1 Introduction

An essential part of perturbation theory in celestial mechanics is based on perturbations of uncoupled Kepler problems. The Kepler problem in \({\mathbb {R}}^{d}, d \ge 2\) is properly-degenerate in the sense that it has d-degrees of freedom while all of its non-singular bounded orbits are closed and thus carries only one non-trivial frequency. Therefore to understand concrete problems as perturbations of Kepler problems typically one has to study higher order effects.

On the other hand, while looking for periodic orbits of a fixed period, a different approach can be taken, as has been shown in [8], which considered a periodically forced Kepler problem. By the third Kepler law, the period of periodic orbits of the Kepler problem changes with respect to the energy and this gives the normal non-degeneracy of the set of periodic orbits with fixed period in the extended phase space. This, combined with regularization techniques in extended phase space, makes it possible to apply some global methods from symplectic geometry to conclude the existence of periodic orbits after regularization, which correspond to periodic orbits in a generalized sense of the original problem.

In this paper, we consider the same type of model as in [8], namely the time-periodically forced Kepler problem in \({\mathbb {R}}^{d}\) for \(d \in {\mathbb {Z}}_{+}\):

$$\begin{aligned} \ddot{q}=-\dfrac{q}{|q|^{3}} +\nabla _{q} U(q, t), \end{aligned}$$
(1.1)

in which \(q \in {\mathbb {R}}^{d}\). The function \(U(q, t, \varepsilon )\) is a time-periodic function which we assume is \(C^{\infty }\)-regular in a neighborhood of the origin. Outside this neighborhood we do not make any further assumptions. Also, when \(U(q, t, \varepsilon )\) is independent of t, we may fix any positive period. By normalization in the variables (q, t) we may assume that \(U(q, t, \varepsilon )\) is 1-periodic in t.

In many concrete models the problem is described via an external parameter. Thus it is also helpful to consider the following interpolating system of the form

$$\begin{aligned} \ddot{q}=-\dfrac{q}{|q|^{3}} +\varepsilon \nabla _{q} U(q, t, \varepsilon ), \end{aligned}$$
(1.2)

in which \(q \in {\mathbb {R}}^{d}\) and \(U(q, t, \varepsilon )\) is a time-periodic function which is \(C^{\infty }\)-regular in a neighborhood of the origin depending continuously on a parameter \(\varepsilon \in [0, 1]\) such that \(U(q, t){:=}U(q, t, 1)\).

This system is a non-autonomous Hamiltonian system with Hamiltonian

$$\begin{aligned} F_{\varepsilon }(p, q, t)=\dfrac{|p|^{2}}{2}-\dfrac{1}{|q|}+ \varepsilon U(q, t, \varepsilon ). \end{aligned}$$

Without loss of generality we may assume

$$\begin{aligned} U(0, t, \varepsilon )=0. \end{aligned}$$
(1.3)

Indeed when this is not the case, it is enough to take \(U(q,t, \varepsilon )-U(0,t, \varepsilon )\) in place of U(q, t) in the above expression. As a consequence, \(U(q, t, \varepsilon )\) admits a Taylor expansion with t-periodic coefficients of the form

$$\begin{aligned} U(q, t) = \sum _{i=1}^{d} c_{i}(t, \varepsilon ) q_{i} +O(|q|^{2}), \end{aligned}$$

and consequently in a neighborhood of the origin there holds

$$\begin{aligned} |U(q, t)| \le C_{0} |q| +O(|q|^{2}, t) \end{aligned}$$
(1.4)

for some constant \(C_{0}>0\).

Generalized periodic solutions are solutions which become periodic after possible isolated double collisions with the attractive center being regularized. Precisely a generalized solution of Eq. (1.2) is a continuous function \({\mathbb {R}}\rightarrow {\mathbb {R}}^{d}, t \mapsto q(t)\) which satisfies Eq. (1.2) except for a discrete set \({\mathcal {Z}}{:=}\{t \in {\mathbb {R}}: q(t)=0\}\) of collision instants. Moreover, for any \(t_{0} \in {\mathcal {Z}}\), the limits

$$\begin{aligned} \lim _{t \rightarrow t_{0}} \dfrac{u(t)}{|u(t)|}, \qquad \hbox { and } \qquad \lim _{t \rightarrow t_{0}} \left( \dfrac{1}{2} |{\dot{q}}(t)|^{2}-\dfrac{1}{|u(t)|}\right) \end{aligned}$$

of collision direction and collision energy exist. The last condition gives a condition for patching solutions which run into and out of the collisions.

The result in [8] concerns the number of generalized periodic orbits of Eq.(1.2). With the remark from [24], it is sufficient that the function \(U(q, t, \varepsilon )\) is well-defined in a neighborhood of the origin for the q-variable as we now assume. The main result is:

For \(d=2,3\), given any positive integer \(l \in {\mathbb {N}}_{+}\), there exists some \(\varepsilon _{*}>0\), such that for all \(\varepsilon \in [0, \varepsilon _{*})\), the equation (1.2) has at least l generalized periodic solutions.

The result was established with regularization techniques and application of a theorem of Weinstein [26] concerning bifurcations of periodic orbits from a non-degenerate periodic manifold when the perturbation is \(C^{2}\)-small in its neighborhood. The external small parameter in Eq. (1.2) was technically needed to control the \(C^{2}\)-norm of the perturbation.

We mention two recent non-perturbative results concerning equations of the form (1.1):

For \(d=2\) and \(U(q, t)=O(|q|^{\alpha })\) for some \(\alpha \in (0, 2)\) when \(|q| \rightarrow \infty \), the existence of at least one generalized periodic orbit has been shown in [7].

For \(d=2,3\) and \(U(q, t)=\langle p(t), q \rangle \), the existence of infinitely many generalized periodic orbits has been established in [6].

These results were obtained via variational methods. Besides these, many results on the existence and multiplicity of periodic solutions with more general singular potentials has been obtained in e.g. [3, 5] with variational methods. Some of these results have been collected in the book [2] in which one may find many further references.

In this article we aim to establish the following theorem which uniformly treats the problem of any dimension with a general external force. In dimension 2 or 3, this enumeratively improves the result from [8] and allows more general forcing with more information about the space locations of the generalized periodic orbits to be found as compared to [6]:

Theorem 1.1

For all \(d \in {\mathbb {Z}}_{+}\), there exists infinitely many generalized periodic orbits in the system (), resp. (1.2) which accumulate at the attractive center.

The result is, in a sense, a typical result of the lunar regime: The Newtonian attraction is singular at the attractive center and therefore in a sufficiently small neighborhood eventually dominates the additional external regular force. Concretely, after embedding the system into the zero-energy hypersurface of an autonomous system and regularization, the generalized 1-periodic orbits of Eq. (1.2) are transformed into certain closed orbits of the regularized system in extended phase space. When the external force is disregarded, these closed orbits form an infinite sequence of periodic manifolds \(\Lambda _{n}\) for those with initial prime period 1/n in the unperturbed system. These periodic manifolds are non-degenerate and are distinguished not only by their prime periods but also by their action values. We shall show that, when the external force is added, some closed orbits bifurcating from \(\Lambda _{n}\) continue to exist, for all sufficiently large n. An estimate on their action values then shows that there are infinitely many of them. The persistence of closed orbits bifurcating from \(\Lambda _{n}\) for n large will be established via a local homotopy-stretching argument from Rabinowitz–Floer homology theory adapted to our situation. Indeed after a rescaling argument one may fix a periodic manifold and use the rescaling parameter to control the \(C^{1}\)-norm of the perturbations. In general this fails to bring a control for the \(C^{2}\)-norm of the perturbations and therefore in general Weinstein’s theorem from [26], which was used in [8], is not applicable in our setting. We consider this as an illustration of how recent developments of symplectic topology enrich the understanding of problems from classical and celestial mechanics.

The models (1.1) and (1.2) are simple general models which contain the well-studied models of the circular and elliptic restricted three-body problems in a fixed reference frame as special cases, see [24] and Appendix C. They already present many essential features of the planetary or lunar three-body problems seen as perturbations of two Keplerian elliptic motions. Theorem 1.1 therefore asserts that

Corollary 1.2

In any circular or elliptic restricted three-body problem with masses of the primaries \(m_{1}, m_{2}>0\), there exist infinitely many generalized periodic solutions accumulating at each of the primaries.

Moreover, Theorem 1.1 gives an indication that a similar phenomenon may as well happen in the non-restricted N-body problems in lunar regimes in which two particles are close to each other while other particles stay sufficiently far away. We shall leave this for further investigations.

We structure this article in the following way: In Sect. 2 we present two regularizations of the problem in the extended phase space, namely extensions of the Levi-Civita regularization for the 2-dimensional case and Moser regularization for all dimensions. We construct proper symplectic coordinates for the regularized systems and discuss the non-degeneracy condition of periodic manifolds of properly-degenerate systems and apply it in our setting to obtain the non-degeneracy in all dimensions in a unified way. In Sect. 3 we present a localization argument which allows us to further localize our analysis near the periodic manifolds \(\Lambda _{n}\). In Sect. 4 we further present a rescaling argument which allows us to transform the problem of perturbing infinitely many periodic manifolds into a problem of perturbing a fixed periodic manifold with perturbations with successively decreasing \(C^{1}\)-norms, which allows us to conclude with Theorem A.1 from the Appendix A, established via a local Rabinowitz Floer Homology argument. In Appendix B, we illustrate the construction of symplectic coordinates by proposing variants of the planar Delaunay variables in which the circular motions are regular. In Appendix C we realize circular and elliptic restricted three-body problems as time-periodically forced Kepler problems.

2 Regularizations of the forced Kepler problem in extended phase space

We deal with the more general system (1.2). By a standard trick we first transform the problem into an autonomous one by embedding the system into the zero-energy hypersurface of the augmented Hamiltonian \(F_{\varepsilon }(p, q, t)+ \tau \) defined on the extended phase space which is the exact symplectic manifold

$$\begin{aligned} \left( T^{*} ({\mathbb {R}}^{d} \setminus O)) \times T^{*} S^{1}, d \left( \sum _{i=1}^{d} p_{i} d q_{i} + \tau d t \right) \right) , \end{aligned}$$

in which \(\tau \) denotes the variable conjugate to t. The variable t is now considered as a space variable, which for now is identical to the (fictitious) time variable up to a shift by integers.

We now continue with two different approaches, the first one works for the 2-dimensional case and the second works for all dimensions.

2.1 Dimension 2: Levi-Civita regularization

2.1.1 Levi-Civita regularization

We change the fictitious time \(s \mapsto t\) according to \(ds/dt=|q|^{-1}\) on the zero-energy level

$$\begin{aligned} \{F_{\varepsilon }(p, q, t) + \tau =0\}. \end{aligned}$$

The resulting system has the same flow as the flow on the zero-energy level of the slowed-down Hamiltonian

$$\begin{aligned} \{|q|F_{\varepsilon }(p, q, t) + \tau |q|=0\}. \end{aligned}$$

We shall use the notation \('=\dfrac{d}{d s}\) to denote the derivative with respect to the new fictitious time s.

In dimension 2 (which contains the 1-dimensional system as a subcase), a regularization is given by further pulling-back the system by the contragradient of the complex square mapping [15, 19, 20]:

$$\begin{aligned} {\mathbb {C}}\setminus \{0\} \times {\mathbb {C}}\rightarrow {\mathbb {C}}\times {\mathbb {C}}, \quad (z, w) \mapsto \left( z^{2}, \dfrac{w}{2 {\bar{z}}}\right) . \end{aligned}$$

The unperturbed regularized Hamiltonian with \(\varepsilon =0\) is found to be

$$\begin{aligned} H_{0}(z, w, \tau , t){:=}\dfrac{|w|^{2}}{8} + \tau |z|^{2} -1. \end{aligned}$$

The regularized Hamiltonian with \(\varepsilon \in [0, 1]\) is found to be

$$\begin{aligned} H_{\varepsilon }(z, w, \tau , t){:=}\dfrac{|w|^{2}}{8} + \tau |z|^{2} +|z|^{2} \varepsilon U(z^{2}, t, \varepsilon ) -1. \end{aligned}$$

All these systems now extend regularly near the subset consisting of collisions \(\{z=0\}\) of their zero-energy hypersurfaces respectively in the regularized extended phase space \(\Bigl (T^{*} {\mathbb {C}}\times T^{*} S^{1}, d (\Re \{{\bar{w}} d z\}+ \tau d t)\Bigr )\). The extended systems are called Levi-Civita regularized systems. In the unperturbed regularized system \(H_{0}\), the variable \(\tau \) is a first integral. When \(\tau \) is fixed, \(H_{0}\) describes a pair of isotropic harmonic oscillators, i.e. a pair of harmonic oscillators with identical frequencies in the (z, w)-variables.

