Some Collision Solutions of the Rectilinear Periodically Forced Kepler Problem

1 Introduction

In the rectilinear periodically forced Kepler problem (studied, e.g., by Lazer and Solimini [3])

generalized solutions were defined by Ortega in [5] to be collision solutions of the system with collisions regularized by elastic bouncing, i.e., when the particle moves backwardly after the collision while keeping the same energy.

Indeed, in this one-dimensional rectilinear problem, if no collision is admitted, the richness of the dynamics reduces greatly (cf. [5]). In the case when the force term p⁢(t) is C1, Ortega showed the existence of many generalized periodic solutions by analyzing a Poincaré section associated to the collisions and also showed that the associated Poincaré map is an exact symplectic twist map.

In this note, we explain a different approach to this fact. We embed the non-autonomous Hamiltonian system (1.1) into the zero-energy level of an autonomous Hamiltonian system and we employ the Levi-Civita regularization (see Levi-Civita [4]; note that this already appeared in Goursat [2]) to regularize the collisions. Since the resulting system is regular at collisions, we may then deduce by standard arguments that the Poincaré map associated to the collision set is exact symplectic.

In expressing the unperturbed regularized system in action-angle coordinates, we observe that the twist property of this Poincaré map follows from the non-degeneracy of this system with respect to the action variables in a region where u is supposed to be small enough. When the function p⁢(t) is smooth enough to allow the application of the KAM theorem, we may obtain the following KAM-type results.

These orbits are obtained from invariant 2-tori lying in the three-dimensional zero-energy hypersurface of the embedded two-degree-of-freedom autonomous system. The existence of these invariant tori prohibits a large change of the corresponding action variables corresponding to the energy and the amplitude of u in (1.1) of the trajectories lying between them. Moreover, by an application of the Poincaré recurrence theorem to the Poincaré map associated to the Poincaré section related to collisions, we conclude that the restricted dynamics lying in the region bounded by two invariant curves (intersections of invariant tori with the Poincaré section) is Poisson stable. When p⁢(t) is sufficiently smooth, this Poisson stability gives, in particular, a partial answer to the question raised in [5] concerning the recurrence of the restricted dynamics.

2 Levi-Civita Regularization of the Rectilinear Kepler Problem

The system (1.1) is a time-periodic Hamiltonian system with Hamiltonian

H ⁢ ( u , y , t ) = y 2 2 - 1 u - p ⁢ ( t ) ⁢ u ,

where y denotes the conjugate momentum of u.

In the case when the perturbation p⁢(t) is of class C1, we may embed this time-periodic Hamiltonian system in the zero-energy hypersurface {H~=0} of the autonomous C1-Hamiltonian

H ~ = τ + H = τ + y 2 2 - 1 u - p ⁢ ( t ) ⁢ u .

The initial time variable t∈𝕋:=ℝ/2⁢π⁢ℤ is now an angle variable in this system and the variable τ, conjugate to t, is just the negative of the energy of H.

On {H~=0}, we change time (which now has nothing to do with the real time t) by multiplying H~ by u and pulling back the resulting function by the usual symplectic 2-to-1 Levi-Civita transformation restricted to T*(ℝ∖{z=0}), i.e.,

The system H~ is thus transformed into the system

K ( z , w , t , τ ) = L . C . * ( u H ~ ) = z 2 τ + w 2 8 - p ( t ) z 4 - 1 ,

which defines a Hamiltonian system on the symplectic manifold

( T * ( ℝ ∖ { z = 0 } ) × T * 𝕋 , d w ∧ d z + d τ ∧ d t ) .

By its expression, this system can be extended analytically to

( T * ℝ ∖ { z = 0 , w = 0 } × T * 𝕋 , d w ∧ d z + d τ ∧ d t ) .

The extended system is regular on the two-dimensional subset 𝒞ol:={z=0} of {K=0}, which is called the collision set. As a matter of easy computation, we see that |w|=2⁢2 on 𝒞⁢ℴ⁢𝓁⁢o⁢l.

3 The Poincaré Section Related to the Collision Set

We now analyze, in our setting, the surface of section associated to collisions used in [5] and we give an alternative approach to [5, Section 6] concerning the exact symplecticity of this mapping.

For τ>0, when p⁢(t)=0, the dynamics of K0=z2⁢τ+w2/8-1 is easily understood. This is a coupled system of a harmonic oscillator in the (z,w)-space together with a rotator in (τ,t)-space. The set 𝒞ol+:=𝒞ol∩{τ>0} is seen to be a global section for the flow restricted to {K0=0,τ>0}.

The Poincaré map 𝒮:𝒞⁢o⁢l+⊂ℝ×𝕋→ℝ×𝕋 sends a pair of collision time-energy (t0,τ0) to the successive pair of collision time-energy (t1,τ1). In [5, Proposition 5.1], the author has proved that there exists a 2⁢π-periodic, lower semi-continuous function ϕ:ℝ→ℝ∪{+∞} such that the mapping is well-defined (i.e., t1<+∞) on

D := { ( t 0 , τ 0 ) ∈ D ~ : τ 0 < ϕ ⁢ ( t 0 ) }

with minℝ⁡ϕ≥-2⁢∥p∥∞1/2.

