Variational approach to second species periodic solutions of Poincar\'e of the 3 body problem

We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincar\'e second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincar\'e only sketched the proof of the existence of second species solutions. Rigorous proofs appeared much later and only for the restricted 3 body problem. We develop a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem. As an application, we give a rigorous proof of the existence of a class of second species solutions.

Let q 1 , q 2 ∈ R 2 be positions of m 1 , m 2 with respect to m 3 (Poincaré's heliocentric coordinates) and p 1 , p 2 ∈ R 2 their scaled momenta. The motion of m 1 , m 2 with respect to m 3 is described by a Hamiltonian system (H µ ) with Hamiltonian where The Hamiltonian H µ is a small perturbation of the Hamiltonian H 0 = H 1 + H 2 describing 2 uncoupled Kepler problems. The configuration space of system The configuration space of the perturbed system (H µ ) is U 2 \ ∆, where ∆ = {q = (q 1 , q 2 ) ∈ U 2 : q 1 = q 2 } represents collisions of m 1 , m 2 .
Let R ⊂ P be the regular domain -the set of points in P such that the corresponding Kepler ellipses do not cross. For small µ > 0 solutions of system (H µ ) in R are O(µ)-approximated by solutions of system (H 0 ) on finite time intervals independent of µ. By the classical perturbation theory, away from resonances the same is true on longer time intervals. Moreover, as proved by Arnold [3], for small µ > 0, system (H µ ) has a large measure of invariant 2-dimensional KAM tori on which solutions are quasiperiodic, and thus well approximated (modulo rotation) by solutions of system (H 0 ) on infinite time intervals.
In the singular domain S ⊂ P , where the corresponding Kepler ellipses cross, the classical perturbation theory does not work. Indeed, for almost any initial condition in S, solution of system (H 0 ) is quasiperiodic with incommensurable frequencies, and so eventually m 1 , m 2 simultaneously approach an intersection point of Kepler ellipses. Then the perturbation in (1.1) becomes large, and so it can not be ignored even in the first approximation in µ.
For small µ > 0 solutions of the 3 body problem (H µ ) in S can be described as follows. The bodies m 1 , m 2 move along nearly Kepler ellipses and after many revolutions they almost collide. Then they start moving along a new pair of nearly Kepler orbits. If the new energies H 1 , H 2 are both negative, so the new Kepler orbits are ellipses, then m 1 , m 2 will again nearly collide after many revolutions, and the process repeats itself. Thus almost collision solutions of the 3-body problem (H µ ) shadow chains of collision orbits of system (H 0 ).
Almost collision periodic solutions of system (H µ ) were first studied by Poincaré in New Methods of Celestial Mechanics. Poincaré named them second species periodic solutions. However, he did not provide a rigorous existence proof. Rigorous proofs appeared much later (see e.g. [16,10,7]) and only for the restricted 3 body problem, circular and elliptic. In [10,8] also chaotic second species solutions of the circular and elliptic restricted problem were studied.
The goal of this paper is to develop a variational approach to almost collision periodic orbits of the nonrestricted 3 body problem. As an application, we will give a rigorous proof of the existence of a class of almost collision periodic orbits. Chaotic almost collision orbits will be studied in another paper. Remark 1.1. It is possible to fix the value of angular momentum G and reduce rotational symmetry. Then we obtain a Hamiltonian system with 3 degrees of freedom. However, since reduction of the rotational symmetry considerably complicates the Hamiltonian, it is simpler to work with the original Hamiltonian system (H µ ) with 4 degrees of freedom. Remark 1.2. We consider only near collisions of small masses m 1 and m 2 and exclude near collisions of m 1 , m 2 with m 3 . In particular, triple collisions are excluded. It is well known that double collisions can be regularized, but the Levi-Civita regularization becomes singular as µ → 0. Understanding this singularity is the base for our methods. Levi-Civita regularization was previously used to study second species solutions for the restricted 3 body problem, see e.g. [16,10,14]. Remark 1.3. Main results of this paper hold for more general Hamiltonians with singularity, for example where all functions are smooth without singularity at q 1 = q 2 .

Main results
A solution of the Hamiltonian system (H µ ) is determined by its projection to the configuration space U 2 \ ∆ which will be called a trajectory. Let L µ be the Lagrangian corresponding to H µ . A T -periodic trajectory γ : R → U 2 \ ∆ is a critical point of the Hamilton action functional on the space of T -periodic W 1,2 loc curves in U 2 \ ∆. We write T explicitly in A µ (T, γ) because later it will become a variable. Any trajectory of system (H 0 ) has the form γ = (γ 1 , γ 2 ), where γ j is a trajectory of the Kepler problem. For µ = 0 the action functional A = A 0 splits: is the action functional of the Kepler problem on the space of T -periodic W 1,2 loc curves σ : R → U . We have to consider also trajectories of system (H 0 ) with collisions. A Tperiodic curve γ = (γ 1 , γ 2 ) : R → U 2 is called a periodic n-collision chain if there exist time moments such that: • γ has collisions at t = t j , so that • γ| [tj ,tj+1] is a trajectory of system (H 0 ) which will be called a collision orbit.