2.1.2 Action-angle variables

We now compute a set of action-angle variables for the unperturbed system given by the Hamiltonian \(H_{0}\). Write \(z=z_{1}+i z_{2}, w=w_{1}+i w_{2}\) and the Hamiltonian is written as

$$\begin{aligned} H_{0}{:=}(w_{1}^{2}+w_{2}^{2})/8+{\tau }(z_{1}^{2}+z_{2}^{2})-1. \end{aligned}$$

Based on the one-dimensional computation from [27], we may set

$$\begin{aligned} z_{1}=2^{-1/4} I_{1}^{1/2} \tau ^{-1/4} \cos \theta _{1}, \qquad w_{1}=-2 \cdot 2^{1/4} I_{1}^{1/2} \tau ^{1/4} \sin \theta _{1}, \\ z_{2}=2^{-1/4} I_{2}^{1/2} \tau ^{-1/4} \cos \theta _{2},\qquad w_{2}=-2 \cdot 2^{1/4} I_{2}^{1/2} \tau ^{1/4} \sin \theta _{2}, \end{aligned}$$
$$\begin{aligned} \tau =\tau , \qquad t={\tilde{t}}+4^{-1} \tau ^{-1} (I_{1} \sin 2 \theta _{1}+I_{2} \sin 2 \theta _{2}). \end{aligned}$$
(2.1)

Consequently, we set

$$\begin{aligned} I_{1}=\dfrac{{\mathcal {L}}+{\mathcal {J}}}{2}, I_{2}=\dfrac{{\mathcal {L}}-{\mathcal {J}}}{2}, \theta _{1}=\delta +\gamma , \theta _{2}=\delta -\gamma . \end{aligned}$$

It is seen that \(({\mathcal {L}}, \delta , {\mathcal {J}}, \gamma , \tau , {\tilde{t}})\) form a set of action-angle variables of \(H_{0}\). The pairs \(({\mathcal {L}}, \delta , \tau , {\tilde{t}})\) are the fast variables and \(H_{0}\) depends only on the fast actions \(({\mathcal {L}}, \tau )\):

$$\begin{aligned} H_{0}=\dfrac{\sqrt{2}}{2} {\mathcal {L}} \sqrt{\tau } -1. \end{aligned}$$

We observe that in a neighborhood of \(\{H_{0}=0\}\), the pair of fast action variables \(({\mathcal {L}}, \tau )\) are globally defined, which generates a free Hamiltonian \({\mathbb {T}}^{2}\)-action which we call the fast \({\mathbb {T}}^{2}\)-action. The fast angles \((\delta , {\tilde{t}})\) are angles associated to this \({\mathbb {T}}^{2}\)-action by our construction of \({\mathcal {J}}\) and \(\gamma \) as long as these latter variables are well-defined.

The set of variables \(({\mathcal {L}}, \delta , {\mathcal {J}}, \gamma , \tau , {\tilde{t}})\) covers a neighborhood of the zero energy hypersurface \(\{H_{0}=0\}\), and in particular each periodic manifold, only open-densely. Indeed it is seen that the variables \(({\mathcal {J}}, \gamma )\) are action-angle coordinates defined on a dense-open subset of the orbit space with fixed \({\mathcal {L}}\):

$$\begin{aligned} \Omega _{{\mathcal {O}}}^{{\mathcal {L}}} \cong S^{2}. \end{aligned}$$

A homeomorphism with \(S^{2}\) of the space \(\Omega _{{\mathcal {O}}}^{{\mathcal {L}}}\) of planar (possibly circular or rectilinear) ellipses with fixed semi-major axis and with a fixed focus has been explained e.g. in [1].

We now argue that we may cover a neighborhood of our energy hypersurface with a finite set of similar coordinate charts.

Indeed, proceeding with a symplectic reduction with respect to the fast Hamiltonian \({\mathbb {T}}^{2}\)-action, we see that the symmetry group SO(3) of the system of two isotropic harmonic oscillators acts in a Hamiltonian way on the orbit space \(\Omega _{{\mathcal {O}}}^{{\mathcal {L}}}\), which in this case corresponds to the natural action of SO(3) on \(S^{2}\). This then restricts to the Hamiltonian action of any \(S^{1}\)-subgroup of SO(3) on \(S^{2}\). This last \(S^{1}\)-action is everywhere free on \(\Omega _{{\mathcal {O}}}^{{\mathcal {L}}}\) outside two antipodal points, and extends the fast \({\mathbb {T}}^{2}\)-action to a Hamiltonian \({\mathbb {T}}^{3}\)-action on the regularized extended phase space. We may then apply the Arnold–Liouville theorem to the Lagrangian torus fibration associated to this Hamiltonian \({\mathbb {T}}^{3}\)-action which then gives us a set of action-angle variables locally around any Lagrangian torus within this fibration. By passing from the action-angle variables locally to Darboux coordinates for variables on \(\Omega _{{\mathcal {O}}}^{{\mathcal {L}}}\), we get a set of well-defined variables \(({\mathcal {L}}, \delta , \tau , {\tilde{t}}, \xi , \zeta )\), where \((\xi , \zeta )\) are Darboux coordinates defined in an open set \(\Omega _{{\mathcal {O}}}^{loc, {\mathcal {L}}}\) of \(\Omega _{{\mathcal {O}}}^{{\mathcal {L}}}\) which may be assumed to be small. Note that the angles \((\delta , {\tilde{t}})\) may have different zero-sections when compared to their analogues in the set of variables \(({\mathcal {L}}, \delta , {\mathcal {J}}, \gamma , \tau , {\tilde{t}})\) that we have previously constructed on overlap of charts, but this will not be a concern to us and we shall keep their notations unchanged in different charts. By taking images of \(\Omega _{{\mathcal {O}}}^{loc, {\mathcal {L}}}\) under the SO(3)-action we see that, since \(\Omega _{{\mathcal {O}}}^{{\mathcal {L}}}\) is compact on which SO(3) acts transitively, there exists a finite cover of \(\Omega _{{\mathcal {O}}}^{{\mathcal {L}}}\) by the SO(3)-images of \(\Omega _{{\mathcal {O}}}^{loc, {\mathcal {L}}}\), which then gives rise to finitely many charts of a neighborhood of the zero-energy hypersurface of \(\{H_{0}=0\}\) in the extended phase space by the SO(3)-invariance of the system.

We now aim at better understanding the dependence of \(\xi , \zeta \) on \({\mathcal {L}}\). By the SO(3)-invariance it is enough to understand this in a particular chart. But we see that with the initial set of variables \(({\mathcal {L}}, \delta , {\mathcal {J}}, \gamma , \tau , {\tilde{t}})\), the quantities \(({\mathcal {J}}/{\mathcal {L}}, \gamma )\) are functions defined on \(\Omega _{{\mathcal {O}}}=\Omega _{{\mathcal {O}}}^{1}\) (the space \(\Omega _{{\mathcal {O}}}^{{\mathcal {L}}}\) with \({\mathcal {L}}=1\)) independent of \({\mathcal {L}}\) and \(\tau \). Passing to Darboux coordinates and with the SO(3)-invariance we conclude that any set of Darboux coordinates \((\xi , \zeta )\) on the normalized orbit space \(\Omega _{{\mathcal {O}}}=\Omega _{{\mathcal {O}}}^{1}\) gives rise to a set \(({\mathcal {L}}, \delta , {\mathcal {J}}, \gamma , \tau , {\tilde{t}}, {\hat{\xi }}, {\hat{\zeta }})\) for \(({\hat{\xi }}, {\hat{\zeta }})=(\sqrt{{\mathcal {L}}} \xi , \sqrt{{\mathcal {L}}} \zeta )\).

We summarize this analysis in the following Proposition:

Proposition 2.1

Any Darboux coordinates \((\xi , \zeta )\) of the space \((\Omega _{{\mathcal {O}}}, \omega _{0})\) gives rise to a set of local symplectic coordinates \(({\mathcal {L}}, \delta , \tau , {\tilde{t}}, {\hat{\xi }}, {\hat{\zeta }})\) of \(T^{*} {\mathbb {C}}\times T^{*} S^{1}\) where \(({\hat{\xi }}, {\hat{\zeta }})=(\sqrt{{\mathcal {L}}} \xi , \sqrt{{\mathcal {L}}} \zeta )\). A neighborhood of the energy hypersurface \(\{H_{0}=0\}\) in the extended phases space is covered by finitely many such symplectic charts.

2.1.3 Periodic manifolds of the regularized Kepler problem in extended phase space

For a Hamiltonian system defined on an exact symplectic manifold

$$\begin{aligned} (M, \omega =d \lambda , H) \end{aligned}$$

we denote by \(\phi _{t}\) its flow. We let the set \(\hbox {Per}_{0}^{H} \subset M \times {\mathbb {R}}\) be the set of points \((m, \eta ) \in M \times {\mathbb {R}}\) such that

$$\begin{aligned} H(m)=0, d H(m) \ne 0, \phi _{\eta }(m)=m. \end{aligned}$$

It follows that for \((m, \eta ) \in \hbox {Per}_{0}^{H}\), the orbit \(\phi _{t} (m)\) is a periodic orbit of the system in \(H^{-1}(0)\) with period \(\eta \).

Following [26], a subset \(\Lambda \subset \hbox {Per}_{E}^{H}\) is called a periodic manifold if \(\Lambda \) is a submanifold of \(M \times {\mathbb {R}}\) and if the restriction of the projection \(\pi : M \times {\mathbb {R}}\mapsto M\) to \(\Lambda \) is an embedding. A periodic manifold is called non-degenerate if at any \((m, \eta ) \in \Lambda \) there holds

$$\begin{aligned} T_{m} \pi (\Lambda )=(D \phi _{\eta }-Id)^{-1} (\hbox {span}\{X_{H}(m)\}). \end{aligned}$$
(2.2)

It is under this non-degeneracy hypothesis that Weinstein’s theorem is applicable.

It is direct to check, see e.g. [8], that there always holds

$$\begin{aligned} T_{m} \pi (\Lambda ) \subset (D \phi _{\eta }-Id)^{-1} (\hbox {span}\{X_{H}(m)\}). \end{aligned}$$

Therefore the non-degeneracy condition is equivalent to state that the two spaces \(T_{m} \pi (\Lambda )\) and \((D \phi _{\eta }-Id)^{-1} (\hbox {span}\{X_{H}(m)\})\) have the same dimension.

We denote by \(\Lambda _{n}\) the periodic manifolds of the regularized Kepler problem \(H_{0}\) in extended phase space with prime period 1/n with respect to the initial time variable t.

