Periodic solutions around figure-eight choreography for the equal mass three-body problem

Transcription

1 Periodic solutions around figure-eight choreography for the equal mass three-body problem Hiroshi FUKUDA Kitasato University ( :30-10:00) SS25:Celestial Mechanics and N-Body Problem National Taiwan University, FC-302 1

2 In this talk we report on Morse index and periodic solutions bifurcated from the figure eight choreography under Homogeneous potential u r = 1 a and Lennard-Johns type potential u r = 1 r r r 12 2

3 Contents 1. Figure eight choreography for equal mass three body problem 2. Simó s H solution 3. Morse index and H solution 4. General H solution 5. Other bifurcated solutions under homogeneous potential 6. Bifurcated solutions under Lennard-Jones type Potential 7. Summary 3

4 Figure-eight choreography for equal mass three-body problem under homogeneous potential and Lennard-Jones potential 4

5 Figure eight choreography under homogeneous potential was found by Moore (1993) numerically, and Chenciner and Montgomery (2000) proved its existence mathematically a=1 (Newtonian Gravity) u r = 1 r Three body move in the same orbit periodically. 5

6 a=6 u r = 1 r 6 6

7 Figure eight choreography under Lennard- Jones(LJ) potential Almost same as under 1/r 6 Inhomogeneous potential u r = 1 r r 12 7

8 Figure eight choreography under Lennard- Jones(LJ) potential Almost same as figure eight under 1/r 6 Inhomogeneous potential u r = 1 r r 12 However not scalable 8

9 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 9

10 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 10

11 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 11

12 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 12

13 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 13

14 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 14

15 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 15

16 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 16

17 Minimum Period T m = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 17

18 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 18

19 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 19

20 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 20

21 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 21

22 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 22

23 T = Figure-eight choreographies of the equal mass three-body problem with Lennard- Jones-type potentials, Hiroshi Fukuda, Toshiaki Fujiwara, Hiroshi Ozaki, J. Phys. A: Math. Theor. 50, (2017). T S q = න L q, qሶ dq 0 Period T 23

24 Simó s H solution 24

25 Simó s H solution On the other hand, Simó found H solution* under homogeneous potential a = 1. The H is similar to figure-eight choreography but consists of three distinct orbits *C. Simó, Proc. Celestial Mechanics Conf. (2000) 25

26 Symmetry of H Both figure-eight choreography and the H are symmetric in x and y axes. Thus the H is considered as (figure-eight choreography) (choreographic symmetry) 26

27 H and Morse index of figure-eight choreography Last year, we found relationships between the H solution and the figure-eight choreography via numerical calculation of Morse index, by which we found many H like solutions. H. Fukuda, T. Fujiwara and H. Ozaki, J. Phys A (2018) 27

28 Morse index and H solution 28

29 Morse index N is a number of independent variational functions δq giving negative second variation S (2) at q of action S q = T 0 L q, qሶ dq: S q + hδq = S (0) + h 2 S (2) + δq 2 S (2) > 0 δq 1 S (2) < 0 q S (1) = 0 since q t is a solution of EQM, and is a critical point of S q 29

30 Morse index N for figure-eight choreography under u r = 1/r a is calculated as 4 0 a < N = 2 ( < a < ) < a ΔN = 2 N changes by ΔN = 2 at a = and a =

31 ΔN = 2 at a = Thus, in two directions, δq 1 and δq 2, S (2) changes sign: a = a = 1 δq 2 δq 1 δq 1 δq 2 S (2) = S (2) =

32 The H solution q H and Morse index We found the H solution, q H is expressed in 10 7 by q H q + hδq(θ), δq(θ) = cos Θ δq 1 + sin Θ δq 2 That is, q H is located in the plane spanned by δq 1 and δq 2. q q H δq 1 δq 2 Since solution of EOM is a critical point of manifold S, we may see shallow maximum at q H. 32

33 Three critical points Moreover there are two more critical points around q, q H with cyclic permutation of bodies. q q H δq 1 δq 1 δq 2 33

34 Choreographic transformation C Since choreography q is invariant under transformation C; cyclic permutations of bodies with time shift by T/3, T/3 T/3 T/3 but q H is not, and Cq H and C 2 q H are also the solutions where C 3 = 1. q H Cq H C 2 q H Algebraically, C rotates δq(θ) by 2π/3; Cδq(Θ) = δq(θ + 2π/3) 34

35 General H solution 35

36 Yielding critical points From the following figures, we can imagine that critical points appear when S (2) changes sign, keeping global structure of the manifold S. This is the reason why the H solution is expressed in the form q H q + hδq(θ), δq(θ) = cos Θ δq 1 + sin Θ δq 2 δq 2 δq 1 δq 1 δq 2 36