• The total energy H 0 does not change at collision: By (2.5), the total angular momentum G = G 1 +G 2 is preserved at collisions: Hence G is constant along γ. By (2.6), the total energy H 0 = H 1 + H 2 = E is also constant along γ, but not the energies H 1 , H 2 of m 1 , m 2 , or their angular momenta G 1 , G 2 . A collision chain γ is a broken trajectory of system (H 0 ) -a concatenation of collision orbits with reflections from ∆. However, unlike for ordinary billiard systems, ∆ has codimension 2 in the configuration space U 2 , so the change of the normal component of the momentum at collision is not uniquely determined. Thus there is no direct interpretation of collision chains as trajectories of a dynamical system. Such an interpretation is given later on.
Collision chains are limits of trajectories of system (H µ ) which approach collisions as µ → 0. Indeed, we have: Proposition 2.1. Let γ µ be a T µ -periodic trajectory of system (H µ ) which uniformly converges, as µ → 0, to a T -periodic curve γ. If γ([0, T ]) ∩ ∆ is a finite set, then γ is a periodic collision chain.
We say that γ µ is an almost collision orbit shadowing the collision chain γ. A similar statement holds for nonperiodic collision chains.
Intuitively, Proposition 2.1 is almost evident: a near collision of m 1 , m 2 lasts a short time during which the influence of the non-colliding body m 3 is negligible. Then m 1 , m 2 form a 2 body problem, so their total momentum y = p 1 + p 2 and total energy H 0 = H 1 + H 2 are almost preserved. This yields (2.5)-(2.6). This can be made into a rigorous proof, see e.g. [1]. A better way to prove Proposition 2.1 is by using the Levi-Civita regularization, see section 7.
Collision chains can be characterized as extremals of Hamilton's action functional (2.2). Let Ω T n be the set of ω = (t, T, γ), where t = (t 1 , . . . , t n ) satisfies (2.4) and γ : R → U 2 is a T -periodic W 1,2 loc curve such that γ(t j ) = (x j , x j ) ∈ ∆. The collision times t j and collision points x j are not fixed. Then Ω T n can be identified with an open set in a Hilbert space (see (2.8) and collision times t j and collision points x j are smooth functions on Ω T n .
Remark 2.1. If γ(t) / ∈ ∆ for t = t j , then time moments t j are determined by the curve γ, i.e. the projection Ω T n → W 1,2 (R/T Z, U 2 ), (t, T, γ) → γ, is injective. Then ω = (t, T, γ) is determined by γ. But t j are not continuous functions of γ, so we have to include the variables t in the definition of Ω T n .
Remark 2.2. In the one-dimensional calculus of variations Hilbert spaces are unnecessary: at least locally function spaces can be replaced by finite dimensional subspaces of broken extremals. We will use this approximation later on. However, in this section we use conventional W 1,2 setting.
The action functional A(ω) = A(T, γ) is a smooth function on Ω T n .
Proof. If ω = (t, T, γ) is a critical point of A on Ω T n , then each segment γ| [ti−1,tj ] is a collision orbit of system (H 0 ). Then by the first variation formula [2], where ∆h j and ∆y j are jumps of energy and total momentum: Since the differentials dt j , dx j are independent, critical points of A satisfy ∆h j = 0 and ∆y j = 0. This implies (2.5)-(2.6). Converse is also evident.
For collision chains with fixed energy H 0 = E we use Maupertuis's variational principle [2]. Let Ω n = ∪ T >0 Ω T n . The period T > 0 is now a smooth function on Ω n . Hamilton's action is replaced by the Maupertuis action functional If γ is a collision chain with energy E, then is the classical Maupertuis action. The Maupertuis principle for collision chains is as follows: Due to time and rotation invariance, critical points of the action functional are degenerate. To eliminate time degeneracy, we identify collision chains which differ by time translation This defines a group action of R on Ω n . The coordinates on the quotient space Ω n = Ω n /R can be defined as follows. Let The map map Ω n → R n + × W n , (t, T, γ) → (s, σ), makes it possible to identify Ω n with R n + × W n . We can represent σ j bŷ Then σ is determined by (x,σ), where x = (x 1 , . . . , x n ) ∈ U n andσ = (σ 1 , . . . ,σ n ). Finally The action functional gives a smooth function A(s, σ) on Ω n , invariant under rotations: A(s, e iθ σ) = A(s, σ). We deal with the rotation degeneracy later on.