Observe that for a periodic orbit, a period for t is automatically a period for \({\tilde{t}}\) as well. We may therefore assume that \({\tilde{t}}\) has prime period 1/n, with correspondingly \({\mathcal {L}}={\mathcal {L}}_{n}, \tau =\tau _{n}\). A direct computation shows that the minimal period for the angle \(\delta \) to go through \( \pi \) (instead of \(2 \pi \), since the Levi-Civita regularization mapping is two-to-one) is \(S_{n}= \pi \sqrt{2}\tau _{n}^{-1/2}\). Therefore by assumption

$$\begin{aligned} S_{n} \cdot {\tilde{t}}'({\mathcal {L}}_{n}, \tau _n)= \pi \sqrt{2}\tau _{n}^{-1/2}\cdot \dfrac{\sqrt{2}}{4} {\mathcal {L}}_{n} /\sqrt{\tau _{n}}=\dfrac{1}{2} \pi {\mathcal {L}}_{n} \tau _{n}^{-1}=1/n. \end{aligned}$$

Together with the energy condition

$$\begin{aligned} \dfrac{\sqrt{2}}{2} {\mathcal {L}}_{n} \sqrt{\tau }_{n}-1=0 \end{aligned}$$

we obtain

$$\begin{aligned} {\mathcal {L}}_{n}=2^{2/3} \pi ^{-1/3} n^{-1/3}, \qquad \tau _{n}=2^{-1/3} \pi ^{2/3} n^{2/3}, \end{aligned}$$
(2.3)

and

$$\begin{aligned} S_{n}=(2 \pi )^{2/3} n^{-1/3}. \end{aligned}$$

In the planar case, these manifolds have been analyzed in [8] in which it is verified that they are non-degenerate.

2.1.4 Non-degeneracy of integrable Hamiltonian in action-angle form in extended phase space

For the purpose of uniformly treating non-degeneracy conditions we now revisit this non-degeneracy with the help of the action-angle forms of an integrable (possibly properly-degenerate) Hamiltonian. We consider an N-degrees of freedom Hamiltonian function in extended phase space of the form

$$\begin{aligned} H(I_{1}, \ldots , I_{n}, \tau )=0, \end{aligned}$$

in which \(n \le N\), within the chart given by the symplectic variables

$$\begin{aligned} (I_{1}, \theta _{1}, \ldots , I_{n}, \theta _{n}, \tau , t, {\hat{\xi }}_{n+1}, {\hat{\zeta }}_{n+1}, \ldots , {\hat{\xi }}_{N}, {\hat{\zeta }}_{N}) \end{aligned}$$

containing the base point m, such that the first (\(n+1\))-pairs are (partial) action-angle variables. The fact that the Hamiltonian is independent of all the variables \(({\hat{\xi }}_{1}, {\hat{\zeta }}_{1}, \ldots , {\hat{\xi }}_{n}, {\hat{\zeta }}_{n})\) implies that the entire periodic orbit through m is contained in this symplectic chart. In particular, it is enough to investigate the non-degeneracy condition in each of these charts.

For a solution to be periodic, the frequency vector

$$\begin{aligned}&(\nu _{1}, \ldots , \nu _{n}, \nu _{\tau })\\&\quad :=\left( \dfrac{\partial H(I_{1}, \ldots , I_{n}, \tau )}{\partial I_{1}}, \ldots , \dfrac{\partial H(I_{1}, \ldots , I_{n}, \tau )}{\partial I_{n}}, \dfrac{\partial H(I_{1}, \ldots , I_{n}, \tau )}{\partial \tau }\right) \end{aligned}$$

need to satisfy that for all \(i=1, \ldots , n\), the corresponding frequency \(\nu _{i}\) is an integer multiple of \(\nu _{\tau }\) and \(\nu _{\tau }\ne 0\). This latter implies in particular that we may at least locally express \(\tau =\tau (I_{1}, \ldots , I_{n})\) as a function of \((I_{1}, \ldots , I_{n})\) by implicit function theorem.

We now consider a periodic manifold of the system given by the constraints \((I_{1}=I_{1}^{0}, \ldots , I_{n}=I_{n}^{0},\tau =\tau _{0})\). We set

$$\begin{aligned} \nu _{\tau }^{0}=\dfrac{\partial H(I_{1}, \ldots , I_{n}, \tau )}{\partial \tau } (I_{1}=I_{1}^{0}, \ldots , I_{n}=I_{n}^{0}). \end{aligned}$$

The tangent space at a point is of dimension \(2 N -n +2\) and is given by

$$\begin{aligned} \hbox {span}\{\partial _{\theta _{1}}, \ldots \partial _{\theta _{n}}, \partial _{t}, \partial _{{\hat{\xi }}_{n+1}}, \partial _{{\hat{\zeta }}_{n+1}}\cdots \partial _{{\hat{\xi }}_{N}}, \partial _{{\hat{\zeta }}_{N}}\}. \end{aligned}$$

The flow direction is given by the vector

$$\begin{aligned} \dfrac{\partial H}{\partial {I_{1}}}(I_{1}^{0}, \ldots , I_{n}^{0}, \tau ^{0}) \partial {\theta _{1}} + \cdots + \dfrac{\partial H}{\partial {I_{n}}}(I_{1}^{0}, \ldots , I_{n}^{0}, \tau ^{0}) \partial {\theta _{n}}+\nu _{\tau }^{0}\partial _{t}. \end{aligned}$$

Moreover we have \(\eta =(\nu _{\tau }^{0})^{-1}\) and the mapping \(\phi _{\eta }\) takes the form

$$\begin{aligned}&\phi _{\eta }: (I_{1}, \ldots I_{n}, \tau , \theta _{1}, \ldots \theta _{n}, t, {\hat{\xi }}_{n+1}, {\hat{\zeta }}_{n+1} \cdots {\hat{\xi }}_{N}, {\hat{\zeta }}_{N}) \\&\quad \mapsto (I_{1}, \ldots I_{n}, \tau , \theta _{1}+ ({\nu _{\tau }^{0}})^{-1} \dfrac{\partial H}{\partial {I_{1}}}(I_{1}, \ldots , I_{n}, \tau ) , \ldots \theta _{n} + ({\nu _{\tau }^{0}})^{-1} \dfrac{\partial H}{\partial {I_{n}}}(I_{1}, \ldots , I_{n}, \tau ),\\&\quad t+({\nu _{\tau }^{0}})^{-1} \dfrac{\partial H}{\partial \tau }(I_{1}, \ldots , I_{n}, \tau ), {\hat{\xi }}_{n+1}, {\hat{\zeta }}_{n+1}\cdots {\hat{\xi }}_{N}, {\hat{\zeta }}_{N}). \end{aligned}$$

Thus in matrix form we may write

$$\begin{aligned}&D \phi _{\eta }: \left( Id_{(2n+2) \times (2n+2)} + \begin{pmatrix} 0 &{} 0 \\ ({\nu _{\tau }^{0}})^{-1} \hbox {Hess}_{(I_{1}, \ldots I_{n}, \tau )} H(I_{1}^{0}, \ldots , I_{n}^{0}, \tau _{0}) &{} 0\end{pmatrix} \right) \\&\quad \times Id_{(2 N-2 n) \times (2 N-2 n)}. \end{aligned}$$

where we have denoted by

$$\begin{aligned} \hbox {Hess}_{(I_{1}, \ldots I_{n}, \tau )} H (I_{1}^{0}, \ldots , I_{n}^{0},\tau _{0}) \end{aligned}$$

the Hessian matrix of H with respect to \((I_{1}, \ldots , I_{n}, \tau )\) evaluated at the periodic manifold \((I_{1}=I_{1}^{0}, \ldots , I_{n}=I_{n}^{0}, \tau =\tau _{0})\). In view of (2.2) we may now conclude:

Proposition 2.2

The periodic manifold \((I_{1}=I_{1}^{0}, \ldots , I_{n}=I_{n}^{0})\) is non-degenerate if and only if the matrix

$$\begin{aligned} \hbox {Hess}_{I_{1}, \ldots I_{n}, \tau } H (I_{1}^{0}, \ldots , I_{n}^{0},\tau _{0}) \end{aligned}$$

is non-degenerate.

We remark that in the non-properly-degenerate case, this non-degeneracy condition corresponds to the Kolmogorov non-degeneracy condition well-known in KAM theory.

Applying this Proposition to the Levi-Civita regularized Kepler Hamiltonian

$$\begin{aligned} H_{0}=\dfrac{\sqrt{2}}{2} {\mathcal {L}} \sqrt{\tau } -1, \end{aligned}$$

we directly obtain the fact that

Proposition 2.3

The periodic manifolds \(\Lambda _{n}\) in the 2-dimensional Levi-Civita regularized Kepler problem are non-degenerate.

2.1.5 Computation of action values of periodic manifolds

On \(\{H_{0}=0\}\), the action \( {\int {\mathcal {L}} d \delta +\tau d {\tilde{t}}}\) of a corresponding periodic solution with prime period 1/n for a period with length 1 (both refer to the initial t-variable) is computed via the integral

$$\begin{aligned}&n \int _{0}^{S_{n}} ({\mathcal {L}}_{n} \delta '({\mathcal {L}}_{n},\tau _{n})+\tau _{n} {\tilde{t}}'({\mathcal {L}}_{n},\tau _{n})) ds \\&\quad =n \int _{0}^{2 \pi \sqrt{2}\tau _{n}^{-1/2}} \left( {\mathcal {L}}_{n} \dfrac{\sqrt{2}}{2} \sqrt{\tau }_{n}+\tau _{n} \dfrac{\sqrt{2}}{4} {\mathcal {L}}_{n} /\sqrt{\tau }_{n}\right) ds. \end{aligned}$$

Now substitute the corresponding values of \({\mathcal {L}}_{n}\) and \(\tau _{n}\) from (2.3), we find directly that

$$\begin{aligned} {\mathcal {A}}_{0}^{(n)}=3 \cdot 2^{-1/3} \pi ^{2/3} n^{2/3}={\mathcal {A}}_{0}^{(1)} n^{2/3}. \end{aligned}$$

From this we may state the following observations:

Proposition 2.4

It holds that

  • \({\mathcal {A}}_{0}^{(n)}\) is monotone in n;

  • \(\lim _{n \rightarrow \infty } {\mathcal {A}}_{0}^{(n)}=\infty \);

  • the gaps between the action values of \(\Lambda _{n+1}\) and \(\Lambda _{n}\) is

    $$\begin{aligned} {\mathcal {A}}_{0}^{(1)} ((n+1)^{2/3}-n^{2/3}) \sim n^{-1/3}. \end{aligned}$$

2.2 All dimensions: Moser regularization

2.2.1 Moser regularization

We recall a regularization of the Kepler problem and its regular perturbations in all dimensions now typically attributed to Moser [23].

We start with the Kepler problem

$$\begin{aligned} \left( T^{*} ({\mathbb {R}}^{d} \setminus \{0\}), \omega _{0}, \dfrac{\Vert p\Vert ^{2}}{2}-\dfrac{1}{\Vert q\Vert }\right) \end{aligned}$$

with coordinates \((q, p) \in {\mathbb {R}}^{d} \setminus \{0\} \times {\mathbb {R}}^{d}\). On the energy level \(-1/2\) we may change time and arrive at the zero-energy level of the system

$$\begin{aligned} \left( T^{*} ({\mathbb {R}}^{d} \setminus \{0\}), \omega _{0}, \dfrac{\Vert q\Vert (\Vert p\Vert ^{2}+1)}{2}-1=0\right) . \end{aligned}$$

Proceeding with the canonical change of variables

$$\begin{aligned} {\mathbb {R}}^{d} \times {\mathbb {R}}^{d} \rightarrow {\mathbb {R}}^{d} \mapsto {\mathbb {R}}^{d}, (-x, y) \mapsto (p,q), \end{aligned}$$

we arrive at the following system on its energy level 1

$$\begin{aligned} \dfrac{\Vert y\Vert (\Vert x\Vert ^{2}+1)}{2}=1. \end{aligned}$$

Following Moser we may treat this as the stereographic projection from the North Pole N of the system

$$\begin{aligned} (T^{*}_{+} (S^{d} \setminus N), \omega _{0}, \Vert v\Vert =1), \end{aligned}$$

where \(T^{*}_{+} S^{d}\) denotes the positive cotangent bundle in \(T^{*} S^{d}\), and v denotes the momentum variables. We see that this is equivalent to the system of geodesic flow

$$\begin{aligned} \left( T^{*}_{+} (S^{d} \setminus N), \omega _{0}, \dfrac{\Vert v\Vert ^{2}}{2}=\dfrac{1}{2} \right) . \end{aligned}$$

The set of collisions, or equivalently, the set corresponding to infinite velocities, are now transformed to the fibre \(T^{*}_{N} S^{d}\). Moser’s regularization is completed by the further addition of this fibre and the system thus defined

$$\begin{aligned} \left( T^{*}_{+} S^{d}, \omega _{0}, \dfrac{\Vert v\Vert ^{2}}{2}=\dfrac{1}{2} \right) \end{aligned}$$

is regular with great circle orbits with energy 1/2. A particle moves on such a great circle with unit velocity, therefore has period \(2 \pi \).

