^2+q_4^2, & y_3 ={1}/{2\rho}\, (p_1 q_1 - p_2 q_2 - p_3 q_3 + p_4 q_4), \nonumber \end{align} $$

which is the $KS$ map. This fact shows that the following diagram is commutative:

(2.9)

where $\Psi ^{-1}(0) =\{(q,p)\in T^*\mathbb {R}^4_0:\; \Psi (q,p)=0\}$ .

Note that this operation is equivalent to reducing the $S^1$ symmetry given by the Hamiltonian action associated with $\Phi (q,p)$ or $\Psi (q,p)$ . A detailed treatment of this process is given in [Reference Crespo and Ferrer2, Reference Ferrer and Crespo6].

The following result explores an alternative time reparametrization. It shows a connection between the 4D oscillator and another remarkable 3D physical system. Namely, employing a suitable reparametrization, the oscillator may be separated into two 1-DOF subsystems being one of them the spherical rotor.

Theorem 2.3 Let us consider the 4D harmonic oscillator expressed in the Projective Euler chart. Then, by fixing the energy level $\mathcal {H}_\omega = h$ and carrying out the time reparametrization

(2.10) $$ \begin{align} d\tau=\rho\,ds, \end{align} $$

the flow is described by the Hamiltonian $\tilde {\tilde {\mathcal {H}}}_\omega =\rho \,(\mathcal {H}_\omega - h)$ on the manifold $\tilde {\tilde {\mathcal {H}}}_\omega =0$ . After this manipulation, the 4D harmonic oscillator becomes separable and includes the dynamics of the spherical rotor.

Proof According to the Poincaré technique with $d\tau =\rho \,ds$ , we obtain the following expression for $\tilde {\tilde {\mathcal {H}}}_\omega $ in the Projective Euler chart:

(2.11) $$ \begin{align} \tilde{\tilde{\mathcal{H}}}_\omega={\rho}\,(\mathcal{H}_\omega-h)=\mathcal{K}_{\rho}+\mathcal{K}_{\theta}, \end{align} $$

where

(2.12) $$ \begin{align} \mathcal{K}_{\rho}=\frac{\omega\rho^2}{2}+ 2{\rho^2P^2}-{h\,\rho} ,\quad \mathcal{K}_{\theta}= {2}\left(\Theta^2 + \frac{\Phi^2+\Psi^2-2\,\Phi\Psi\,\cos\theta}{\sin^2\theta} \right). \end{align} $$

That is to say, $\mathcal {K}$ has been separated to two 1-DOF subsystems being the function $\mathcal {K}_{\theta }=\mathcal {Z}$ , the Hamiltonian defining the system associated to the spherical rotor.   ▪

The connection of the 4D oscillator and the spherical rotor may appear less important than the previous one with the Kepler system. However, considering the expression of $\mathcal {K}_{\theta }$ in rectangular coordinates, the link with the Moser regularization is established. Precisely, we have

(2.13) $$ \begin{align} {\mathcal{K}}_{\theta}(\mathbf{q},\mathbf{p})=\mathcal{Z}&=\dfrac{1}{2}\left(|\mathbf{q}|^2|\mathbf{p}|^2-\langle \mathbf{q},\mathbf{p}\rangle^2 \right), \nonumber \\ {\mathcal{K}}_{\rho}(\mathbf{q},\mathbf{p})&=\dfrac{1}{2}\left(\omega|\mathbf{q}|^4+\langle \mathbf{q},\mathbf{p}\rangle^2 -2h|\mathbf{q}|^2\right). \end{align} $$

The Hamiltonian function ${\mathcal {Z}}$ is precisely the one considered by Moser and Zehnder in [Reference Moser and Zehnder15] to regularize the Keplerian flow, which after constraining to $|\mathbf {q}|=1$ and $\langle \mathbf {q},\mathbf {p}\rangle =0$ , leads to the geodesic flow on the sphere $S^3$ . This fact suggests that the harmonic oscillator may also lead to the Kepler system through the Moser procedure. The next section investigates this issue.