Often it is convenient to use parametrization independent Jacobi's form of the Maupertuis action functional -the length of γ in the Jacobi's metric ds E : Then . Thus up to parametrization, extremals of J E and A E are the same. For a collision chain γ corresponding to (s, σ) ∈ Ω n , is a function on W n . We obtain Proposition 2.4. If γ is a periodic n-collision chain of energy E then the corresponding σ ∈ W n is an extremal of the Jacobi action J E . If σ is an extremal of J E , then if each σ j is reparametrized so that H 0 ≡ E, the corresponding γ is a periodic n-collision chain.
We will not use Jacobi's variational principle since J E is not a smooth function. A discrete version of Jacobi's action is smooth and we will use it later on.
Similar variational principles hold for collision chains periodic in a rotating coordinate frame: γ(t + T ) = e iΦ γ(t) for some quasiperiod T and phase Φ. We call γ periodic modulo rotation. If Φ / ∈ 2πQ, then γ is quasiperiodic in a fixed coordinate frame.
The corresponding function space is defined as follows. LetΩ n be the set of all (t, T, γ, Φ), where t, T are as before, Φ ∈ R and the W 1,2 loc curve γ : R → U 2 satisfies γ(t + T ) = e iΦ γ(t) and γ(t j ) ∈ ∆. ThenΩ n can be identified with an open set in a Hilbert space and t j , x j , T, Φ are smooth functions onΩ n . In fact Ω n ∼ = Ω n ×R. Indeed,γ(t) = e −itΦ/T γ(t) is a T -periodic curve, so (t, T,γ) ∈ Ω n .
Define the Maupertuis-Routh action functional onΩ n by This is a smooth function onΩ n and we have: γ is a periodic modulo rotation collision chain with energy E and angular momentum G iff (t, T, γ, Φ) is a critical point of the functional A EG onΩ n .
Remark 2.3. It seems natural to take Φ ∈ T = R/2πZ since Φ + 2π gives the same collision chain. But then the functional A EG will be multivalued: defined modulo 2πG.
Remark 2.4. The name of the functional A EG is motivated as follows. One can perform Routh's reduction [2] of the rotational symmetry for fixed G replacing the configuration space U 2 by U 2 = U 2 /T ∼ = R 2 + × T and the Lagrangian by the so called Routh function. Then the functional A EG becomes the Maupertuis functional for the reduced Routh system. Probably this observation is due to Birkhoff [5]. However, Routh's reduction makes the Lagrangian more complicated, so we do not use it in this paper.
There are several other possible variational principles for collision chains (for example, we may fix the phase Φ), but in the present paper we will use only the ones given above.
Sufficient condition for the existence of a periodic orbit of system (H µ ) shadowing a given collision chain γ requires that the chain is nontrivial in the following sense. Let relative speed is preserved at collision. We impose two essentially equivalent conditions: Direction change condition. Relative collision velocity changes direction at collision: If the direction change condition is not satisfied at some t j , thenγ(t j − 0) = γ(t j + 0), and so γ| [tj−1,tj+1] is a smooth trajectory of system (H 0 ). Deleting the collision time moment t j we obtain a (n − 1)-collision chain violating no early collisions condition.
Conversely, if γ is a n-collision chain violating no early collisions condition, then adding an extra collision time moment, we obtain a (n + 1)-collision chain violating the changing direction condition. From now on we add these two equivalent conditions to the definition of a collision chain.
Remark 2.5. The changing direction condition implies that almost collision orbits γ µ shadowing the collision chain γ come O(µ)-close to collision. Often almost collision orbits discussed in Astronomy come close to collision, but not too close, for example O(µ ν )-close with ν ∈ (0, 1), see e.g. [15,18,14]. Such orbits change direction at near collision, but this change is small as µ → 0. Then the corresponding collision chains do not satisfy the changing direction condition: Our methods do not work for such almost collision orbits.
The changing direction condition makes it possible to construct a shadowing orbit γ µ of system (H µ ), but it does not prevent γ µ from having regularizable double collisions of m 1 , m 2 . To exclude such collisions we need to impose an extra condition: . . , n. But this condition is not as crucial as the changing direction condition, so we do not include it in the definition of a collision chain.
To construct shadowing orbits we also need some nondegeneracy assumptions. We say that a T -periodic n-collision chain γ with energy E is nondegenerate if ω = (t, T, γ) ∈ Ω n is a nondegenerate modulo symmetry critical point of the Maupertuis action A E on Ω n .
Due to time translation and rotation invariance, critical points of A E are all degenerate: the group action γ(t) → e iθ γ(t − τ ) of R × T preserves A E . We say that ω = (t, T, γ) ∈ Ω n is nondegenerate modulo symmetry if the nullity of the quadric form d 2 A(ω) on T ω Ω n is 2 -the lowest possible. Equivalently, the manifold M ⊂ Ω n obtained from ω by the action of the group R × T is a nondegenerate critical manifold. Nondegeneracy modulo symmetry is equivalent to nondegeneracy of the corresponding critical point on Ω n /(R × T) ∼ = Ω n /T.