By properly rescaling the constant we arrive at a regularization for all negative energies. We now work out more details.

We take a d-dimensional centered sphere \(S_{r}^{d}\) with radius r in \({\mathbb {R}}^{d+1}\) and consider the plane \({\mathbb {R}}^{d} \times \{0\}\) to be the plane to be projected to from the north pole \({\mathcal {N}}=(0,\ldots ,0,r)\) of the sphere. Let

$$\begin{aligned} u=(u_{1}, \ldots , u_{d+1}) \in S_{r}^{d} \setminus {\mathcal {N}} \end{aligned}$$

be a point on this sphere different from \({\mathcal {N}}\). We let \(v=(v_{1}, \ldots , v_{d+1}) \in {\mathbb {R}}^{d+1}\) be a (co-)tangent vector at this point, so explicitly we have

$$\begin{aligned} u \ne {\mathcal {N}}, u\cdot u=r^{2}, u \cdot v=0. \end{aligned}$$

We now let \(x=(x_{1}, \ldots , x_{d}) \in {\mathbb {R}}^{d}\) be such that u projects to (x, 0) by stereographic projection from \({\mathcal {N}}\) and \(y=(y_{1}, \ldots , y_{d})\) be a (co-)vector at the point x, such that the 1-form

$$\begin{aligned} \sum _{j=1}^{d+1} v_{j} d u_{j} =\sum _{i=1}^{d} y_{i} d x_{i} \end{aligned}$$

is preserved.

We therefore have

$$\begin{aligned} x_{i}=\dfrac{r u_{i}}{r-u_{d+1}}, y_{i}=\dfrac{r-u_{d+1}}{r} v_{i} + \dfrac{u_{i} v_{d+1}}{r}, i=1,\ldots , d, \end{aligned}$$
(2.4)

with inverse

$$\begin{aligned} u_{i}= & {} \dfrac{2 r^{2} x_{i}}{r^{2}+\Vert x\Vert ^{2}}, u_{d+1}=\dfrac{\Vert x\Vert ^{2}-r^{2}}{\Vert x\Vert ^{2}+r^{2}}r, \\ v_{i}= & {} \dfrac{\Vert x\Vert ^{2}+r^{2}}{2 r^{2}} y_{i} - \dfrac{(x \cdot y) x_{i}}{r^{2}}, v_{d+1}=\dfrac{x \cdot y}{r}, i=1,\ldots , d. \end{aligned}$$

From these equalities it follows that

$$\begin{aligned} \Vert v\Vert =\left( \dfrac{\Vert x\Vert ^{2}+r^{2}}{2 r^{2}}\right) \Vert y\Vert . \end{aligned}$$

Therefore for the Kepler problem, once we fix a negative energy \(-f<0\), after a proper time reparametrization, we arrive at the system

$$\begin{aligned} \left( T^{*}_{+} ({\mathbb {R}}^{d} \setminus \{0\}), \omega _{0}, 2 f\dfrac{\Vert q\Vert (\Vert p\Vert ^{2}+2 f)}{2 \cdot (2f)}-1=0\right) , \end{aligned}$$

which after the symplectic change of variables \((-x, y) \mapsto (p,q)\) is the stereographic projection of the system

$$\begin{aligned} (T^{*}_{+} (S^{d}_{\sqrt{2 f}} \setminus \{{\mathcal {N}}\}), \omega _{0}, 2 f \Vert v\Vert =1). \end{aligned}$$

This is equivalent to the reparametrized geodesic flow system

$$\begin{aligned} \left( T^{*} (S^{d}_{\sqrt{2 f}} \setminus \{{\mathcal {N}}\}), \omega _{0}, \dfrac{(2 f \Vert v\Vert )^{2}}{2}=\dfrac{1}{2}\right) , \end{aligned}$$

so the orbits are great circles with radius \(\sqrt{2 f}\) and the velocities have norm 1/(2f). The period is therefore \(2 \pi (\sqrt{2 f})^{3}=4 \sqrt{2} \pi f^{3/2}\).

Unlike Levi-Civita and Kustaanheimo–Stiefel regularizations as have been used in [6, 8], this regularization does not create additional symmetry of the regularized system and is valid for all dimensions in a uniform way. On the other hand, here we have to work on curved space instead of Euclidean spaces. In [18], the relationship between the Levi-Civita/Kustaanheimo–Stiefel regularization and the Moser regularization has been established, and Moser regularization can be regarded in the planar case as a quotient of Levi-Civita regularization by a \({\mathbb {Z}}_{2}\)-symmetry, and in the spatial case as a symplectic quotient of the Kustaanheimo–Stiefel regularization by a Hamiltonian \(S^{1}\)-symmetry respectively.

2.2.2 Extension to the forced Kepler problem

Moser regularization extends to the forced Kepler problem in the extended phase space in the following way: Starting from the system

$$\begin{aligned} \left( T^{*} ({\mathbb {R}}^{d} \setminus \{0\}) \times T^{*} S^{1}, \omega _{0}, \dfrac{\Vert p\Vert ^{2}}{2}-\dfrac{1}{\Vert q\Vert } + \varepsilon U(q, t, \varepsilon )+\tau =0\right) \end{aligned}$$

with coordinates \((q, p, t, \tau ) \in {\mathbb {R}}^{d} \setminus \{0\} \times {\mathbb {R}}^{d} \times S^{1} \times {\mathbb {R}}\), we change time so that we obtain the system

$$\begin{aligned} \left( T^{*} ({\mathbb {R}}^{d} \setminus \{0\}), \omega _{0}, \dfrac{\Vert q\Vert (\Vert p\Vert ^{2}+2\tau )}{2}+ \varepsilon \Vert q\Vert U(q, t, \varepsilon )-1=0\right) . \end{aligned}$$

with energy zero. We set \((x, y) = (-p,q)\) as in previous discussions.

It follows from (2.4) that

$$\begin{aligned} \Vert p\Vert ^{2}=\Vert x\Vert ^{2}=\dfrac{2 \tau (2 \tau -u_{d+1}^{2})}{(\sqrt{2 \tau }-u_{d+1})^{2}}, \end{aligned}$$

and therefore

$$\begin{aligned} \dfrac{2\tau }{\Vert p\Vert ^{2}+2\tau }=\dfrac{\sqrt{2 \tau }-u_{d+1}}{2 \sqrt{2 \tau }}, \end{aligned}$$

which extends to a regular function on the sphere \(S^{d}_{\sqrt{2 \tau }}\). Therefore

$$\begin{aligned} \Vert y\Vert =\Vert q\Vert =\Vert v\Vert \dfrac{\sqrt{2 \tau }-u_{d+1}}{\sqrt{2 \tau }}. \end{aligned}$$
(2.5)

Therefore by stereographic projection the system extends to a system

$$\begin{aligned} \left( \{T^{*}_{+} (S^{d}_{\sqrt{2 \tau }}):(\tau , t) \in T^{*} S_{1}\}, \omega _{0}, 2 \tau \Vert v\Vert -1+ \varepsilon \Vert v\Vert \dfrac{\sqrt{2 \tau }-u_{d+1}}{ \sqrt{2 \tau }} U(q(v, u), t, \varepsilon ) =0 \right) \end{aligned}$$

for \(u \in S^{d}_{\sqrt{2 \tau }}\). The space is a (trivial) fibre bundle with fibres \(T^{*}_{+} (S^{d}_{\sqrt{2 \tau }})\) over \(T^{*} S^{1}\), with fibres depending on \(\tau \) and independent of t.

To normalize the fibres to be \(\tau \)-independent we make a symplectic rescaling

$$\begin{aligned} v={\tilde{v}} /\sqrt{2 \tau } , \qquad u=\sqrt{2 \tau } {\tilde{u}}, \end{aligned}$$

and accordingly take a shift of the t-variable

$$\begin{aligned} {\tilde{t}}=t-\phi (\tau , {\tilde{u}}, {\tilde{v}}) \end{aligned}$$

for some function \(\phi \) for the purpose of keeping the change of variables canonical. The function \(\phi \) is implicitly determined by the relationship (after lifting \(t, {\tilde{t}}\) to be defined on \({\mathbb {R}}\)):

$$\begin{aligned} {\sum _{i} v_{i} d u_{i}-t d \tau =\sum _{i} {\tilde{v}}_{i} d {\tilde{u}}_{i}-{\tilde{t}} d \tau .} \end{aligned}$$
(2.6)

We shall only need the form of \(\phi \) in local coordinates which we shall determine later.

We thus have

$$\begin{aligned} \Vert q\Vert =\Vert {\tilde{v}}\Vert \tau ^{-1/2} (1-u_{d+1}). \end{aligned}$$
(2.7)

The resulting system is equivalent to

$$\begin{aligned} (T^{*}_{+} S^{d}_{1} \times T^{*} S^{1}, \omega _{0}, \sqrt{2 \tau } \Vert {\tilde{v}}\Vert -1+\varepsilon \Vert {\tilde{v}}\Vert (1-{\tilde{u}}_{d+1}) (2 \tau )^{-1/2} U(q({\tilde{v}}/\sqrt{2 \tau }, u),{\tilde{t}}, \varepsilon ) =0). \end{aligned}$$

When \(\varepsilon =0\), we see that the corresponding energy hypersurface of the regularized Kepler problem is \(T^{*}_{(2 \tau )^{-1/2}} S^{d}_{1} \times T^{*} S^{1}\).

In the above system the perturbation term takes the form

$$\begin{aligned} \varepsilon U=\varepsilon \Vert {\tilde{v}}\Vert (1-{\tilde{u}}_{d+1}) (2 \tau )^{-1/2} U(q({\tilde{v}}/\sqrt{2 \tau }, u),{\tilde{t}}, \varepsilon ). \end{aligned}$$

In view of (2.5), (1.4) and the energy constraint, by passing to a local chart of the sphere \(S^{d}\) around the north pole given by \(({\tilde{u}}_{1}, \ldots , {\tilde{u}}_{d})\) and using the identity

$$\begin{aligned} 1-{\tilde{u}}_{d+1}=\dfrac{{\tilde{u}}_{1}^{2}+\cdots +{\tilde{u}}_{d}^{2}}{1+{\tilde{u}}_{d+1}} \end{aligned}$$

we get that in a sufficiently small neighborhood \(\{{\tilde{u}}_{1}^{2}+\cdots +{\tilde{u}}_{d}^{2} \le {\tilde{\varepsilon }}\},\) for some sufficiently small, \(0<{\tilde{\varepsilon }}<<1\) of the north pole, and for \(\tau \ge \tau _{*}>0\), the perturbation term \(\varepsilon U\) has the estimate

$$\begin{aligned} \Vert \varepsilon U\Vert \le {\tilde{C}}_{0} ({\tilde{u}}_{1}^{2}+\cdots +{\tilde{u}}_{d}^{2})^{2}, \end{aligned}$$

for some constant \({\tilde{C}}_{0}\) which depends only on \(\tau _{*}\) and \({\tilde{\varepsilon }}\).

From these discussions we conclude

Proposition 2.5

The perturbation term \(\varepsilon U\) is regular near the set of collisions and vanishes on this set together with its first, second and third derivatives.