Indeed, constraints $|\mathbf {q}|=1$ and $\langle \mathbf {q},\mathbf {p}\rangle =0$ are equivalent to $\rho =1$ and $\rho P=0$ . Moreover, for the case $\omega =h=1$ , and taking into account expression (2.12), the constrained manifold obtained from imposing $\rho =1$ and $\rho P=0$ is an invariant manifold of the Hamiltonian system given by (2.11). This can be checked by noting that $\rho $ and $\rho P$ commute with $\mathcal {K}_{\theta }$ , and considering the equations of motion associated to ${\mathcal {K}}_{\rho }$ given by

$$ \begin{align*}\dot\rho=4\rho^2P,\quad \dot P=\omega\rho+4\rho P^2-h=\rho+4\rho P^2-1.\end{align*} $$

3 Alternative Moser regularizations

In this section, we investigate a broad family of Hamiltonian functions giving the geodesic flow when restricted to $TS^3$ . Note that Moser used two different Hamiltonians to regularize the Keplerian flow. Indeed, in [Reference Moser13, Reference Moser and Zehnder15], the Kepler system connection was made through the Hamiltonian system defined by the following functions:

(3.1) $$ \begin{align} \mathcal{Z}=\dfrac{1}{2}\left(|\mathbf{q}|^2|\mathbf{p}|^2-\langle \mathbf{q},\mathbf{p}\rangle^2 \right),\quad \mathcal{M}=\dfrac{1}{2}\left(|\mathbf{q}|^2|\mathbf{p}|^2\right). \end{align} $$

Of course, both systems define the same vector field on the tangent bundle of the sphere $TS^3$ , the Hamiltonian version of the geodesic flow. That is to say, given a Riemannian manifold $(M,g)$ , the geodesic flow on M may be given by the restriction to the configuration space of any of the Hamiltonian flows on $TM$ or $T^*M$ defined, respectively, by the Hamiltonian functions

$$ \begin{align*}H(x,y)=\dfrac{1}{2}{g}_x(y),\quad \tilde{H}(x,p_y)=\dfrac{1}{2}\tilde{g}_x(p_y),\end{align*} $$

where $\tilde {g}$ is the inverse of the metric tensor. Considering $M=\mathbb {R}^4$ , with the metric tensor given by the usual Euclidean inner product, we have that the geodesic flow is defined in the submanifold $|\mathbf {q}|=r$ and $\langle \mathbf {q},\mathbf {p}\rangle =0$ , together with the energy surface ${\tilde {H}(\mathbf {q},\mathbf {p})}=|\mathbf {p}|^2=p^2$ . We are using the terminology of geodesic flow in a broad sense, allowing the flow to traverse geodesics at constant arbitrary velocity, rather that imposing $p=1$ .

The reparametrization and splitting of the 4-DOF harmonic oscillator given in Theorem 2.3 suggest that the regularization procedure given by Moser in [Reference Moser13] may also work for the oscillator. In this regard, we shall characterize a wide family of Hamiltonians allowing the Kepler regularization through the Moser procedure.

Any Hamiltonian flow in $T^*\mathbb {R}^4$ containing the geodesic flow of $S^3(r)$ (at constant velocity) must satisfy that the submanifold

(3.2) $$ \begin{align} \Omega_{r,p}=\{(\mathbf{q},\mathbf{p})\in T^*\mathbb{R}_0^4:\;|\mathbf{q}|=r,\,|\mathbf{p}|=p,\,\langle \mathbf{q},\mathbf{p}\rangle=0\}, \end{align} $$

is invariant, where $\Omega _{r,p}$ is a submanifold of $TS^3(r)$ containing the geodesics travelled at constant rate p. Moreover, the functions $|\mathbf {q}|^2$ , $|\mathbf {p}|^2$ , and $\langle \mathbf {q},\mathbf {p}\rangle $ span the following Lie algebra.