37 q H for a 1 Thus for a 1, when a , q H is also represented as q H q + hδq(θ) with h 0 and S (2) 0. a = a = S (2) = S (2) = δq 1 δq 2 δq 1 δq 2 37

38 Six q H bifurcate at a = In the both sides of a = , q H looks a = a = a = h < 0 h = 0 h > 0 Thus 6=3+3 solutions, C n (q ± hδq), n = 0,1,2, bifurcate at a = Vanderbauwhede and Fujiwara (2005) 38

39 Cyclic permutation of bodies C 2 C 1 Six congruent H; C n (q ± hδq) q hδq q + hδq 39

40 Other bifurcated solutions under homogeneous potentials 40

41 Morse index N for figure-eight choreography under u r = 1 ra also change by ΔN = 2 at a = a < N = 2 ( < a < ) < a ΔN = 2 Thus like a = , a = has bifurcation represented by q + hδq but its symmetry is different. 41

42 Bifurcation at a = yields two kind of solutions with different symmetry: D x ; Symmetric in x-axis only and q ± hδq(θ x ) are congruent. D 2 ; Symmetric at origin only and q ± hδq(θ 2 ) are congruent. 42

43 Twelve solutions from a = As H, each of them consists of three solutions with cyclic permutation of bodies, and their inversion; C n (q ± hδq Θ x ) and C n (q + hδq Θ 2 ). Thus there are six D x and six D 2 and totally twelve solutions bifurcate. 43

44 Cyclic permutation of bodies C 2 C 1 Six congruent D x ; C n (q ± hδq Θ x ) q + hδq q hδq 44

45 Cyclic permutation of bodies C 2 C 1 Six congruent D 2 ; C n (q ± hδq Θ 2 ) q + hδq q hδq 45

46 Bifurcated solutions under Lennard-Johns type potentials 46

47 Morse index N(T) for figure-eight under LJ Morse index under LJ is a function of period T, N(T), which monotonically increases from figure-eight shape α at large T to gourd-shape α +, from 0 to 12. N(T) α + α 47

48 Morse index N(T) for figure-eight under LJ There are eight steps, that is eight bifurcation points, four ΔN = 2, four ΔN = 1. N(T) ΔN = 2 ΔN = 1 α + α 48

49 Two D xy under LJ Two points with ΔN = 2 bifurcate D xy type solutions, q ± hδq, as the H solution q + hδq q hδq q + hδq q hδq 49

50 Two D x, D 2 under LJ The other two points with ΔN = 2 bifurcate D x, D 2 type solutions, q ± hδq(θ x ), q ± hδq(θ 2 ), as bifurcation at a = for homogeneous system. q + hδq(θ x ) q + hδq(θ 2 ) q + hδq(θ x ) q + hδq(θ 2 ) 50

51 C 2 under LJ A point with ΔN = 1 bifurcates C 2 type solution, q ± hδq, which is D 2 with choreographic symmetry. q + hδq q hδq 51

52 C x under LJ Another point with ΔN = 1 bifurcates C x type solutions, q ± hδq, which is D x with choreographic symmetry. q + hδq q hδq 52

53 C y under LJ The third point with ΔN = 1 bifurcates C y type solutions, q ± hδq, which is choreographic and symmetric in y-axis only q + hδq q hδq 53

54 Morse index N for figure-eight under LJ Finally a point at minimum T with ΔN = 1 bifurcates figureeight choreography itself each other. α + q hδq q + hδq α 54

55 C y T = C 2 T = D xy T = C x T = D xy T = C e = α ± T = D x, D 2 T = D x, D 2 T =

56 Summary 56

57 We observed followings numerically The point with ΔN 0 of Morse index N for figure eight choreography q in the homogeneous and LJ system corresponds to bifurcation to periodic solutions. Periodic solution bifurcated, q b, is written q b q + hδq where δq is the linear combination of ΔN independent function which changes sign of S (2) If ΔN = 1, q b is choreographic. If ΔN = 2, q b q + h cos Θ δq 1 + sin Θ δq 2 is chosen to be ΔS(q b ) critical. ΔN is the number of congruent class of functions bifurcating. 57

58 We expect There is no other bifurcation point than ΔN 0 to the periodic orbits. At each bifurcation point Euler characteristic χ for S χ = q 1 N q conserves where q is a periodic solution and N q index. its Morse 58

59 Example: C 2 bifurcation at T = For T < there is no bifurcated solution χ = ( 1) 11. For T > there are two mirror symmetric solutions χ = ( 1) 12 +2( 1) N(C2). Thus if N C 2 =odd χ conseves. 59