2.2.3 A set of symplectic coordinates

As in the two-dimensional case, the space \(T^{*}_{+} S^{d}_{1} \times T^{*} S^{1}\) carries a free Hamiltonian \({\mathbb {T}}^{2}\)-action associated to the pair of variables \(({\mathcal {I}}, \tau )\), in which \({\mathcal {I}}\) is a moment map of the Hamiltonian \(S^{1}\)-action on \(T^{*}_{+} S^{d}_{1}\) given by the geodesic flow on \(S^{d}_{1}\).

When we restrict the system \(H_{0}\) with any fixed \(\tau >0\) to the cotangent bundle \(T^{*} S^{1}\) of a great circle \(S^{1}\) in \(S^{d}\), we observe that both \({\mathcal {I}}\) and \(\Vert {\tilde{v}}\Vert \) generates the same \(S^{1}\)-action on \(T^{*} S^{1}\), thus in this case \({\mathcal {I}}\) and \(\Vert {\tilde{v}}\Vert \) agree up to an additive constant, which may be discarded by shifting \({\mathcal {I}}\) by this constant. We may thus set \({\mathcal {I}}=\Vert {\tilde{v}}\Vert \) by the SO(d)-invariance of the system \(H_{0}\) . Therefore we have

$$\begin{aligned} H_{0}=\sqrt{2 \tau } {\mathcal {I}} -1. \end{aligned}$$

The symplectic quotient of the space \(T^{*}_{+} S^{d}_{1} \times T^{*} S^{1}\) by this free Hamiltonian \({\mathbb {T}}^{2}\)-action at a level with fixed \({\mathcal {I}}\) and with total energy zero is the space \(\Omega _{{\mathcal {O}}}^{{\mathcal {I}}} \cong T_{1}^{*} S^{d}/S^{1}\) that we refer to as the orbit space, which is a symplectic manifold of dimension \(2(d-1)\) on which \(SO(d+1)\) acts transitively in a Hamiltonian way.Footnote 1 We apply Darboux’ theorem locally on \(\Omega _{{\mathcal {O}}}^{{\mathcal {I}}}\), which ensures the existence of a set of local symplectic coordinates \(({\mathcal {I}}, \theta , \tau , {\tilde{t}}, {\hat{\xi }}_{1}, {\hat{\zeta }}_{1}, \ldots , {\hat{\xi }}_{{d-1}}, {\hat{\zeta }}_{{d-1}})\). The coordinates \(({\hat{\xi }}_{1}, {\hat{\zeta }}_{1}, \ldots , {\hat{\xi }}_{{d-1}}, {\hat{\zeta }}_{{d-1}})\) are defined on an open subset \(\Omega _{{\mathcal {O}}}^{{\mathcal {I}}, loc} \subset \Omega _{{\mathcal {O}}}^{{\mathcal {I}}}\) and by compactness of \(\Omega _{{\mathcal {O}}}^{{\mathcal {I}}}\) and transitivity of the action of \(SO(d+1)\) we conclude that we may cover \(\Omega _{{\mathcal {O}}}^{{\mathcal {I}}}\) with finitely many images of \(\Omega _{{\mathcal {O}}}^{{\mathcal {I}}, loc}\) under the action of \(SO(d+1)\).

In the case \(d=2\), by comparing both forms of the regularized Hamiltonians by Moser and Levi-Civita regularizations respectively and viewing the 2-to-1 cover of the angles, we find that

$$\begin{aligned} {\mathcal {I}}={\mathcal {L}}/2, \qquad \theta =2 \delta . \end{aligned}$$

We denote by \(\Omega _{{\mathcal {O}}}=\Omega _{{\mathcal {O}}}^{1}\) and call it the normalized orbit space. As in the \(d=2\) case with Levi-Civita regularization, we have the following:

Proposition 2.6

Any Darboux coordinates \((\xi _{i}, \zeta _{i})\) on \((\Omega _{{\mathcal {O}}}, \omega _{0})\) and the symplectic form of \((T^{*}_{+} (S^{d}_{1} \setminus \{{\mathcal {N}}\}) \times T^{*} ({\mathbb {R}}/ {\mathbb {Z}})\) can be written as

$$\begin{aligned} d {\mathcal {I}} \wedge d \theta + d \tau \wedge d {\tilde{t}} + \sum _{i=1}^{d-1} d (\sqrt{ {\mathcal {I}}} \xi _{i}) \wedge d (\sqrt{ {\mathcal {I}}} \zeta _{i})=d {\mathcal {I}} \wedge d \theta + d \tau \wedge d {\tilde{t}} + \sum _{i=1}^{d-1} d {\hat{\xi }}_{i} \wedge d {\hat{\zeta }}_{i}, \end{aligned}$$

for \(({\hat{\xi }}_{i}, {\hat{\zeta }}_{i})=(\sqrt{{\mathcal {I}}} \xi _{i},\sqrt{{\mathcal {I}}} \zeta _{i})\).

In this set of Darboux coordinates we have from (2.7) that

$$\begin{aligned} q_{i} ={\mathcal {I}} \tau ^{-1/2} f_{i}(\theta , \xi _{i}, \zeta _{i}) \end{aligned}$$

for some regular functions \(f_{i}\) independent of \({\mathcal {I}}\) and \(\tau \). Consequently we may write

$$\begin{aligned} |q|U(q, t)={\mathcal {I}}^{2} \tau ^{-1} {\tilde{U}}({\mathcal {I}}, \theta , \xi _{i}, \zeta _{i}, t), \end{aligned}$$

where by (1.4), the function \( {\tilde{U}}({\mathcal {I}}, \theta , \xi _{i}, \zeta _{i}, t)\) satisfies

$$\begin{aligned} {\tilde{U}}( {\mathcal {I}}, \theta , \xi _{i}, \zeta _{i}, t)= {\hat{U}}( \theta , \xi _{i}, \zeta _{i}, t) + O({\mathcal {I}} \tau ^{-1/2}). \end{aligned}$$

Moreover, since the functions \({\tilde{t}}, t, {\mathcal {I}}, \tau \) are SO(d)-invariant, by comparing to the formula (2.1) in the 2-dimensional case, we conclude in the same way that the change of time variable takes the form

$$\begin{aligned} {\tilde{t}}=t+{\mathcal {I}} \tau ^{-1} {\tilde{\phi }}(\theta , \xi _{i}, \zeta _{i}) \end{aligned}$$
(2.8)

for some smooth function \({\tilde{\phi }}\).

To be in consistence in the 2-dimensional case with Levi-Civita regularization, we may as well further set

$$\begin{aligned} {\mathcal {I}}={\mathcal {L}}/2, \theta =2 \delta . \end{aligned}$$

The unperturbed Hamiltonian takes the form

$$\begin{aligned} H_{0}=\dfrac{\sqrt{2}}{2} {\mathcal {L}} \sqrt{\tau } -1 \end{aligned}$$

which is the same as the unperturbed Levi-Civita regularized system in dimension 2. This expression is now valid in all dimensions.

2.2.4 Periodic manifolds of the Moser-regularized Kepler problem and their non-degeneracy

In terms of the variables \(({\mathcal {L}}, \tau )\), the periodic manifold \(\Lambda _{n}\) of the Moser-regularized Kepler problem with prime period 1/n is characterized by the condition

$$\begin{aligned} \{{\mathcal {L}}={\mathcal {L}}_{n}, \tau =\tau _{n}\}. \end{aligned}$$

Since all the changes of variables are canonical, the action value is constant on \(\Lambda _{n}\). By restricting to the 2-dimensional case, we conclude that the corresponding action value is \({\mathcal {A}}_{0}^{(n)}\) which has been computed in Sect. .

Applying Proposition 2.2 we conclude that

Proposition 2.7

For any \(d \in {\mathbb {Z}}_{+}\), the periodic manifolds \(\Lambda _{n}\) of the Moser-regularized Kepler problem in extended phase space are non-degenerate.

3 Localization in space

In this section we shall explain a localization procedure of the regularized system. The argument is presented with the Levi-Civita regularized system in dimension 2 for fixing the ideas, but it works for the Moser regularized system in any dimension without much change, which we shall explain at the end of this section.

3.1 The L.C. regularized systems in action-angle form

From previous discussions we have seen that the regularized extended Hamiltonian is written in a proper chart as

$$\begin{aligned} H_{\varepsilon }({\mathcal {L}}, \delta , \xi , \zeta , \tau , t, \varepsilon ){:=} \dfrac{\sqrt{2}}{2} {\mathcal {L}}\sqrt{\tau }-1+ \varepsilon {\mathcal {L}}^{2} \tau ^{-1} \cdot {\tilde{U}}({\mathcal {L}}, \delta , \xi , \zeta , t, \varepsilon ), \end{aligned}$$
(3.1)

in which the function

$$\begin{aligned} {\tilde{U}}({\mathcal {L}}, \delta , \xi , \zeta , t, \varepsilon ) \end{aligned}$$

is \(C^{\infty }\) regular in all of its variables and depends continuously on the parameter \(\varepsilon \in [0, 1]\). Moreover, we have that

$$\begin{aligned} {\tilde{U}}({\mathcal {L}}, \delta , \xi , \zeta , t, \varepsilon )= {\hat{U}}(\delta , \xi , \zeta ,\tau , t, \varepsilon ) + O({\mathcal {L}} \tau ^{-1/2}). \end{aligned}$$

A priori the terms in \(O({\mathcal {L}} \tau ^{-1/2})\) may depend on all of the listed variables.

In order to work within the corresponding symplectic chart \(({\mathcal {L}}, \delta , {\hat{\xi }}, {\hat{\zeta }}, \tau , t),\) we shall treat \(\xi , \zeta , t\) as functions defined in this chart given by the following expressions

$$\begin{aligned} \xi ={\mathcal {L}}^{-1/2} {\hat{\xi }},\quad \zeta ={\mathcal {L}}^{-1/2} {\hat{\zeta }}, \quad t ={\tilde{t}}+{\mathcal {L}} \tau ^{-1} {\tilde{\phi }} (\xi , \zeta ), \end{aligned}$$
(3.2)

in which \({\tilde{\phi }}\) is a regular function of \((\xi , \zeta )\) as defined in (2.1).

Recall that we use \('=\dfrac{d}{ds}\) to denote the time derivative of a quantity with respect to the reparametrized fictitious time s.

The non-trivial Hamiltonian equations associated to the unperturbed system \(H_{0}=\dfrac{\sqrt{2}}{2} {\mathcal {L}} \sqrt{\tau }-1\) reads

$$\begin{aligned} \delta '=\dfrac{\sqrt{2}}{2} \sqrt{\tau },\,\, {\tilde{t}}'=\dfrac{\sqrt{2}}{4}{\mathcal {L}}/ \sqrt{\tau }, \end{aligned}$$

with other variables conserved. Moreover the 0-energy hypersurface is given by the relationship \(\dfrac{\sqrt{2}}{2}{\mathcal {L}} \sqrt{\tau }-1=0\), or equivalently \(\tau =2 {\mathcal {L}}^{-2}\). The equation for \({\tilde{t}}\) thus becomes \({\tilde{t}}'=\dfrac{1}{4} {\mathcal {L}}^{2}\). Recall that the negative of the Keplerian energy satisfies \(\tau =\dfrac{1}{2 a}\), where a is the semi-major axis of the instantaneous Keplerian elliptic orbit. We therefore have \({\mathcal {L}}=2 \sqrt{a}\).

3.2 The \(\kappa \)-localization

In this section we restrict the system to a fixed open set

$$\begin{aligned} {\mathcal {U}}{:=}\{0<{\mathcal {L}} < {\tilde{L}}_{*}, \tau > {\tilde{\tau }}_{*}\} \end{aligned}$$

of the regularized extended phase space, for some fixed \(L_{*}>0\) and \({\tilde{\tau }}_{*}>0\). The argument below shows that in (1.2), when a and \(\tau ^{-1}\) are small enough, any orbit of \(H_{\varepsilon }\) in \(H^{-1}_{\varepsilon }(0)\) starting with such semi major axis a and \(\tau \) will stay in this neighborhood for the fictitious time \(s \in [0, S]\), in which S is the fictitious time required for the \({\tilde{t}}\)-variable to change from 0 to 1. All the norms of U (or rather \({\tilde{U}}\)) and its partial derivatives, which depend also on the external parameter \(\varepsilon \in [0, 1]\) are taken as \(L^{\infty }\) norms in \({\mathcal {U}} \times [0, 1]\). By properly shrinking the symplectic chart when necessary, we may assume that these norms all take bounded values.