As usual in the classical calculus of variations, the Hessian operator corresponding to d 2 A E (ω) is a sum of invertible and compact operators on the Hilbert space T ω Ω n , so nondegeneracy modulo symmetry implies that the Hessian has bounded inverse on T ω Ω n /T ω M . In fact, at least locally, A E can be reduced to a finite dimensional discrete action functional (see section 5), so all Hilbert spaces involved are essentially finite dimensional. Now two main results will be formulated.
Next we consider collision chains with fixed energy E and angular momentum G. Again we say that a periodic modulo rotation collision chain γ is nondegenerate if the corresponding (t, T, γ, Φ) ∈Ω n is a nondegenerate modulo symmetry critical point of the functional A EG onΩ n . Thus it has only degeneracy coming from rotation and time translation invariance. Theorem 2.2. Let γ be a T -periodic modulo rotation nondegenerate collision chain with energy E and angular momentum G. Then for small µ > 0 there exists a periodic modulo rotation orbit γ µ of system (H µ ) with energy E and angular momentum G which O(µ)-shadows γ: where An estimate (2.12) holds also here. Even if γ is periodic (Φ ∈ 2πQ), in general the shadowing orbit γ µ will be periodic only modulo rotation, and thus quasiperiodic in a fixed coordinate frame.
To use Theorems 2.1-2.2, we need to find nondegenerate modulo symmetry collision chains. In general this is not easy. A simple application of Theorem 2.2, based on a perturbative approach, is given in section 3. More complex applications will be given in a future publication.
In section 4 a description of nondegenerate collision orbits is given. Using this description, in section 5 we reduce the action functionals to their discrete versions. Then in section 6 we formulate a local connection result -Theorem 6.1 -and use it to prove Theorem 2.1. The proof of Theorem 2.2 is similar. In section 7 we use Levi-Civita regularization to reduce Theorem 6.1 to Theorem 7.2 which is a generalization of the Shilnikov Lemma [21] to Hamiltonian systems with a normally hyperbolic critical manifold.
Remark 2.6. In this paper we do not attempt to use global variational methods. The reason is that although one can use global methods to find critical points of the action functionals, in general it is hard to check that the critical points satisfy the changing direction condition.

Restricted elliptic limit
Suppose that one of the small masses m 1 , m 2 is much smaller than the other: α 1 ≪ α 2 . In the formal limit α 1 → 0 we obtain the restricted elliptic 3 body problem for which many second species periodic solutions were obtained in [7]. These results do not immediately extend to the case of small α 1 > 0. However, we will show that they can be used to obtain many second species periodic solutions for the nonrestricted 3 body problem.
Let us fix energy E and angular momentum G. For α 1 ≪ α 2 , the Maupertuis-Routh action functional onΩ n is a small perturbation of the Maupertuis-Routh action functional for the Kepler problem. Indeed, the functional A EG 0 = A EG | α1=0 does not depend on γ 1 : The condition γ(t j ) ∈ ∆ imposes no restrictions on γ 2 . Thus the functional B EG is defined on the set Π of (T, σ, Φ), where σ : The functional B EG is very degenerate, because all orbits of the Kepler problem with energy E < 0 are periodic with the same period τ = 2π(−2E) −3/2 . Suppose E, G are such that there exists an elliptic orbit Γ of Kepler's problem with energy E and angular momentum G. For definiteness let G > 0. Then 0 < (−2E)G < 1. The major semiaxis and eccentricity of Γ are The Maupertuis action is The counterclockwise elliptic orbit Γ : R → U is defined uniquely modulo rotation and time translation.
Then all critical points ω = (T, σ, Φ) of the functional B EG on Π belong to one of the nondegenerate critical manifolds M m ⊂ Π, m ∈ N, obtained from (mτ, Γ, 0) by rotation and time translation of Γ. We have Proof. Let (T, σ, Φ) ∈ Π be a critical point of B EG . Then σ is a solution of the Kepler problem with energy E and angular momentum G and hence σ is a time translation and rotation of Γ. Since Γ is a non-circular orbit, quasiperiodicity condition σ(t + T ) = e iΦ σ(t) implies that Φ = 0 mod 2πZ and T = mτ for some m ∈ N.
Next we need to check that M m is a nondegenerate critical manifold of B EG . Essentially this is the same statement, but now we need to consider the linearized Kepler problem.
The second variation d 2 B EG (ω) at ω = (mτ, Γ, 0) is a bilinear form on the tangent space T ω Π which is the set of η = (θ, ξ, φ), where θ, φ ∈ R and ξ : R → R 2 is a vector field such that The standard calculus of variations implies that if η ∈ T ω Π belongs to the kernel of d 2 B EG (ω), then ξ is a solution of the variational equation for Γ which lie on the zero levels of the linear first integrals corresponding to the integrals of angular momentum and energy. The linear approximations at Γ to the integrals of energy and angular momentum arė Since Γ is noncircular, θ = φ = 0 and so ξ(t + mτ ) = ξ(t). It follows that η = (0, ξ, 0) is tangent to M m , i.e. the variation ξ(t) is obtained by time translation and rotation of Γ(t).