We now follow [7] to show that this map is exact symplectic. For any small disk α in D and its image α′=𝒮⁢(α), we denote by Σ the cylinder formed by the integral curves of K connecting ∂⁡α and ∂⁡α′. By the Stokes formula and the closeness of ω=d⁢λ with λ=w⁢d⁢z+τ⁢d⁢t, we have

∫ α ω = ∫ α ′ ω + ∫ Σ ω .

Since Σ is formed by the integral curves of K and is two-dimensional, we deduce from the fact that the vector field XK lies in the kernel of the restriction of ω to {K=0} that the further restriction of ω to Σ vanishes. Therefore,

∫ α 𝑑 τ ∧ d ⁢ t = ∫ α ′ 𝑑 τ ∧ d ⁢ t .

On the other hand, for any loop γ in D and γ′=𝒮⁢(γ), we deduce by the Stokes formula that

∫ γ λ - ∫ γ ′ λ = ∫ Σ ω = 0

and, therefore,

∫ γ τ ⁢ 𝑑 t = ∫ γ ′ τ ⁢ 𝑑 t .

We thus have the following proposition.

4 Some Periodic and Quasi-Periodic Solutions

Given the function p⁢(t), when restricted to a region of the phase space of K where z is small enough and τ>0 is bounded away from zero, the system K⁢(z,w,t,τ) is a perturbation of the integrable system

K 0 = z 2 ⁢ τ + w 2 8 - 1 .

The “integrable approximating system” K0 reads

2 2 ⁢ τ 1 / 2 ⁢ I - 1

in action-angle form by calculating the action-angle coordinates (I,θ,τ,t′) defined by the relations

{ z = 2 - 1 / 4 ⁢ I 1 / 2 ⁢ τ - 1 / 4 ⁢ cos ⁡ θ , w = - 2 ⋅ 2 1 / 4 ⁢ I 1 / 2 ⁢ τ 1 / 4 ⁢ sin ⁡ θ , τ = τ , t = t ′ + 4 - 1 ⁢ I ⁢ τ - 1 ⁢ sin ⁡ 2 ⁢ θ .

The calculation goes in the following way. We first fix τ and reduce the system K0 by the 𝕊1-symmetry of shifting t, and then we calculate the action variable I=A⁢(h)/2⁢π by calculating the enclosed area A⁢(h) of the ellipse {K0=h} in the (z,w)-plane. Inverting the mapping h↦I⁢(h) gives

K 0 = h = 2 2 ⁢ τ 1 / 2 ⁢ I - 1 .

We thus set

z = 2 - 1 / 4 ⁢ I 1 / 2 ⁢ τ - 1 / 4 ⁢ cos ⁡ θ , w = - 2 ⋅ 2 1 / 4 ⁢ I 1 / 2 ⁢ τ 1 / 4 ⁢ sin ⁡ θ .

In the unreduced phase space with variables (I,θ,τ,t), we have

d ⁢ w ∧ d ⁢ z + d ⁢ τ ∧ d ⁢ t = d ⁢ I ∧ d ⁢ θ + d ⁢ τ ∧ d ⁢ ( t - 4 - 1 ⁢ I ⁢ τ - 1 ⁢ sin ⁡ 2 ⁢ θ ) .

Therefore, the set of variables (I,θ,τ,t′=t-4-1⁢I⁢τ-1⁢sin⁡2⁢θ) forms a set of action-angle coordinates of K0.

The subset {τ>0,I>0} of the phase space of K0 is seen to be foliated by the invariant 2-tori of K0 obtained by fixing I and τ, and the associated Poincaré map 𝒮ˇ=𝒮|D is an exact twist map. The perturbation takes the form

P ⁢ ( I , θ , τ , t ′ ) = p ⁢ ( t ⁢ ( I , θ , τ , t ′ ) ) ⁢ I 2 ⁢ τ - 1 ⁢ cos 4 ⁡ θ .

We now have a choice to study the dynamics of K as a perturbation of K0 (which is the case for given p⁢(t) when |z| is small enough). We may either work directly with the Hamiltonian or with the Poincaré map 𝒮ˇ. The objects between the two approaches are naturally related. Fixed and periodic points of 𝒮ˇ give rise to periodic solutions of K and invariant curves of 𝒮ˇ give rise to invariant tori of K. They give rise to generalized periodic and quasi-periodic solutions of (1.1), respectively. Recall that by “generalized solutions” we simply refer to those collision solutions along which the collisions are regularized by elastic bouncing, which is exactly what the Levi-Civita regularization implies for the initial system.

We now prove Theorem 1.1.

The existence of these two-dimensional KAM tori in the three-dimensional {K=0} entails that these periodic orbits are “stable” in the sense that there are no large changes of the action variables τ and I, which translates into the energy and the amplitude of (1.1), respectively.

In [5], the existence of families of generalized periodic solutions is shown by an application of the Poincaré–Birkhoff theorem. In our case, these periodic solutions, which correspond to periodic points of the Poincaré map lying between invariant curves (which are themselves intersections of invariant 2-tori with the constructed Poincaré section), are stable à la Lagrange in the same sense that there are no large changes of the energy and the amplitude. Moreover, these invariant curves bound the positive but finite |d⁢τ∧d⁢t|-measure in D preserved by 𝒮, which allows us to apply the Poincaré recurrence theorem (see [1, Section 2.6]) to confirm that |d⁢τ∧d⁢t|-almost all points in the bounded regions (what is called Birkhoff’s region of instability) are recurrent. Therefore, the dynamics confined to these bounded regions is Poisson stable.