We set \({\mathcal {L}}(0)=\kappa \) for \(\kappa >0\) sufficiently small. We shall explain in this subsection that any sufficiently small \(\kappa \) gives a localized system which gives several a priori bounds on the corresponding periodic solutions of period 1 in the t variable.

First, we have the energy constraint

$$\begin{aligned} \dfrac{\sqrt{2}}{2} \kappa \sqrt{\tau (0)} + \kappa ^{2} \tau ^{-1}(0) \varepsilon {\tilde{U}}=1. \end{aligned}$$
(3.3)

When \(\lim _{\kappa \rightarrow 0} \sqrt{\tau (0)} \kappa = 0\), it follows from this energy constraint that \(\tau (0) \sim \kappa ^{2} \varepsilon U\) which is not allowed for small \(\kappa \) by our definition of the neighborhood \({\mathcal {U}}\).

We therefore necessarily have \(\lim _{\kappa \rightarrow 0} \sqrt{\tau (0)} \kappa \ne 0\). We then deduce from (3.3) that

$$\begin{aligned} \lim _{\kappa \rightarrow 0} \sqrt{\tau (0)} \kappa =\sqrt{2}, \end{aligned}$$

therefore \(\dfrac{1}{\sqrt{\tau (0)}}=\dfrac{\sqrt{2}}{2} \kappa +o(\kappa )\).

We thus have \(\kappa ^{2} \tau ^{-1}(0)=O(\kappa ^{4})\). By plugging these into (3.3), we find that actually \(\dfrac{1}{\sqrt{\tau (0)}}=\dfrac{\sqrt{2}}{2} \kappa +O(\kappa ^{3})\). Thus

$$\begin{aligned} \dfrac{\sqrt{2}}{\sqrt{ \tau (0)}} \in (\kappa -\kappa ^{2}, \kappa +\kappa ^{2}) \end{aligned}$$

for \(\kappa \) sufficiently small.

The region

$$\begin{aligned} \left\{ {\mathcal {L}}(s), \dfrac{\sqrt{2}}{\sqrt{\tau (s)}} \in (\kappa -\kappa ^{3/2}, \kappa +\kappa ^{3/2})\right\} \end{aligned}$$

on \(\{H_{\varepsilon }=0\}\) is referred to as the \(\kappa \)-localization. We shall show that

Lemma 3.1

When \(\kappa \) is sufficiently small, any 0-energy solution of \(H_{\varepsilon }\) with \({\mathcal {L}}(0)=\kappa \) lies entirely in the \(\kappa \)-localization for \(s \in [0, S]\) such that \(\int _{0}^{S} {\tilde{t}}' d s=1\).

We take these as a localization ansatz for the following estimates.

To prove Lemma 3.1, we shall need the weaker inclusion

$$\begin{aligned} {\mathcal {L}}(s), \dfrac{\sqrt{2}}{\sqrt{\tau (s)}} \in \left( \dfrac{1}{2} \kappa , \dfrac{3}{2} \kappa \right) , s \in [0, S], \end{aligned}$$

which we refer to as the weak localization ansatz. When the weak localization ansatz holds, we have that for sufficiently small \(\kappa \) for all \(\varepsilon \in [0, 1]\), that

$$\begin{aligned} {\mathcal {L}}^{'}=-{\mathcal {L}}^{2} \tau ^{-1} \varepsilon \left( \dfrac{\partial {\tilde{U}}}{\partial \delta }+\dfrac{\partial {\tilde{U}}}{\partial t} \dfrac{\partial t}{\partial \delta }\right) . \end{aligned}$$

Note that \(\dfrac{\partial t}{\partial \delta }={\mathcal {L}} \tau ^{-1} \dfrac{\partial {\tilde{\phi }}}{\partial \delta }=O(\kappa ^{3})\), therefore there exists \(C_{1}>0\) such that

$$\begin{aligned} |{\mathcal {L}}'| \le \dfrac{3^{4}}{2^{5}}\kappa ^{4} \left( \Vert \dfrac{\partial {\tilde{U}}}{\partial \delta } \Vert _{\infty }+ \dfrac{3^{3}}{2^{4}}\kappa ^{3} \Vert \dfrac{\partial {\tilde{U}}}{\partial t}\Vert _{\infty } \Vert \dfrac{\partial {\tilde{\phi }}}{\partial \delta }\Vert _{\infty }\right) \le C_{1} \kappa ^{4}. \end{aligned}$$

We also have

$$\begin{aligned} \delta '=\dfrac{\sqrt{2}}{2} \sqrt{\tau } + {\mathcal {L}} \tau ^{-1} \varepsilon \left( 2 {\tilde{U}} +{\mathcal {L}} \dfrac{\partial {\tilde{U}}}{\partial {\mathcal {L}}}\right) . \end{aligned}$$

Note that

$$\begin{aligned} \dfrac{\partial {\tilde{U}}}{\partial {\mathcal {L}}}=\dfrac{\partial {\hat{U}}}{\partial t} \dfrac{\partial t}{\partial {\mathcal {L}}}+O(\tau ^{-1/2})=O(\kappa ), \end{aligned}$$

thus for sufficiently small \(\kappa \) there holds

$$\begin{aligned} |\delta '| \le 3 \kappa ^{-1}. \end{aligned}$$

Next we consider \({\hat{\xi }}^{'}\) and \({\hat{\zeta }}'\). We have

$$\begin{aligned} {\hat{\xi }}^{'}=-{\mathcal {L}}^{2} \tau ^{-1} \varepsilon \left( \dfrac{\partial {\tilde{U}}}{\partial \zeta }+\dfrac{\partial {\tilde{U}}}{\partial t} \dfrac{\partial t}{\partial \zeta }\right) \dfrac{\partial \zeta }{\partial {\hat{\zeta }}}, \end{aligned}$$

therefore

$$\begin{aligned} |{\hat{\xi }}'| \le \dfrac{3^{4}}{2^{3}}\kappa ^{7/2} \left\| \dfrac{\partial {\tilde{U}}}{\partial \zeta }+\dfrac{\partial {\tilde{U}}}{\partial t} \dfrac{\partial t}{\partial \zeta } \right\| _{\infty } {:=} C_{2} \kappa ^{7/2}, C_{2}>0; \end{aligned}$$

similarly

$$\begin{aligned} {\hat{\zeta }}^{'}={\mathcal {L}}^{2} \tau ^{-1} \varepsilon \left( \dfrac{\partial {\tilde{U}}}{\partial \xi }+\dfrac{\partial {\tilde{U}}}{\partial t} \dfrac{\partial t}{\partial \xi }\right) \dfrac{\partial \xi }{\partial {\hat{\xi }}}, \end{aligned}$$

thus

$$\begin{aligned} |{\hat{\xi }}'| \le \dfrac{3^{4}}{2^{3}}\kappa ^{7/2} \left\| \dfrac{\partial {\tilde{U}}}{\partial \xi }+\dfrac{\partial {\tilde{U}}}{\partial t} \dfrac{\partial t}{\partial \xi } \right\| _{\infty } {:=} C_{3} \kappa ^{7/2}, C_{3}>0. \end{aligned}$$

Next we have

$$\begin{aligned} \tau '=-{\mathcal {L}}^{2} \tau ^{-1} \varepsilon \dfrac{\partial {\tilde{U}}}{\partial t} \dfrac{\partial t}{\partial {\tilde{t}}} \hbox { thus } |\tau '| \le \dfrac{3^{4}}{2^{4}}\kappa ^{4} \left\| \dfrac{\partial {\tilde{U}}}{\partial {\tilde{t}}} \right\| _{\infty } {:=} C_{4} \kappa ^{4}, C_{4}>0. \end{aligned}$$

The right hand sides of the equations for \(\mathcal {L'}\) and \(\tau '\) are both of order \(O(\kappa ^{4})\). Therefore in view of our estimates for \({\mathcal {L}}(0) \) and \(\tau (0)\), we conclude that for sufficiently small \(\kappa \), the weak localization ansatz holds for all \(s \in [0, \kappa ^{-3}]\).

Moreover, the equation for \({\tilde{t}}'\) takes the form:

$$\begin{aligned} {\tilde{t}}'=\dfrac{\sqrt{2}}{4} \dfrac{{\mathcal {L}}}{\sqrt{\tau }}-{\mathcal {L}}^{2} \tau ^{-1} \varepsilon \left( \tau ^{-1}{\tilde{U}}-\dfrac{\partial {\tilde{U}}}{\partial \tau }\right) . \end{aligned}$$
(3.4)

Note that \(\dfrac{\partial {\tilde{U}}}{\partial \tau }=\dfrac{\partial {\tilde{U}}}{\partial t} \dfrac{\partial t}{\partial \tau }\) is of order \({\mathcal {L}} \tau ^{-2}\) by (3.2). Therefore the last term in the right hand side of the above formula is of order \(O(\kappa ^{6})\).

We therefore have

$$\begin{aligned} |{\tilde{t}}'-\dfrac{\kappa ^{2}}{4} | \le C_{5 }\kappa ^{6}. \end{aligned}$$

The time S is given implicitly by the integral \(\int _{0}^{S} {\tilde{t}}' ds=1\). It follows from the above estimate for \({\tilde{t}}'\) that

$$\begin{aligned} |S-4 \kappa ^{-2}| \le C_{6} \kappa ^{2}, C_{6} >0. \end{aligned}$$

In particular the weak localization ansatz is valid for all \(s \in [0, S]\) for \(\kappa \) sufficiently small. We may therefore estimate

$$\begin{aligned} \max _{s \in [0, S]}|{\mathcal {L}}(s)-{\mathcal {L}}(0)| \le \int _{0}^{S} |{\mathcal {L}}'| d s \le C_{1} \kappa ^{4} \cdot 5 \kappa ^{-2}=5 C_{1} \kappa ^{2}, \end{aligned}$$

and therefore for \({\mathcal {L}}(0)=\kappa \) it holds that

$$\begin{aligned} {\mathcal {L}}(s) \in (\kappa -\kappa ^{3/2},\kappa +\kappa ^{3/2}), \qquad \hbox { for } s \in [0, S] \end{aligned}$$

for \(\kappa \) sufficiently small. Likewise, for \(\dfrac{\sqrt{2}}{\sqrt{{\tau }(0)}} \in (\kappa -\kappa ^{3}, \kappa +\kappa ^{3})\) it holds that

$$\begin{aligned} \dfrac{\sqrt{2}}{\sqrt{{\tau }(s)}} \in (\kappa -\kappa ^{3/2}, \kappa +\kappa ^{3/2}), \qquad s \in [0, S] \end{aligned}$$

for \(\kappa \) sufficiently small. This proves Lemma .

Moreover, we have

$$\begin{aligned} |{\hat{\xi }}(0)|, |{\hat{\zeta }}(0)| = O(\sqrt{{\mathcal {L}}})=O(\kappa ^{1/2}) \end{aligned}$$

and

$$\begin{aligned}&\max _{s \in [0, S]}|{\hat{\xi }}(s)-{\hat{\xi }}(0)| \le \int _{0}^{S} |{\hat{\xi }}'| d s \le C_{2} \kappa ^{7/2} \cdot 5 \kappa ^{-2}=5 C_{2} \kappa ^{3/2}, \\&\max _{s \in [0, S]}|{\hat{\zeta }}(s)-{\hat{\zeta }}(0)| \le \int _{0}^{S} |{\hat{\zeta }}'| d s \le C_{3} \kappa ^{7/2} \cdot 5 \kappa ^{-2}=5 C_{3} \kappa ^{3/2}. \end{aligned}$$

Thus for sufficiently small \(\kappa >0\) we may indeed argue within a local symplectic chart as we do now.