The critical manifold N m = π −1 (M m ) ⊂Ω n of A EG 0 corresponding to M m is (up to time translation and rotation) This is an infinite dimensional nondegenerate critical manifold of A EG 0 . For nonzero α 1 , by (3.1), By a standard property of nondegenerate critical manifolds [17], any nondegenerate modulo symmetry critical point ω ∈ N m of A EG | Nm for small α 1 > 0 gives a nondegenerate modulo symmetry critical point of A EG , and hence a nondegenerate modulo symmetry collision chain with energy E and angular momentum G.
Up to an additive constant and a constant multiple, A EG | Nm is Hamilton's action B(mτ, σ) for the Kepler problem. It is defined on the set Π Γ,m of (t, σ), where σ : R → U is an mτ -periodic curve such that σ(t j ) = Γ(t j ). Thus B(mτ, σ) = B Γ,m (t, σ) is precisely the action functional whose critical points are collision chains of the elliptic restricted 3 body problem. This functional was studied in [7], and many its nondegenerate critical points were found for small eccentricity (almost circular Γ), i.e. (−2E)G close to 1. Also the changing direction and no early collisions condition was verified in [7], and this carries out for small α 1 > 0. We obtain Theorem 3.1. Let 0 < (−2E)|G| < 1 be close to 1. Then for sufficiently small α 1 > 0 there exist many collision chains γ such that for sufficiently small µ > 0, γ is O(µ)-shadowed by a second species periodic modulo rotation solution γ µ of the nonrestricted 3 body problem with given E, G.
This result can be improved by using a more quantitative statement from [7]. The obtained second species solutions are periodic in a rotating coordinate frame and quasiperiodic in a fixed coordinate frame. Proper periodic orbits will be obtained in a future publication; for them reduction to the restricted elliptic problem is impossible.

Collision action function
Collision chains can be represented as critical points of a function of a finite number of variables -discrete action functional. This is needed for the proof of Theorems 2.1-2.2 and in subsequent publications. Since collision chains are concatenations of collision orbits, we need to describe collision orbits first.
A collision orbit γ = (γ 1 , γ 2 ) of system (H 0 ) is a pair of Kepler orbits joining the points x − , x + ∈ U . Thus description of collision orbits is reduced to the classical Lambert's problem [22] of joining the points x − , x + by a Kepler orbit.
First we join the points x − , x + by a Kepler orbit Γ : [0, τ ] → U with fixed energy E < 0, or, equivalently, fixed major semi axis a = (−2E) −1 . Due to scaling invariance of Kepler's problem without loss of generality set a = 1. Then a Kepler ellipse passing through x − , x + is determined by the second focus F such that The solution F = F (x − , x + ) of these equations exists and smoothly depends on x ± if the corresponding circles intersect transversely, i.e. (x − , x + ) lie in the set For (x − , x + ) ∈ X there exist two solutions F of equations (4.1), and we take one of them, for definiteness the one on the left side of the segment x − x + . Let Γ(x − , x + ) be the counter clock wise simple arc of the constructed Kepler ellipse joining the points x − and x + . Let be the Maupertuis action of Γ. This is a smooth rotation invariant function on X: f (e iθ x − , e iθ x + ) = f (x − , x + ).
Due to scaling invariance of the Kepler problem, for arbitrary negative energy E < 0, the Maupertuis action of a simple counter clock wise arc Γ = Γ(E, x − , x + ) connecting the points ( • Γ smoothly depends on E, x − , x + .
• The Maupertuis action of Γ is The orbits Γ n (E, x − , x + ) are nondegenerate (have non-conjugate end points) and any nondegenerate connecting orbit with E < 0 is obtained in this way.
For the classical Lambert's problem [22], when Γ is a simple elliptic arc, n = 0 or n = −1 depending on if Γ is a counterclockwise or clockwise. We set sgn 0 = 1.
The first term in (4.3) is the Maupertuis action for n complete revolutions around the Kepler ellipse, and the second is the action of a simple elliptic arc. Equation (4.4) follows from the first variation formula; it is essentially Kepler's time equation. So only inequality (4.5) is non-evident. It is enough to check it for the classical Lambert's problem with n = 0, 1. Then (4.5) can be deduced from the explicit formula (4.6), although the computation is not trivial. An equivalent statement was proved in [19].
Next we consider Lambert's problem for fixed time τ > 0. This problem involves solving the transcendental Kepler's equation so there is no explicit formula for the solution. Let D n ⊂ R + × U 2 be the open set which is the image of the diffeomorphism • Γ smoothly depends on τ, x − , x + .