Proposition 3.1 shows the key role that the function $\mathcal {Z}$ plays. It is the Hamiltonian function of the spheric rotor and the generator of the center of G. Thus, every submanifold given by the combination of any of the constraints $|\mathbf {q}|^2=k_1$ , $|\mathbf {p}|^2=k_2$ , or $\langle \mathbf {q},\mathbf {p}\rangle =k_3$ is an invariant manifold for the Hamiltonian system associated to $\mathcal {Z}$ . In this regard, it is no wonder that Moser chose it to define the geodesic flow after the corresponding constraints are imposed. However, there are many functions $f(x,y,z)$ , evaluated in $x=|\mathbf {q}|^2$ , $y=|\mathbf {p}|^2$ , and $z=\langle \mathbf {q},\mathbf {p}\rangle $ , which may also be suitable for this purpose. To be precise, such a function must satisfy that when constrained to $\Omega _{r,p}$ , the following brackets vanish:

$$ \begin{align*}\{F( \mathbf{q}, \mathbf{p}),|\mathbf{q}|^2\}=\{F( \mathbf{q}, \mathbf{p}),|\mathbf{p}|^2\}=\{F( \mathbf{q}, \mathbf{p}),\langle \mathbf{q},\mathbf{p}\rangle\}=0,\quad \mathbf{q}, \mathbf{p}\in \Omega_{r,p},\end{align*} $$

for some r and p, and $F( \mathbf {q}, \mathbf {p})=f(|\mathbf {q}|^2,|\mathbf {p}|^2,\langle \mathbf {q},\mathbf {p}\rangle )$ . These conditions impose the following relations involving partial derivatives:

(3.3) $$ \begin{align} |\mathbf{q}|^2f_x=|\mathbf{p}|^2f_y,\quad |\mathbf{p}|^2f_z+2\langle \mathbf{q},\mathbf{p}\rangle f_x=0,\quad |\mathbf{q}|^2f_z+2\langle \mathbf{q},\mathbf{p}\rangle f_y=0. \end{align} $$

Note that any f in the center of G satisfies the above conditions. Nevertheless, there are functions, not in the center of G, for which (3.3) holds. For instance, the harmonic oscillator $\mathcal {H}_\omega $ , associated to $f(x,y)=1/2(y^2+\omega x^2)$ and $\omega>0$ , is a possible choice. Precisely, for $r={p}/\sqrt {\omega }$ and $\langle \mathbf {q},\mathbf {p}\rangle =0$ , we have that

$$ \begin{align*}\{\mathcal{H}_\omega,|\mathbf{q}|^2\}=-2\langle \mathbf{q},\mathbf{p}\rangle=0,\quad \{\mathcal{H}_\omega,|\mathbf{p}|^2\}=2\omega\langle \mathbf{q},\mathbf{p}\rangle=0,\end{align*} $$

and

$$ \begin{align*}\{\mathcal{H}_\omega,\langle \mathbf{q},\mathbf{p}\rangle\}=\omega|\mathbf{q}|^2-|\mathbf{p}|^2=\omega\, r^2-p^2=0.\end{align*} $$

Therefore, among all the submanifolds $\Omega _{r,p}$ in $T^*\mathbb {R}^4$ , the harmonic oscillator only leaves invariant that one given by $|\mathbf {q}|=r=p/\sqrt {\omega }$ , $|\mathbf {p}|=p$ and $\langle \mathbf {q},\mathbf {p}\rangle =0$ . According to Theorem 2.3, the Hamiltonian vector fields generated by the functions $\mathcal {H}_\omega $ and $\mathcal {Z}$ agree along this manifold. Furthermore, as it was showed by Moser in [Reference Moser and Zehnder15], the constrained Hamiltonian $\mathcal {Z}$ leads to the geodesic flow, hence so does $\mathcal {H}_\omega $ .

