Free Group Element of the Collisionless Periodic Orbits in the free-fall three-body problem

Xiaoming LI and Shijun LIAO

Shanghai Jiaotong University, China


Parameters:

Body mass: m1, m2, m3

Newton's gravitational constant: G = 1

Initial positions: (-0.5, 0), (0.5, 0), (x, y)

Initial velocities: (0,0), (0,0), (0,0)

Free Group Element
Fi(m1,m2,m3) Free group element(word)
F1(1,1,1) AbABbaBa
F2(1,1,1) ABaBbAba
F3(1,1,1) AbAbABaAbaBaBa
F4(1,1,1) AbABBaaAAbbaBa
F5(1,1,1) AbabABaAbaBABa
F6(1,1,1) AbaBAbaABabABa
F7(1,1,1) AbaaBAbbBBabAABa
F8(1,1,1) AbaBBabAaBAbbABa
F9(1,1,1) AbAbABaaAAbaBaBa
F10(1,1,1) AbaBBBabAaBAbbbABa
F11(1,1,1) AAbABBAbaABabbaBaa
F12(1,1,1) AAbABBBaaAAbbbaBaa
F13(1,1,1) AbABBBaaBaAbAAbbbaBa
F14(1,1,1) AbABAbaBabBAbABabaBa
F15(1,1,1) AbaBBAbaBaBAbBabAbABabbABa
F16(1,1,1) AbaBBABaaaBAbBabAAAbabbABa
F17(1,1,1) AbABBAbaBaBAbBabAbABabbaBa
F18(1,1,1) AAbABBBBBBBBAbAAaaBabbbbbbbbaBaa
F19(1,1,1) AbABBBBBAbaaaaabAbBaBAAAAABabbbbbaBa
F20(1,1,1) AbABBBBBAbaaaaaabAbBaBAAAAAABabbbbbaBa
F21(1,1,1) AAAbABBBBBBBBBBBAbAAaaBabbbbbbbbbbbaBaaa
F22(1,1,1) AAAbABBBBBBBBBBAbAAAaaaBabbbbbbbbbbaBaaa
F23(1,1,1) AbABBBBBBBAbaaaaabAbBaBAAAAABabbbbbbbaBa
F24(1,1,1) AbABBBBBAbaaaaaaabAbBaBAAAAAAABabbbbbaBa
F25(1,1,1) AAAbABBBBBBBBBBBBAbAAAaaaBabbbbbbbbbbbbaBaaa
F26(1,1,1) AbABBBBBBAbaaaaaaabAbbBBaBAAAAAAABabbbbbbaBa
F27(1,1,1) AbaBABAbabABABabaBABAbaBbABababABAbabaBABababABa
F28(1,1,1) AbABBBBBAbaaaaaaaaaaabAbbBBaBAAAAAAAAAAABabbbbbaBa
F29(1,1,1) AAAbABBBBBBBBBBBBBBBBBBAbAAAaaaBabbbbbbbbbbbbbbbbbbaBaaa
F30(1,1,1) AAAAbABBBBBBBBBBBBBBBBBAbAAAAaaaaBabbbbbbbbbbbbbbbbbaBaaaa
F1(1,0.8,0.8) BaBbAb
F2(1,0.8,0.8) AbaBbABa
F3(1,0.8,0.8) AAbABbaBaa
F4(1,0.8,0.8) AbaBBbbABa
F5(1,0.8,0.8) AbABBBbbbaBa
F6(1,0.8,0.8) AbaBAbaABabABa
F7(1,0.8,0.8) AbaBAbbBBabABa
F8(1,0.8,0.8) AbABBBBbbbbaBa
F9(1,0.8,0.8) AbaBBabAaBAbbABa
F10(1,0.8,0.8) AbaBBAbaABabbABa
F11(1,0.8,0.8) AbABBBBBbbbbbaBa
F12(1,0.8,0.8) AbaBBBabAaBAbbbABa
F13(1,0.8,0.8) AbaBBBAbaABabbbABa
F14(1,0.8,0.8) AbaBAbaBBaAbbABabABa
F15(1,0.8,0.8) AAAbaBBAbaABabbABaaa
F16(1,0.8,0.8) AAAbABBAbaABabbaBaaa
F17(1,0.8,0.8) BaBAAAABaBBbbAbaaaabAb
F18(1,0.8,0.8) AAAbaBBBabAaBAbbbABaaa