• Hamilton's action is a smooth function on D n and • All nondegenerate connecting orbits with E < 0 are Γ n (τ, x − , x + ) for some n ∈ Z and (τ, Proof. We need to find the energy E < 0 such that the connecting orbit Γ = Γ n (E, x − , x + ) in Proposition 4.1 has given time τ = τ n (E, x − , x + ). Then Hence J n and −F n are Legendre transforms of each other: is convex in τ and smooth.
The initial and final total momenta of γ are given by the first variation formula Now it is easy to describe nondegenerate collision orbits γ = (γ 1 , γ 2 ) of system (H 0 ). Denote by k = [γ] = (k 1 , k 2 ) ∈ Z 2 , k j = [γ j ], the rotation vector of γ. We obtain Proposition 4.3. For any k = (k 1 , k 2 ) ∈ Z 2 and any (τ, x − , x + ) ∈ V k = D k1 ∩ D k2 : • There exists a nondegenerate collision orbit γ : • Hamilton's action of γ is By the first variation formula [2], where E is the total energy of the collision orbit γ, and y ± = y(t ± ) are total momenta at collisions. Thus (4.10) Remark 4.2. We will not need this in the present paper, but for almost all (τ, x − , x + ) ∈ V k the collision action satisfies the twist condition Thus S k is the generating function of a symplectic collision map (τ, x − , y − ) → (τ, x + , y + ). This will be important for the study of chaotic collision chains.

Remark 4.3.
It can happen that the collision orbit γ has early collisions: γ(t) ∈ ∆ for some t ∈ (0, τ ). To avoid this, we may need to delete from V k a zero measure set, see [7].
Let us now fix energy E < 0 and look for collision orbits of system (H 0 ) with energy E. The map Let L E k be the Legendre transform of −S k with respect to τ : where τ is obtained by solving the last equation (4.10). Then L E k is a smooth function on is a smooth function on W E k . The total momenta at collision are In terms of actions functions (4.3) for the Kepler problem, Remark 4.4. Due to homogeneity of the Kepler problem, where L k corresponds to energy E = −1/2.
Remark 4.5. The action functions S k and L E k can not be expressed in elementary functions. However they admit simple asymptotic representation for large k. This will be done in a subsequent publication.
In the next section we use the action functions S k and L E k to represent collision chains as critical points of discrete action functionals.

Discrete variational principles
For a given sequence k = (k 1 , . . . , k n ) ∈ Z 2n , k j ∈ Z 2 , define a discrete Hamilton's action by where s = (s 1 , . . . , s n ) ∈ R n + , x = (x 1 , . . . , x n ) ∈ U n and S k j is the action function on V k j defined by (4.8). The domain of A k is Any (s, x) ∈ V k defines (t, T, γ) ∈ Ω n as follows. Take t = (t 1 , . . . , t n ) so that s j = t j+1 − t j and set where γ(k j , s j , x j , x j+1 ) : [0, s j ] → U 2 is the collision orbit in Proposition 4.3. Then γ = γ k (s, x) is a broken trajectory of system (H 0 ) with period T = n j=1 s j and Hamilton's action A k (s, x) = A(T, γ). Of course (t, γ) is defined modulo time translation, so we identify curves which differ by time translation. Thus we defined an embedding ι : V k → Ω n = Ω n /R and A k = A • ι.
For collision chains with fixed period T , we restrict A k to

Indeed, critical points of
where −E is the Lagrange multiplier. By (4.10), E is the energy of the corresponding collision chain γ. The total momentum at collision is Thus γ satisfies (2.5)-(2.6).
Proposition 5.1 follows also from Proposition 2.2. Indeed, the functional A k is the restriction of Hamilton's action A to the set ι(V k ) ⊂ Ω n of broken extremals. This set is obtained by equating to zero the differential of A for fixed t, T, x.