4 Symplectic stereographic-type transformation

This section is devoted to the transformation connecting the dynamics of the Kepler and oscillator flows. We define our transformation based on Moser’s work, but it is not the same as we are going to see. Our variation allows us to extend Moser’s regularization to the positive energy case. In our view, the reason this work was not finished in [Reference Moser13] is that Moser was not concerned about parabolic or hyperbolic motions, as it is indicated in the title of the paper [Reference Moser13]. On the contrary, he was more focused on small perturbations of bounded Keplerian orbits related to big questions as the stability of the solar system [Reference Moser14].

First, we define the transformation in the configuration space. For this purpose, we consider the following stereographic-type transformation, which was dubbed by Moser in [Reference Moser and Zehnder15] as the “homogeneous” version of the stereographic mapping:

$$ \begin{align*}\Sigma\Pi:\mathbb{R}_0^4-L_0\longrightarrow \Delta\subset \mathbb{R}_0^4,\quad\;\;\;\mathbf{q}\rightarrow \mathbf{x},\end{align*} $$

and defined as follows

(4.1) $$ \begin{align} x_0=\,&|\mathbf{q}|,\quad x_j=\dfrac{q_j}{|\mathbf{q}|-q_0}, \end{align} $$

where $L_0$ is the closed positive $q_0$ -axis and $\Delta $ is the open set of $\mathbb {R}_0^4$ defined by $x_0>0$ . Notice that this map is related to the stereographic projection, but it is not the same. Indeed, the following geometrical interpretation was given in [Reference Moser and Zehnder15, Section 1.6]. “For $x_0=1$ , the transformation (4.1) is the stereographic projection, while for arbitrary $x_0$ , it maps rays through the origin into vertical half lines on $\mathbb {R}_0^4$ such that the heights $x_0$ agree with the distance on the ray.” Now, we carry out the canonical extension by considering the corresponding generating function

$$ \begin{align*}W(\mathbf{x},\mathbf{p})=p_0 x_0\,\dfrac{|\tilde{\mathbf{x}}|^2-1}{|\tilde{\mathbf{x}}|^2+1}+\dfrac{2x_0}{|\tilde{\mathbf{x}}|^2+1}\,(p_1x_1+p_2x_2+p_3x_3).\end{align*} $$

Thus, the canonical extension is given by

$$ \begin{align*}\mathcal{ST}:T^*(\mathbb{R}_0^4-L_0)\longrightarrow \Delta\times\mathbb{R}^4\subset T^*\mathbb{R}_0^4,\quad(\mathbf{q},\mathbf{p})\rightarrow(\mathbf{x},\mathbf{y}),\end{align*} $$

(4.2) $$ \begin{align} x_0=\,&|\mathbf{q}|,\quad\qquad\!\!\! x_j=\dfrac{q_j}{|\mathbf{q}|-q_0}, \nonumber \\ y_0=\,&\dfrac{\langle \mathbf{q},\mathbf{p}\rangle}{|\mathbf{q}|},\quad y_i=(|\mathbf{q}|-q_0)p_i-\left(\dfrac{\langle \mathbf{q},\mathbf{p}\rangle}{|\mathbf{q}|}-p_0\right)q_i. \end{align} $$

This transformation has been named symplectic stereographic-type transformation due to the geometrical relation with the stereographic projection in configuration space. Next, for the sake of completeness, we also provide the inverse transformation of the stereographic-type map given above

$$ \begin{align*}\mathcal{ST}^{-1}: \Delta\times\mathbb{R}^4\subset T^*\mathbb{R}_0^4 \longrightarrow T^*(\mathbb{R}_0^4-L_0),\quad(\mathbf{x},\mathbf{y})\rightarrow(\mathbf{q},\mathbf{p}). \end{align*} $$

