ON CHENCINER-MONTGOMERY S ORBIT IN THE THREE-BODY PROBLEM. Kuo-Chang Chen. (Communicated by Antonio Ambrosetti)

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1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS Volume 7, Number, January pp ON CHENCINER-MONTGOMERY S ORBIT IN THE THREE-BODY PROBLEM Kuo-Chang Chen School of Mathematics University of Minnesota, Minneapolis, MN 55455, USA (Communicated by Antonio Ambrosetti) Abstract. Recently A. Chenciner and R. Montgomery found a remarkable periodic orbit for a three-body problem by variational methods. On this orbit all masses chase each other along a figure-eight circuit without any collision, and the solution curve is indeed a minimizer of the action functional on a properly chosen path space. One technical difficulty, where numerical integration had been used in their proof, is to show that the minimizing orbit does not experience any collision. In this paper a short analytical proof will be presented.. Introduction. Recently A. Chenciner and R. Montgomery [] found a remarkable periodic orbit for the planar three-body problem with equal masses. On this orbit all masses chase each other along a figure-eight circuit without any collision, and the solution curve as a loop on the shape sphere (see []) passes through Euler s configurations and is symmetric with respect to those circles representing isosceles triangle configurations. It turns out that each piece of the curve joining Euler s configuration to the nearby isosceles triangle configurations minimizes the action functional constrained on some properly chosen path spaces. More precisely, let E i represents Euler s configurations with ith mass in the middle and M i represents isosceles triangles with ith mass equally distant from the other two, the piece of solution curve goes from E to M (for instance) is the minimum of T A(x) := ẋ dt r r r on Λ := {x H ([,T], X ):x() E,x(T) M }, where r ij = x i x j and X := {x =(x,x,x ) (R ) : x + x + x =}. The functional A restricted on Λ is coercive and a standard argument in calculus of variation shows that the infimum of A on Λ is attained. In order to prove that the action minimizing orbit does not experience any collision, A. Chenciner and R. Montgomery compare the action of orbits with the action for two-body problems, then use a result due to W. Gordon [] to obtain a lower bound for the action over all paths in Λ with collisions, and then they choose a particular equipotential test path that has even lower action. Those two estimates are dangerously close so that the proof has to rely on numerical integration (although they mention that this can be avoided by perturbation methods, but the proof is significantly longer). 99 Mathematics Subject Classification. 4C5, 49J7, 7F7, 7F5. Key words and phrases. three-body problem, variational methods, action minimizing orbit. 85

2 86 KUO-CHANG CHEN Let Λ c be the collection of collision orbits in Λ; that is, Λ c := {x Λ:x i (τ) = x j (τ) for some i j, τ [,T]}. The purpose of this paper is to provide an analytic proof for the fact that no minimizer of A on Λ is in Λ c. The idea of proof is to, instead of comparing only with Keplerian orbits, take all collision orbits into consideration to get a better lower bound for the action over these paths. To avoid numerical integration, a test path that is simpler than the equipotential path is chosen, and direct computation shows that its action is lower than the improved lower bound for admissible collision orbits.. Some results from Chenciner-Montgomery []. The main theorem in [] states: Theorem.. (Chenciner-Montgomery) Fix T >. There exists an eight-shaped planar loop q : R/ T Z R, q() = (, ), with the following properties: (a) q(t)+q(t + T/) + q(t + T/) = for any t; (b) q(σ t) =σ q(t) and q(τ t) =τ q(t) for any t R, whereσ t = t + T/, τ t = t + T/, σ (q,q )=( q,q ), τ (q,q )=(q, q ). (c) x(t) =(q(t),q(t + T/),q(t + T/)) is a zero angular momentum T -periodic solution of the planar three-body problem with equal masses. t = t = T t = T t = T Figure. Chenciner-Montgomery s orbit (T = T/). The motion can be easily visualized by projecting it onto shape spheres (see figure ), which are obtained by first introducing the Jacobi s coordinates ( ) (z,z ):= (x x ), (x (x + x )) C, and then quotient out rotations by the mapping (z,z ) ( z z, z z )=:(u,u + iu ) R C. A level set I (c), c>, of the moment of inertial I := x x is a -sphere and is mapped onto a -sphere u + u + u = c. By using spherical coordinates (u,u,u )=(r cos φ cos θ, r cos φ sin θ, r sin φ)

3 CHENCINER-MONTGOMERY S ORBIT 87 and a lemma by W.-Y. Hsiang (see also []), the potential energy restricted on I = has the following nice representation: U(θ, φ) = + cos φ cos θ + + cos φ cos(θ + π ) + + cos φ cos(θ + 4π ). Using this representation the equipotential path on I = joining E (i.e. φ = θ =)andm (i.e. θ = π or 4π 5 ) is implicitly defined by U(θ, φ) =. Numerical estimates by C. Simó (see []) indicates that the action of a equipotential path on some shape sphere I = I which moves from E to M at constant speed is very close to the actual minimum of A on Λ. M M M E E E Figure. The unit shape sphere I = and the level curve U(θ, φ) = 5. A. Chenciner and R. Montgomery s proof for Theorem. consists of three parts. Firstly they reduce the minimization problem of A to the minimization of reduced action functional A red, where A red (x) = T K red (x)+u(x)dt, and K red (x) = ẋ ω I (ω is the angular momentum) comes from Saari s decomposition of kinetic energy. A minimizer of A on Λ has zero angular momentum, and therefore it also minimizes A red on Λ. The quotient metric corresponding to K red is given by ds = dr + r 4 (cos φdθ + dφ ).