In Proposition 5.1 the period T is fixed. For collision chains with fixed energy E < 0 we consider a discrete Maupertuis action functional on V k : Now T is a function on V k , and A E k (s, x) is the Maupertuis action (2.7) of the broken trajectory γ = γ k (s, x). We obtain Proposition 5.2. To any critical point (s, x) of A E k on V k there corresponds a periodic collision chain γ with energy E. All nondegenerate collision chains with energy E are obtained in this way from nondegenerate modulo rotation critical points of some A E k . Remark 5.1. Hamilton's action is invariant under rotations: A k (s, e iθ x) = A k (s, x). Thus every critical point of the functional A E k is degenerate. To obtain nondegenerate critical points we should consider the quotient functional A E k on the quotient space Let us now fix energy E < 0 and angular momentum G and consider periodic modulo rotation collision chains γ with given E, G. We obtain the discrete Maupertuis-Routh action functional The independent variables are s = (s 1 , . . . , s n ), x = (x 1 , . . . , x n ) and Φ, so the domain of A EG there corresponds a periodic modulo rotation collision chain γ = γ k (s, x, Φ) with energy E and angular momentum G. Any nondegenerate periodic modulo rotation collision chain with energy E and angular momentum G is obtained from a nondegenerate modulo rotation critical point of some A EG k . To construct orbits of system (H µ ) shadowing the collision chain γ corresponding to a critical point (s, x), we need to verify the changing direction condition. For k = (k 1 , By (4.7) the relative collision velocities (2.10) of a collision orbit γ = γ(k, τ, x − , x + ) : [0, τ ] → U 2 are given bẏ Thus the changing direction condition for the collision chain corresponding to (s, x) can be expressed as follows: We have where k = (k 1 , k 2 ) with k j = (k 1 j , . . . , k n j ) ∈ Z n and is the discrete action functional for the Kepler problem. If (s, x) is a critical point of A k with respect to x, then by (5.1), the changing direction condition (5.3) is equivalent to Next we reformulate the shadowing Theorems 2.1-2.2.
Theorem 5.1. Let (s, x) ∈ V k be a nondegenerate modulo rotation critical point of A E k satisfying the changing direction condition (5.6). Then for sufficiently small µ > 0 the corresponding T -periodic collision chain γ is O(µ)-shadowed modulo time translation by an almost collision T µ -periodic orbit γ µ of the 3 body problem with period T µ = T + O(µ).
Theorem 5.2. Let (s, x, Φ) ∈V k be a nondegenerate modulo rotation critical point of A EG k satisfying the changing direction condition (5.6). Then for sufficiently small µ > 0, the corresponding collision chain γ is O(µ)-shadowed modulo rotation and time translation by an almost collision periodic modulo rotation orbit γ µ of the 3 body problem with energy E and angular momentum G.
These discrete versions of Theorems 2.1-2.2 are most suitable for applications. In a future publication we will use them in [9] to find many nontrivial second species solutions.
For a dynamical systems reformulation, it is convenient to introduce Jacobi's discrete action functional It is defined on Equating to 0 the derivatives of A EG k (s, x) with respect to s, we obtain: Proposition 5.4. Any nondegenerate modulo symmetry periodic collision chain with energy E corresponds to a nondegenerate modulo rotation critical point x of some J E k . A critical point x = (x 1 , . . . , x n ) of J E k is a n-periodic trajectory of a discrete Lagrangian system (L E ) with a multivalued discrete Lagrangian Thus description of second species solutions is reduced to the dynamics of a discrete Lagrangian system (L E ). Under a twist condition, a periodic trajectory of system (L E ) corresponds to a periodic trajectory (x j , y j ), , of a sequence of symplectic twist maps (x j , y j ) → (x j+1 , y j+1 ) with generating functions L E k j . We postpone this reformulation to a future paper, where we deal with chaotic almost collision orbits.
The minimal degeneracy of a critical point of J E k is at least 1 due to rotational symmetry J E k (e iθ x) = J E k (x). This implies that the discrete Lagrangian system (or the corresponding symplectic map) has an integral of angular momentum G = ix j · y j . One can perform Routh's reduction in this discrete Lagrangian system reducing it to one degree of freedom [12], but this complicates the discrete Lagrangian.
For periodic modulo rotation collision chains with fixed E, G we have: Proposition 5.5. Any nondegenerate periodic modulo rotation collision chain with energy E corresponds to a nondegenerate modulo rotation critical point (x, Φ) of the discrete Jacobi-Routh action functional The proofs of Theorems 5.1 and 5.2 are modifications of the proof of Theorem 2.1 in [6]. They are based on the Levi-Civita regularization and shadowing. The proof of Theorem 5.1 will be given in the next section. The proof of Theorem 5.2 is similar and will be omitted.

Proof of Theorem 5.1
For µ = 0, the action functional (2.1) of system (H µ ) is singular when γ approaches ∆. We will formulate a variational problem for almost collision orbits of system (H µ ) with given energy E which has no singularity at ∆.
Let us fix energy E < 0. Trajectories of system (H µ ) with energy E are extremals of the Jacobi action functional Away from ∆, the functional J E µ is a regular perturbation of the Jacobi functional J E for system (H 0 ). Regularizing J E µ near ∆ requires some preparation. First we describe local behavior of trajectories of system (H 0 ) colliding with ∆. We will use the variables (this is a version of Jacobi's variables) x = α 1 q 1 + α 2 q 2 , y = p 1 + p 2 , u = q 2 − q 1 , v = α 1 p 2 − α 2 p 1 .
• γ + smoothly depends on (x 0 , y 0 , u + ) ∈ M × S ρ and • The Maupertuis action of γ + has the form Remark 6.1. On S ρ we use the polar coordinate θ, where u = ρe iθ . Thus O(ρ 2 ) means a function of x 0 , y 0 , θ whose C 2 norm is bounded by cρ 2 with c independent of ρ.