The explicit expression of the inverse is obtained by considering the inverse of the configuration space transformation given in (4.1). Then, by using the generating function

$$ \begin{align*}\mathcal{W}(\mathbf{q},\mathbf{y})=y_0 |\mathbf{q}|+\dfrac{1}{|\mathbf{q}|-q_0}\,(y_1q_1+y_2q_2+y_3q_3),\end{align*} $$

we arrive to the inverse stereographic-type transformation

(4.3) $$ \begin{align} q_0=\,&x_0\,\dfrac{|\tilde{\mathbf{x}}|^2-1}{|\tilde{\mathbf{x}}|^2+1}, & q_j&=x_0\,\dfrac{2x_j}{|\tilde{\mathbf{x}}|^2+1},& \nonumber\\ p_0=\,&\frac{\langle \tilde{\mathbf{x}},\tilde{\mathbf{y}}\rangle}{x_0}+y_0\dfrac{|\tilde{\mathbf{x}}|^2-1}{|\tilde{\mathbf{x}}|^2+1}, & p_j&=\dfrac{|\tilde{\mathbf{x}}|^2+1}{2 \,x_0}\,y_j-\langle \tilde{\mathbf{x}},\tilde{\mathbf{y}}\rangle\,\frac{ x_j}{x_0}+\dfrac{2y_0x_j}{|\tilde{\mathbf{x}}|^2+1}, \end{align} $$

where $\mathbf {z}=(z_0,z_1,z_2,z_3)$ , $\tilde {\mathbf {z}}=(z_1,z_2,z_3)$ , and $j=1,\,2,\,3$ .

The direct stereographic-type transformation is the same as the one given by Moser in [Reference Moser and Zehnder15, Section 1.6]. However, by comparison of (4.3) with the formulas (1.82) given in [Reference Moser and Zehnder15], the reader may observe a slight difference in the inverse stereographic-type transformation, allowing us to consider the positive energy case. Precisely, while Moser in [Reference Moser13, Reference Moser and Zehnder15] was focused on the restriction to the tangent bundle of the sphere, we define a canonical extension between open sets in $T^*\mathbb {R}_0^4$ .

5 Proof of Theorem 1.1

This section demonstrates Theorem 1.1 by introducing a suitable change of variables and distinguishing the cases of positive and negative energies.

5.1 Differential system in stereographic variables

First, we consider the inverse symplectic stereographic-type transformation given in (4.3). After applying this change of variables, we can replace the Hamiltonian ${\mathcal {H}}_\omega (\mathbf {q},\mathbf {p})$ given in (1.2) by

(5.1) $$ \begin{align} {\mathcal{G}}_\omega(\mathbf{x},\mathbf{y})=\dfrac{\omega x_0^2+y_0^2}{2}+\dfrac{1}{2x_0^2}\dfrac{1}{4}\left(|\tilde{\mathbf{x}}|^2+1 \right)^2|\tilde{\mathbf{y}}|^2. \end{align} $$

Previous to the regularization, we prepare the Hamiltonian system associated to (5.1). For this purpose, we consider a function $\mathcal {F }(\tilde {\mathbf {x}},\tilde {\mathbf {y}})$ satisfying

(5.2) $$ \begin{align} \mathcal{F }^2(\tilde{\mathbf{x}},\tilde{\mathbf{y}})=\dfrac{1}{4}\left(|\tilde{\mathbf{x}}|^2+1 \right)^2|\tilde{\mathbf{y}}|^2. \end{align} $$

Of course, we have two options, which will be used for the cases of positive and negative energies. Thus, ${\mathcal {G}}_\omega $ is given by

(5.3) $$ \begin{align} {\mathcal{G}}_\omega(\mathbf{x},\mathbf{y})=\dfrac{\omega x_0^2+y_0^2}{2}+\dfrac{1}{2x_0^2}\mathcal{F }^2. \end{align} $$

Moreover, the function $\mathcal {F }$ is an integral of the Hamiltonian ${\mathcal {G}}_\omega $ , which associated equations of motion are given by

(5.4) $$ \begin{align} \dot{x}_0(t)=&\,y_0(t),\qquad\quad \dot{y}_0(t)=-\omega x_0(t)+\dfrac{1}{x_0(t)^3}\mathcal{F }^2, \nonumber\\ \dot{\tilde{\mathbf{x}}}(t)=&\,\dfrac{1}{x_0^2(t)}\mathcal{F }{\mathcal{F}}_{\tilde{\mathbf{y}}},\quad\dot{\tilde{\mathbf{y}}}(t)=-\dfrac{1}{x_0^2(t)}\mathcal{F }{\mathcal{F}}_{\tilde{\mathbf{x}}}. \end{align} $$