4 88 KUO-CHANG CHEN In particular the unit shape sphere I = is isometric to the standard sphere of radius. The second part of their proof, where numerical integration had been used, is to prove the inequality inf A(x) < inf A(x). ( ) x Λ x Λ c The upper bound they got for the left hand side is approximately.56t /,and the right hand side has a lower bound ( π 4 )/ T /.558T /. The last part describes the shape of the orbit, including showing that each lobe of the figure-eight is starshaped.. An analytic proof of ( ). Lemma.. Assume T =.ThenA(x) >.87 for any x Λ c. Proof. Fix any x Λ c and define δ = δ x := max x(s ) x(s ). s,s [,T ] Case :. δ. First observe that α + β (α + β ) for all α, β R. If the initial distance between mass m (or m )andm is d, and the initial distance between m and m is d, then the assumption that x Λ c experience collision at some τ [, ] by some m i and m j implies δ x i (τ) x i () + x j (τ) x j () d. The first inequality holds with τ replaced by any t [, ]. With these observations we obtain r (t) x (t) x () + x () x () + x () x (t) ( x (t) x () + x (t) x () )+d δ. Similarly, r (t) δ, and r (t) ( x (t) x () + x (t) x () )+d δ. Then, by Cauchy-Schwartz inequality, A(x) = ẋ dt dt r r r ( ) ẋ dt + δ + δ. δ + δ + δ Note that the function of δ on the last row is strictly increasing for δ,and hence its absolute minimum over the smaller interval [., + ) occurs at δ =. where its value (.8889) is bigger than.87. Case : δ<.. Assume for now x Λ c experiences collision(s) by masses m and m. Other cases can be handled in the same way.

5 CHENCINER-MONTGOMERY S ORBIT 89 and ( /, Let w = ẋ dt) then for any s [, ] x (s) x () ẋ (t) dt w, δ x (s) x () + x (s) x () + x (s) x () ( x (s) x () + x (s) x () ) + x (s) x () x (s) x () + x (s) x () + x (s) x () = x (s) x (). In the last line we used the center of mass condition to replace x + x by x. Similar estimates hold for x and x replacing x, yielding δ x i(s) x i () for i =,,. Instead of using the estimates for r and r in the previous case the following version will give us a better bound for the action functional: r (t) x (t) x () + x () x () + x () x (t) w + d + δ w + ( + )δ, and similarly, r (t) w +d + δ w + ( + )δ. Then ẋ + + dt w r r + w + ( + )δ + w + ( + )δ > w + w + ( + ). + w + ( + ). The function on the second row with w has minimum γ(.97448) >.97 at w Let A be the action of the collinear Keplerian orbit starting at zero velocity and collide at time t =. It is not hard to derive the explicit formula for A : A = (π )/. By Gordon s theorem [], A(x) = ẋ + ẋ + A + ẋ + + dt r r > = This completes the proof. dt + r ẋ + r + r dt

6 9 KUO-CHANG CHEN Corollary.. For general T>, we have A(x) >.87T / for any x Λ c. Proof. This can be easily obtained by scaling r and t properly. More precisely, the action of any y H ([,T], X ) differs from the action of the scaled path ỹ(t) := T / y(tt ) H ([, ], X ) by a multiple of T /. Lemma.. There is a path x Λ with A(x) <.64T /. Proof. Consider the collection of paths lying on fixed shape spheres I = I which move from E to M at constant speed along the great circle {φ = arctan( sin θ)}. In particular on I = the potential energy on the circle {φ = arctan( sin θ)} is V (θ) = + + cos θ + + cos(θ+ π ) + cos(θ+ 4π ) +4 sin θ +4 sin θ +4 sin θ which has maximum value κ(.5574) <.56 on [, π ]. Let x(i) Λbethe path on I = I that moves from E (φ = θ =)tom (φ = θ = π/) at constant speed. Let η be the length of the path x () (as a path on the standard sphere of radius with standard metric). It is easy to see that η = arccos( )( ) < The action of x (I) satisfies the inequality: T ( A(x (I) η I )= T ) + I V (θ(t))dt < ( η ) I + κt. T I ( The minimum of the right hand side over all I > isati = this particular I, ( )( ) η κt / ( ) κt / + κt ) / κt η. For A(x (I) ) < T η η = (κη)/ T / (.66494T / ) <.64 T /. The inequality ( ) now follows directly from Corollary. and Lemma.. Acknowledgements. The author would like to express his gratitude to his advisor R. Moeckel for his guidance and many stimulating discussions, to the referee for valuable suggestions, and to A. Chenciner, R. Montgomery for their inspiring discovery and helpful comments. REFERENCES [] Chenciner, A.; Montgomery, R., A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math. (to appear). [] Gordon, W., A minimizing property of Keplerian orbits, Amer.J.Math.99. (977), [] Moeckel, R., Some qualitative features of the three-body problem, Contemporary Math. Vol. 8, (988), Received June ; revised September. address: kchen@math.umn.edu