The proof is obtained by a simple shooting argument, because H 0 has no singularity at ∆: It remains to solve the equation u(τ + ) = u + for τ + and v 0 , where |v 0 | = 2αλ(x 0 , y 0 ).
• The Maupertuis action of γ has the form Proof. We find x 0 from the equation where η ± is defined in (6.6). Differentiating (6.5), we see that the Hessian matrix is nondegenerate. By the implicit function theorem, the solution x 0 = ξ(q − , q + ) is smooth.
• The Maupertuis action of γ has the form The proof of Lemma 6.3 is given in section 7. It is based on Levi-Civita regularization and a generalization of Shilnikov's Lemma [20], see also [21], to normally hyperbolic critical manifolds of a Hamiltonian system.
Thus the action J E µ (γ) = g E µ (q − , q + ) has a limit g E (q − , q + ) as µ → 0 which is the action of the reflection orbit in Proposition 6.1. The condition that the distance to ∆ is attained at t = 0 is needed only to exclude time translations, so that t ± are uniquely defined.
Proof. We need to find (x 0 , y 0 ) such that x ± µ (x 0 , y 0 , u − , u + ) = x ± . Since the implicit function theorem worked in the proof of Proposition 6.1, by (6.12), for small µ > 0 it will work also here.
Proof of Theorem 5.1. Let γ be a nondegenerate n-collision chain with energy E. Let t = (t 1 , . . . , t n ) be collision times, x = (x 1 , . . . , x n ), γ(t j ) = (x j , x j ), the corresponding collision points and y j the collision total momenta. Take δ > 0 so small that the collision points and the collision speeds v ± j = v(t j ± 0) satisfy Then (x j , y j ) ∈ M and x j ∈ D. Take small ρ > 0 and let t ± j = t j ± s ± j be the closest to t j times when q ± j = γ(t ± j ) ∈ Σ ρ . Since γ satisfies the changing direction condition, (q − j , q + j ) ∈ Q ρ if C > 0 is taken sufficiently large and ρ > 0 sufficiently small. Moreover for ξ ± j ∈ Σ ρ close to q ± j , we have (ξ − j , ξ + j ) ∈ Q ρ . Thus by Theorem 6.1 for small µ > 0 the points ξ ± j can be joined in N ρ by a trajectory γ j µ of system (H µ ) with energy E and the Maupertuis action J E µ (γ j µ ) = g E µ (ξ − j , ξ + j ). Since γ| [tj,tj+1] is nondegenerate and, by no early collisions condition, does not come near ∆, for ξ + j close to q + j and ξ − j+1 close to q − j+1 and small µ > 0, the points ξ + j and ξ − j+1 can be joined by a trajectory σ j µ of system (H µ ) with energy E and the Maupertuis action . This trajectory smoothly depends on µ also for µ = 0, and h E 0 (ξ + j , ξ − j+1 ) is the Maupertuis action of a connecting trajectory of system (H 0 ).
Combine the trajectories γ j µ , σ j µ in a broken trajectory γ µ with energy E and Maupertuis action The function f µ has a limit f 0 as µ → 0 and is independent of µ.
Looking for critical points with respect to ξ ± j with fixed ξ j we obtain f 0 = J E k (ξ 1 , . . . , ξ n ) for some k ∈ Z 2n , and J E k has a nondegenerate modulo rotation critical point x. Thus f 0 has a nondegenerate critical point q ∈ Σ 2n ρ . Then for small µ > 0 the function f µ (ξ) has a nondegenerate modulo rotation critical point ξ µ close to q. The corresponding broken trajectory γ µ has no break of velocity at intersection points ξ ± j with Σ ρ and hence γ µ is a periodic trajectory of system (H µ ) with energy E.

Levi-Civita regularization
In this section we prove Lemma 6.3. In the Jacobi variables (6.1), the Hamiltonian H µ takes the form Let us perform the Levi-Civita regularization on the fixed energy level H µ = E. We identify u, v ∈ R 2 = C with complex numbers and make a change of variables u = ξ 2 , v = η/2ξ.
Collisions of m 1 , m 2 with nonzero relative velocity correspond to the solutions asymptotic to where M is as in (6.2). This is is a compact normally hyperbolic symplectic critical manifold for H E 0 . We obtain Theorem 7.1. Collision orbits of system (H 0 ) with energy E correspond to orbits of system (H E 0 ) doubly asymptotic to M. Orbits of system (H µ ) with energy E passing O(µ)-close to the singular set ∆ correspond to orbits of system (H E µ ) on the level Γ E µ passing O( √ µ)-close to M.
Since the stable and unstable manifolds are smooth, J − is a smooth function on M × S r . Similarly we define the function J + on M × S r as the action of a solution ζ + asymptotic to M as t → −∞.