The action functional along a solution curve (not necessarily periodic) from \({\tilde{t}}=0\) to \({\tilde{t}}=1\) with energy zero in these variables is computed as

$$\begin{aligned} {\mathcal {A}}{:=}\int _{0}^{S} ({\mathcal {L}} \delta '+ {\hat{\xi }} {\hat{\zeta }}'+\tau {\tilde{t}}' )d s. \end{aligned}$$

For the unperturbed system \(H_{0}\), if we write \(\kappa \) in place of \({\mathcal {L}}={\mathcal {L}}_{n}=2^{2/3} \pi ^{-1/3} n^{-1/3}\), we obtain the order estimate \({\mathcal {A}}_{0} \sim \kappa ^{-2}\).

With the estimates established above, we also have that

Lemma 3.2

When \(\kappa >0\) is sufficiently small, it holds that \({\mathcal {A}}_{\varepsilon } \sim \kappa ^{-2}\) for any \(\varepsilon \in [0, 1].\)

Note that due to the additional shift in the transformation \(t \mapsto {\tilde{t}}\), in general it does not hold that

$$\begin{aligned} t(S)-t(0)=1 \Leftrightarrow {\tilde{t}}(S)-{\tilde{t}}(0)=1. \end{aligned}$$

Nevertheless it is enough to observe from (3.2) that this equivalence holds for S-periodic solutions (in the time variable s) of the regularized system.

We therefore draw the following conclusions for periodic orbits of the forced system with t-period 1, which justify that our analysis is valid in any prescribed small neighborhood of the origin as long as \(\kappa >0\) is sufficiently small.

Proposition 3.3

For all \(\varepsilon \in [0, 1]\), when \(\kappa >0\) is sufficiently small, the action \({\mathcal {A}}_{\varepsilon }\) and \(\tau \) as computed along a periodic solution from an initial value with \({\mathcal {L}}(0)=\kappa \) for the time interval [0, S] such that

$$\begin{aligned} t(0)=0, t(S)=1 \end{aligned}$$

are of the same order \(\kappa ^{-2}\). The quantity |q(s)| is of the order \(O(\kappa ^{2})\) for \(s \in [0, S]\), and is thus confined to a sufficiently small neighborhood of the origin.

In particular it is enough for our analysis to have that the function U(q, t) or \(U(q, t, \varepsilon )\) is regularly defined in a sufficiently small neighborhood of the origin.

3.3 Extension to Moser-regularized systems in all dimensions

To show the validity of Lemmas 3.1, 3.2 and Proposition 3.3 within the context of Moser regularization in any dimension, we observe that in dimension 2 this is obtained by simply going through the 2-to-1 cover.

In higher dimensions, with the set of symplectic variables as constructed in Sect. 2.2.3, we see that the estimates for the variables \(({\mathcal {I}}, \theta , \tau , {\tilde{t}})\) are valid in the same way in all dimensions, and the additional variables \(({\hat{\xi }}_{i}, {\hat{\zeta }}_{i})\) satisfies the same estimates as the pair \(({\hat{\xi }}, {\hat{\zeta }})\) in the Levi-Civita regularized system. With this argument we conclude that

Proposition 3.4

Lemmas 3.1, 3.2 and Proposition 3.3 hold true for the Moser regularized system of (1.2) in any dimensions.

4 Rescalings of periodic manifolds and proof of Theorem 1.1

4.1 The Rabinowitz action functional and rescaling

Let \((X, \omega = d \lambda )\) be an exact symplectic manifold. Set

$$\begin{aligned} ({\hat{X}}, {\hat{\omega }}=d {\hat{\lambda }})=(X \times T^{*} S^{1}, \omega \oplus d \tau \wedge d t), \end{aligned}$$

where we have denoted by \((\tau , t)\) the variables in \(T^{*} S^{1}\). On \(({\tilde{X}}, {\tilde{\omega }})\) is defined a Hamiltonian \(H \in C^{\infty } ({\hat{X}}, {\mathbb {R}})\). By a periodic orbit we mean a closed orbit of \(({\hat{X}}, {\hat{\omega }}, H)\) with energy 0 along which the t-variable winds exactly once in the corresponding \(S^{1}\)-factor.

Let \(u \in C^{\infty } (S^{1}, {\hat{X}})\) be a smooth loop in \({\hat{X}}\), where \(S^{1}={\mathbb {R}}/ {\mathbb {Z}}\). In this subsection, we normalize the (fictitious) time along a closed orbit in the extended phase space to have period 1 in this section, and denote it by \({\underline{t}}\). The (original, non-normalized) fictitious time in the extended phase space will not appear in this section.

For \((u, \eta ) \in C^{\infty } (S^{1}, {\hat{X}}) \times (0, \infty )\), the Rabinowitz action functional of u with respect to a Hamiltonian function H is defined as

$$\begin{aligned} {\mathcal {A}}^{H} (u, \eta )=-\int _{0}^{1} u^{*} {\hat{\lambda }} +\eta \int _{0}^{1} H(u) d \underline{t}, \end{aligned}$$

with critical point equations

$$\begin{aligned} \dfrac{d u}{d {\underline{t}}}=\eta X_{H}(u), \qquad \int _{0}^{1} H(u)d \underline{t} =0 \Leftrightarrow H(u) \equiv 0. \end{aligned}$$

In other words, the critical points of the Rabinowitz action functional are 0-energy periodic orbits of H with period \(\eta \).

A critical manifold, i.e. a manifold consisting of critical points of the action functional, is called Morse–Bott if at each point the kernel of the Hessian of the action functional agrees with the tangent space at the point. An explicit computation of the kernel of the Hessian of the Rabinowitz action functional in [11, Chapter 7, Section 3] directly shows that

Proposition 4.1

A non-degenerate periodic manifold in the sense of Condition (2.2) is exactly a Morse–Bott critical manifold of the associated Rabinowitz action functional.

Therefore, in any dimensions, the periodic manifolds \(\Lambda _{n}\) of \(H_{0}\) are Morse–Bott critical manifolds.

We now investigate how the Rabinowitz action functional and its critical point equations change under rescaling. We take a conformal symplectic mapping

$$\begin{aligned} \phi _{\kappa }: {\hat{X}} \rightarrow {\hat{X}} \end{aligned}$$

such that

$$\begin{aligned} {\tilde{\lambda }}=\kappa ^{2} \phi _{\kappa }^{*} {\hat{\lambda }}, \qquad {\tilde{\omega }}=\kappa ^{2} \phi _{\kappa }^{*} {\hat{\omega }} \end{aligned}$$

for \(\kappa >0\).

We set

$$\begin{aligned} {\tilde{u}}{:=}\phi _{\kappa }^{-1} \circ u \in C^{\infty }(S^{1}, {\hat{X}}). \end{aligned}$$

We set \({\tilde{H}}_{\kappa }({\tilde{u}}): =\phi _{\kappa }^{*} H(u)\). Accordingly we define \(\eta =\kappa ^{-2} {\tilde{\eta }}\).

We compute

$$\begin{aligned} \int _{0}^{1} u^{*} {\hat{\lambda }}=\int _{0}^{1} (\phi _{\kappa } \circ {\tilde{u}})^{*} (\kappa ^{-2} {\phi _{\kappa }^{-1}}^{*}{\tilde{\lambda }})=\kappa ^{-2} \int _{0}^{1} {\tilde{u}}^{*} {\tilde{\lambda }}. \end{aligned}$$

Thus we have that

$$\begin{aligned} {\mathcal {A}}^{H} (u, \eta )=-\kappa ^{-2} \int _{0}^{1} {\tilde{u}}^{*} {\tilde{\lambda }} + \kappa ^{-2} {\tilde{\eta }} \int _{0}^{1} {\tilde{H}}_{\kappa }({\tilde{u}}) d \underline{t}, \end{aligned}$$

or equivalently \({\mathcal {A}}^{H}(u, \eta )=\kappa ^{2} {\mathcal {A}}^{{\tilde{H}}_{\kappa }} ({\tilde{u}}, {\tilde{\eta }})\) with critical point equations

$$\begin{aligned} \dfrac{d}{d {\underline{t}}} {\tilde{u}}={\tilde{\eta }} X_{{\tilde{H}}_{\kappa }}({\tilde{u}}), \qquad \int _{0}^{1} {\tilde{H}}_{\kappa }({\tilde{u}}) d \underline{t} =0 \Leftrightarrow {\tilde{H}}_{\kappa }({\tilde{u}}) \equiv 0. \end{aligned}$$

Therefore \({\tilde{u}}\) is a critical point of the rescaled Rabinowitz action functional, with the rescaled multiplier \({\tilde{\eta }}\).

4.2 Rescaled \(\kappa \)-localization

In our problem we have

$$\begin{aligned} {\hat{X}}=T^{*} S^{d} \times T^{*} S^{1}. \end{aligned}$$

We take a \(\kappa \) localization for \(\kappa =\kappa _{n}=2^{2/3} \pi ^{-1/3} n^{-1/3}\) sufficiently small in the phase space. Precisely this means that on the zero-energy hypersurface there holds

$$\begin{aligned} {\mathcal {L}}, \dfrac{\sqrt{2}}{\sqrt{\tau }} \in (\kappa _{n}-\kappa _{n}^{3/2}, \kappa _{n}+\kappa _{n}^{3/2}). \end{aligned}$$

We make a conformal change of the symplectic variables as

$$\begin{aligned} ({\mathcal {L}} =\kappa _{n} \tilde{{\mathcal {L}}}, \mathcal {\delta } = \kappa _{n}^{-3} {\tilde{\delta }}, {\hat{\xi }}_{i}=\kappa _{n}^{-1} {\tilde{\xi }}_{i}, {\hat{\zeta }}_{i}=\kappa _{n}^{-1} {\tilde{\zeta }}_{i}, \tau =\kappa _{n}^{-2} {\tilde{\tau }}, {\tilde{t}}={\tilde{t}}). \end{aligned}$$
(4.1)

We have

$$\begin{aligned} {\tilde{\lambda }}= \left( \tilde{{\mathcal {L}}} d {\tilde{\delta }}+\sum _{i=1}^{d-1} {\tilde{\xi }}_{i} d {\tilde{\zeta }}_{i} + {\tilde{\tau }} d {\tilde{t}})=\kappa _{n}^{2}({\mathcal {L}} d \delta +\sum _{i=1}^{d-1} {\tilde{\xi }}_{i} d {\tilde{\zeta }}_{i}+ \tau d {\tilde{t}}\right) =\kappa _{n}^{2} {\hat{\lambda }}, \end{aligned}$$

thus

$$\begin{aligned} {\tilde{\omega }}=d {\tilde{\lambda }}=\kappa _{n}^{2} d {\hat{\lambda }}=\kappa _{n}^{2} \hat{\omega .} \end{aligned}$$

In other words, the mapping

$$\begin{aligned} \phi _{\kappa _{n}}: {\hat{X}} \rightarrow {\hat{X}}, ({\tilde{L}}, {\tilde{\delta }}, {\tilde{\xi }}, {\tilde{\zeta }}, {\tilde{\tau }}, {\tilde{t}}) \mapsto (L, \delta , {\hat{\xi }}, {\hat{\zeta }}, \tau , {\tilde{t}}) \end{aligned}$$

satisfies

$$\begin{aligned} {\tilde{\lambda }}=\kappa _{n}^{2} \phi _{\kappa _{n}}^{*} {\hat{\lambda }}, \qquad {\tilde{\omega }}=\kappa _{n}^{2} \phi _{\kappa _{n}}^{*} {\hat{\omega }}. \end{aligned}$$