5.2 Part 1. Regularization of bounded orbits

For the regularization of the bounded orbits, we consider the case of $\omega>0$ . Then, we restrict the 4D oscillator to the invariant submanifold $\Omega _r\subset T^*(\mathbb {R}_0^4-L_0)$ given by

(5.5) $$ \begin{align} \Omega_r=\{(\mathbf{q},\mathbf{p})\in TS^3(r)\subset T^*\mathbb{R}_0^4:\;|\mathbf{q}|=r,\,|\mathbf{p}|=p=r\sqrt{\omega},\,\langle \mathbf{q},\mathbf{p}\rangle=0,\,q_0\neq r\}. \end{align} $$

We already proved that the flow of the harmonic oscillator and $\mathcal {Z}$ agree in $\Omega _r$ , and hence they both agree with the geodesic flow. Thus, we only need to prove that the transformation (4.3) connects the negative energy orbits of the Kepler system\break with $\mathcal {H}_\omega $ .

The invariant manifold $\Omega _r$ corresponds in the $\mathbf {x}\mathbf {y}$ -space to

(5.6) $$ \begin{align} \Omega^*_r=\{(\mathbf{x},\mathbf{y})\in T^*\mathbb{R}_0^4:\;x_0=r,\;y_0=0,\;\left(|\tilde{\mathbf{x}}|^2+1 \right)^2|\tilde{\mathbf{y}}|^2=4r^4\omega\}, \end{align} $$

where r is a positive real constant. By imposing this restriction to the invariant manifold $\Omega ^*_r$ , and taking into account that $x_0=r$ and $\mathcal {F }=r^2\sqrt {\omega }$ , the equations of motion (5.4) are given as follows:

(5.7) $$ \begin{align} \dot{x}_0=&\,0,\ \quad\dot{y}_0=0,\nonumber\\ \dot{\tilde{\mathbf{x}}}=&\,\tilde{\mathcal{F}}_{\tilde{\mathbf{y}}}, \quad\dot{\tilde{\mathbf{y}}}=-\tilde{\mathcal{F}}_{\tilde{\mathbf{x}}}, \end{align} $$

where $\tilde {\mathcal {F}}(\tilde {\mathbf {x}},\tilde {\mathbf {y}})=\sqrt {\omega }/{2}\left (|\tilde {\mathbf {x}}|^2+1 \right )|\tilde {\mathbf {y}}|$ , and we are restricted to the energy manifold $\tilde {\mathcal {F}}(\tilde {\mathbf {x}},\tilde {\mathbf {y}})={\omega }\,r^2$ .

Now, we perform another change of variables, and by abuse of notation, we use the symbols $({\mathbf {x}},{\mathbf {y}})$ in the following way:

(5.8) $$ \begin{align} \tilde{\mathbf{x}}=r\,\mathbf{y},\quad \tilde{\mathbf{y}}=-\dfrac{1}{r}\,\mathbf{x}, \end{align} $$

which leads to the Hamiltonian

$$ \begin{align*}{\tilde{\tilde{\mathcal{F}}}}({\mathbf{x}},{\mathbf{y}})=\dfrac{\sqrt{\omega}}{2}\left(r^2|{\mathbf{y}}|^2+1 \right)\dfrac{1}{r}\,|{\mathbf{x}}|-\omega\,r^2.\end{align*} $$

However, the context makes clear what objects are we referring to in each case. Then, after this symplectic change of variables, the introduction of a new independent variable

$$ \begin{align*}dt=(\sqrt{\omega}\,r\,|{\mathbf{x}}|)^{-1}\,d\tau\end{align*} $$

and the elimination of constant terms lead to the system given by the Hamiltonian function

(5.9) $$ \begin{align} {\mathcal{F}}_\mu({\mathbf{x}},{\mathbf{y}})=\dfrac{1}{2}|{\mathbf{y}}|^2 -\dfrac{\mu}{|{\mathbf{x}}|}, \end{align} $$

in the manifold $ {\mathcal {F}}_\mu =-1/(2r^2)$ , where $\mu =\sqrt {\omega }\,r$ . That is to say, the Kepler system at the level energy $-\sqrt {\omega }/(2r^2)$ is connected with the 4D isotropic oscillator at the level energy $\omega \, r^2$ .