Recall that we have \(H_{0}=\dfrac{\sqrt{2}}{2} {\mathcal {L}} \sqrt{\tau }-1\). It is instructive to write

$$\begin{aligned} H_{\varepsilon }=H_{0}+O(\kappa _{n}^{4};\varepsilon ), \end{aligned}$$

as it follows from (1.4) that the perturbation term is of order \(O(\kappa _{n}^{4})\) in \(\kappa _{n}\) and tends to zero when \(\varepsilon \rightarrow 0\). After rescaling, we have that

$$\begin{aligned} {\tilde{H}}_{\varepsilon , \kappa _{n}}=\dfrac{\sqrt{2}}{2} \tilde{{\mathcal {L}}} \sqrt{{\tilde{\tau }}}-1+O(\kappa _{n}^{4};\varepsilon ). \end{aligned}$$
(4.2)

We now apply this argument for each n sufficiently large to normalize the corresponding Keplerian periodic manifolds \(\Lambda _{n}\) of \(H_{0}\), which we describe in local charts, to

$$\begin{aligned} \Lambda _{1}^{(n)}{:=}\{\tilde{{\mathcal {L}}}={\mathcal {L}}_{1}, \tau =\tau _{1}, {\tilde{\delta }} \in {\mathbb {R}}/ (2 \pi /n) {\mathbb {Z}}\}. \end{aligned}$$

Globally \(\Lambda _{1}^{(n)}\) is realized in the fibre bundle \(B^{(n)}\) with fibre \(T_{+}^{*} ({\mathbb {R}}/ (2 \pi /n) {\mathbb {Z}}\times S^{1})\) over the normalized orbit space \(\Omega _{{\mathcal {O}}}\) as

$$\begin{aligned} \Lambda _{1}^{(n)}{:=}\{\tilde{{\mathcal {L}}}={\mathcal {L}}_{1}, \tau =\tau _{1}\} \subset B^{(n)}. \end{aligned}$$

To further uniformize the periodic manifolds we now lift the systems \(H_{0}\) and \(H_{\varepsilon }\) defined on \(B^{(n)}\) to a fibrewise n-cover of \(B^{(n)}\) by lifting \(T_{+}^{*} ({\mathbb {R}}/ (2 \pi /n) {\mathbb {Z}}\times S^{1})\) to \(T_{+}^{*} {\mathbb {T}}^{2}\) in a fibrewise way. The fibres \(T_{+}^{*} ({\mathbb {R}}/ (2 \pi /n) {\mathbb {Z}}\times S^{1})\) and \(T_{+}^{*} {\mathbb {T}}^{2}\) are both quotients of \(T_{+}^{*} {\mathbb {R}}^{2}\) by \({\mathbb {Z}}^{2}\)-actions on \({\mathbb {R}}^{2}\), which can be identified by means of an affine transformation on \({\mathbb {R}}^{2}\). Therefore both \(B^{(n)}\) and its fibrewise n-cover are quotients of the same \(T_{+}^{*} {\mathbb {T}}^{2}\)-bundle with respect to the (different but identifiable) actions of \({\mathbb {Z}}^{2}\) which act in a fibrewise fashion. They are therefore isomorphic as fibre bundles. On the other hand, all \(B^{(n)}\)’s are isomorphic. Therefore for all n, the fibrewise n-covers of \(B^{(n)}\) are isomorphic to the fibre bundle \(B{:=}B^{(1)}\) with fibres \(T^{*}_{+} {\mathbb {T}}^{2}\) over the base \(\Omega _{{\mathcal {O}}}\). Moreover, the equivariant lift of the symplectic structure on \(B^{(n)}\) agrees with the canonical symplectic structure on \(B^{(1)}\). We may therefore take \(B{:=}B^{(1)}\) with its canonical symplectic structure as a uniform representative. The periodic manifolds \(\Lambda _{1}^{(n)}\) are now lifted to \(\Lambda _{1}=\Lambda _{1}^{(1)}\) in B.

The unperturbed function \(H_{0}\) is independent of the angles and can be lifted directly. The perturbation term can be written in a chart as a function of the rescaled variables

$$\begin{aligned} (\tilde{{\mathcal {L}}}, n {\tilde{\delta }}, {\hat{\xi }}_{i}, {\hat{\zeta }}_{i}, {\tilde{\tau }}, {\tilde{t}} ), \end{aligned}$$

in which in particular the angle \({\tilde{\delta }}\) is defined in \({\mathbb {R}}/ (2 \pi /n) {\mathbb {Z}}\). which then lifts to a function depending on the variables

$$\begin{aligned} (\tilde{{\mathcal {L}}}, [n {\tilde{\delta }}], {\hat{\xi }}_{i}, {\hat{\zeta }}_{i}, {\tilde{\tau }}, {\tilde{t}} ), \end{aligned}$$

while in the latter

$$\begin{aligned}{}[n {\tilde{\delta }}]=n {\tilde{\delta }} \mod 2 \pi \end{aligned}$$

and the angle \({\tilde{\delta }}\) is extended to be defined in \({\mathbb {R}}/ (2 \pi ) {\mathbb {Z}}\).

In an overlap region of two charts given respectively by

$$\begin{aligned} (\tilde{{\mathcal {L}}}, {\tilde{\delta }}_{1}, {\tilde{\xi }}_{i, 1}, {\tilde{\zeta }}_{i, 1}, {\tilde{\tau }}, {\tilde{t}}_{1} ), \end{aligned}$$

and

$$\begin{aligned} (\tilde{{\mathcal {L}}}, {\tilde{\delta }}_{2}, {\tilde{\xi }}_{i, 2}, {\tilde{\zeta }}_{i, 2}, {\tilde{\tau }}, {\tilde{t}}_{2} ), \end{aligned}$$

the transition map is induced from the corresponding transition map of the charts given by the non-rescaled variables

$$\begin{aligned} {\bar{\phi }}: ({{\mathcal {L}}}, {\delta }_{1}, {\hat{\xi }}_{i, 1}, {\hat{\zeta }}_{i, 1}, {\tau }, {\tilde{t}}_{1} ) \mapsto ({{\mathcal {L}}}, {\delta }_{2}, {\hat{\xi }}_{i, 2}, {\hat{\zeta }}_{i, 2}, {\tau }, {\tilde{t}}_{2}) \end{aligned}$$

which in particular implies that

$$\begin{aligned} ({{\mathcal {L}}}, n {\tilde{\delta }}_{2}, {\hat{\xi }}_{i, 2}, {\hat{\zeta }}_{i, 2}, {\tau }, {\tilde{t}}_{2})={\bar{\phi }} ({{\mathcal {L}}}, n {\tilde{\delta }}_{1}, {\hat{\xi }}_{i, 1}, {\hat{\zeta }}_{i, 1}, {\tau }, {\tilde{t}}_{1} ) \end{aligned}$$

in which \({\tilde{\delta }}_{1}, {\tilde{\delta }}_{2} \in {\mathbb {R}}/ (2 \pi /n) {\mathbb {Z}}\). This equality extends to

$$\begin{aligned} ({{\mathcal {L}}}, [n {\tilde{\delta }}_{2}], {\hat{\xi }}_{i, 2}, {\hat{\zeta }}_{i, 2}, {\tau }, {\tilde{t}}_{2})={\bar{\phi }} ({{\mathcal {L}}}, [n {\tilde{\delta }}_{1}], {\hat{\xi }}_{i, 1}, {\hat{\zeta }}_{i, 1}, {\tau }, {\tilde{t}}_{1} ) \end{aligned}$$

in which we now regard \({\tilde{\delta }}_{1}, {\tilde{\delta }}_{2} \in {\mathbb {R}}/ 2 \pi {\mathbb {Z}}\).

This shows that the lift of the perturbation term takes consistent value on the overlap region of two lifted charts and is thus globally defined in a neighborhood of \(\Lambda _{1}\) in B. Moreover it still assumes the form \(O(\kappa _{n}^{4};\varepsilon )\). In particular, this lift construction does not change its \(C^{1}\)-norm.

We now remark on the \(C^{1}\)-norm of the perturbation term, which assumes the form \(O(\kappa _{n}^{4};\varepsilon )\). We see that due to the rescaling of the variables (4.1), in particular the fast angle \(\delta \), the partial derivative of the perturbation with respect to \({\tilde{\delta }}\) is only of order \(O(\kappa _{n})\) while other partial derivatives remain to be of order \(O(\kappa _{n}^{4})\). As a result, the \(C^{1}\)-norm of the perturbation term is of order \(O(\kappa _{n})\). In general, we may only have an estimate of the \(C^{2}\)-norm of the perturbation as \(O(\kappa _{n}^{-2})\). This estimate is not bounded when \(\kappa _{n} \rightarrow 0\). Since the theorem of Weinstein [26] requires a proper smallness of the \(C^{2}\)-norm of the perturbation, this can only be applied to obtain our result in special cases. To obtain the result in full generality we shall instead conclude with the local Rabinowitz–Floer homology argument from Appendix A.

4.3 Proof of Theorem 1.1 and Corollary 1.2

Proof of Theorem 1.1

We apply Theorem A.1 to the rescaled system (4.2) near the normalized periodic manifold \(\Lambda _{1}\) of (4.2) with \(\varepsilon =0\). The period of periodic orbits from \(\Lambda _{1}\) is \(S_{1}=(2 \pi )^{2/3}\). We take \(T^{\pm }=S_{1} \pm {{\tilde{\beta }}}\) for a sufficiently small \({\tilde{\beta }}>0\). Moreover, we have \(K^{+}, K^{-} =O(\kappa _{n}^{4})\). In this setting Theorem A.1 is applicable for all sufficiently small \(\kappa _{n}\), which then assures the existence of a periodic orbit with action \({\mathcal {A}}_{0}^{(1)}+O(\kappa _{n}^{4})\) of the system (4.2) for all sufficiently large n. After rescaling back this corresponds to a periodic orbit of the system (1.2) with action of the order \(\kappa _{n}^{-2}+O(\kappa _{n}^{2})\). Viewing from Proposition 2.4 that the action gaps of the regularized Kepler problem in extended phase space \(H_{0}\) is of order \({{\tilde{\kappa }}}_{n}\) for sufficiently small \(\kappa _{n}\), the periodic orbits thus obtained take infinitely many distinct action values. Since the perturbations to the Moser-regularized Kepler flow vanish at the collisions, any of these orbits passes transversally through the set of collisions along which the variable \(\tau \) is continuous, we conclude that this orbit gives rise to a generalized periodic orbit of the system (1.2) resp. (1.1) with the same action value. Consequently we obtain infinitely many generalized periodic orbits of the system (1.2) resp. (1.1). \(\square \)

Indeed the argument shows that there exists \(N^{*}>0\) which depends only on the \(C^{1}\)-norm of U, such that for all \(n >N^{*}\) there bifurcates at least one periodic orbit from each periodic manifold \(\Lambda _{n}\). For \(n \le N^{*}\) we may control the smallness of the \(C^{1}\)-norm of the perturbation by choosing small \(\varepsilon >0\) as in [8]. We thus obtain the following corollary:

Corollary 4.2

There exists \(\varepsilon ^{*}>0\) depending only on the \(C^{1}\)-norm of U, such that for \(\varepsilon \in [0, \varepsilon ^{*}]\) the system (1.2) has infinitely many periodic orbits bifurcating from each of the periodic manifolds \(\{\Lambda _{n}\}_{n=1,2,\ldots }\) .

Remark 4.3

We have seen that in general the \(C^{2}\)-norm of \(\varepsilon U\) is of the order \(O(\kappa ^{-2};\varepsilon ) \), caused by the rescaling of the fast angle \(\delta \). This can be improved provided that the perturbation is independent of \(\delta \), or when the perturbation is of the order \(o(\kappa ^{6}; \varepsilon )\). The latter is achieved, for example, when the function \(U(q, t, \varepsilon )\) is super-cubic in |q|, i.e. \(U(q, t, \varepsilon )=O(|q|^{4}; \varepsilon )\) in a neighborhood of the origin. In these cases we may as well conclude Theorem 1.1 and Corollary 4.2 with Weinstein’s theorem.

Finally we remark that by realizing circular or elliptic restricted three-body problems as periodically forced Kepler problems as in Appendix C, we obtain infinitely many generalized periodic orbits accumulating at each of the primaries, which proves Corollary 1.2.