5.3 Part 2. Regularization of unbounded orbits

The regularization of the unbounded orbits is related to the case $\omega <0$ and positive energy $\mathcal {H}_\omega =h>0$ . The assumption of negative frequencies implies that $\Omega ^*_r$ is no longer an invariant manifold. Moreover, we choose the integral function $\mathcal {F}$ in (5.2) to be

(5.10) $$ \begin{align} \mathcal{F }(\tilde{\mathbf{x}},\tilde{\mathbf{y}})=-\dfrac{1}{2}\left(|\tilde{\mathbf{x}}|^2+1 \right)|\tilde{\mathbf{y}}|, \end{align} $$

with fixed value $\mathcal {F }=f<0$ .

Equations of system (5.4) can be separated in two parts. Variables $(x_0,y_0)$ define a 1-DOF subsystem, and by using the Hamiltonian invariance ${\mathcal {G}}_\omega (\mathbf {x},\mathbf {y})=h$ , the variable $y_0$ is expressed as follows:

$$ \begin{align*}y_0^2=2h+|\omega|x_0^2-{f^2}/x_0^2.\end{align*} $$

Therefore, the differential equation for $\dot {x}_0$ is integrated by rewriting it as

(5.11) $$ \begin{align} \dot{x}_0=\pm\sqrt{ 2h+|\omega|x_0^2-{f^2} /x_0^2}. \end{align} $$

Once the subsystem $(x_0(t),y_0(t))$ is solved, we introduce the following new independent variable:

$$ \begin{align*}ds={f^2}/x_0^2(t)\, dt.\end{align*} $$

Then, subsystem $({\tilde {\mathbf {x}}},{\tilde {\mathbf {y}}})$ becomes

(5.12) $$ \begin{align} \dot{\tilde{\mathbf{x}}}(t)=&\,{\mathcal{F}}_{\tilde{\mathbf{y}}},\quad \dot{\tilde{\mathbf{y}}}(t)=-{\mathcal{F}}_{\tilde{\mathbf{x}}}. \end{align} $$

By restricting to the energy level $\mathcal {F}=f$ and introducing the following symplectic change of coordinates

(5.13) $$ \begin{align} \tilde{\mathbf{x}}=h\,\mathbf{y},\quad \tilde{\mathbf{y}}=-\dfrac{1}{h}\,\mathbf{x}, \end{align} $$

we obtain that system (5.12) is given by the Hamiltonian

(5.14) $$ \begin{align} \hat{\mathcal{F}}({\mathbf{x}},{\mathbf{y}})=-\dfrac{1}{2}\left(h^2|{\mathbf{y}}|^2+1 \right)|{\mathbf{x}}|/h-f. \end{align} $$

Finally, we introduce a new independent variable $\tau ,$

$$ \begin{align*}ds=-(h\,|{\mathbf{x}}|)^{-1}\,d\tau,\end{align*} $$

which leads to the system defined by the Hamiltonian function

(5.15) $$ \begin{align} {\mathcal{F}}_\mu({\mathbf{x}},{\mathbf{y}})=\dfrac{1}{2}|{\mathbf{y}}|^2 -\dfrac{\mu}{|{\mathbf{x}}|}, \end{align} $$

in the manifold $ {\mathcal {F}}_\mu =1/(2h^2)$ , where $\mu =-f/h>0$ . That is to say, the Kepler system at the positive energy level $1/(2h^2)$ is connected with the system given by $\mathcal {H}_\omega $ at the level energy h.