F19(1,0.8,0.8) AbaBABababABaAbaBABAbabABa
F20(1,0.8,0.8) AbaBBBAbbAbABaAbaBaBBabbbABa
F21(1,0.8,0.8) AAbaBBAbabaBAbBabABABabbABaa
F22(1,0.8,0.8) AbaBABababABBabAaBAbbaBABAbabABa
F23(1,0.8,0.8) AbaBABababABABabAaBAbabaBABAbabABa
F24(1,0.8,0.8) AbaBABAbabABABabAaBAbabaBABababABa
F25(1,0.8,0.8) AbaBABAbabABABababABaAbaBABAbabaBABababABa
F26(1,0.8,0.8) AbaBAbbAbaBBabAbaBBaBAbaBbABabAbbABaBAbbABaBBabABa
F27(1,0.8,0.8) AbaBABAbabaBABababABABabAaBAbabaBABAbabABABababABa
F28(1,0.8,0.8) AbaBABababABABaBABAbabaBABabAaBAbabABABababAbabaBABAbabABa
F29(1,0.8,0.8) AbaBABababAbabaBABababABAbaBABAbabABaAbaBABababABabaBABAbabABABaBABAbabABa
F1(1,0.8,0.6) AbBa
F2(1,0.8,0.6) AbaABa
F3(1,0.8,0.6) AbAaBa
F4(1,0.8,0.6) AbaaAABa
F5(1,0.8,0.6) AbAbBaBa
F6(1,0.8,0.6) AAbaABaa
F7(1,0.8,0.6) BaBaaAAbAb
F8(1,0.8,0.6) AbaaaAAABa
F9(1,0.8,0.6) AbAbaABaBa
F10(1,0.8,0.6) AbAbbBBaBa
F11(1,0.8,0.6) BaBaaaAAAbAb
F12(1,0.8,0.6) AbAbbbBBBaBa
F13(1,0.8,0.6) ABaBaaAAbAba
F14(1,0.8,0.6) AABaBaAbAbaa
F15(1,0.8,0.6) AAbAbbBBaBaa
F16(1,0.8,0.6) AAAbAbBaBaaaA
F17(1,0.8,0.6) AbAbbbbBBBBaBa
F18(1,0.8,0.6) AbABAbbBBabaBa
F19(1,0.8,0.6) AAbAbaaAABaBaa
F20(1,0.8,0.6) AAABaBaAbAbaaa
F21(1,0.8,0.6) AbABaaaAAAbaBa
F22(1,0.8,0.6) AbAbAbAbBaBaBaBa
F23(1,0.8,0.6) AbAbbbbbBBBBBaBa
F24(1,0.8,0.6) AAbAbbbbBBBBaBaa
F25(1,0.8,0.6) AbAbaaaaAAAABaBa
F26(1,0.8,0.6) AAAbAbbbBBBaBaaa
F27(1,0.8,0.6) AAbAbaaaAAABaBaaA
F28(1,0.8,0.6) AbAbabABaAbaBABaBa
F29(1,0.8,0.6) AbAbbaBAbBabABBaBa
F30(1,0.8,0.6) AbAbbabAbBaBABBaBa
F31(1,0.8,0.6) AbaBABAbAaBababABa
F32(1,0.8,0.6) AbAbBaBAbBabAbBaBa
F33(1,0.8,0.6) AbaBABabAaBAbabABa
F34(1,0.8,0.6) AAbAbAbaBbABaBaBaa
F35(1,0.8,0.6) AbAbAbABaAbaBaBaBa
F36(1,0.8,0.6) AbABBAbAbBaBabbaBa
F37(1,0.8,0.6) AbAbaaaaaAAAAABaBa
F38(1,0.8,0.6) AbABABaBaAbAbabaBa
F39(1,0.8,0.6) AAbAbbbbbBBBBBaBaa
F40(1,0.8,0.6) AAAbAbbbbBBBBaBaaaA
F41(1,0.8,0.6) AAbAbaaaaAAAABaBaaA
F42(1,0.8,0.6) AbAbabABaaAAbaBABaBa
F43(1,0.8,0.6) AbABABaBaaAAbAbabaBa
F44(1,0.8,0.6) AAbAbabAbAaBaBABaBaa
F45(1,0.8,0.6) AbAbaabAbAaBaBAABaBa
F46(1,0.8,0.6) AAbAbbaBAbBabABBaBaa
F47(1,0.8,0.6) AAbAbabABaAbaBABaBaa
F48(1,0.8,0.6) AbAbBBaBAbBabAbbBaBa
F49(1,0.8,0.6) AbABBBAbAbBaBabbbaBa
F50(1,0.8,0.6) AbAbaabABaAbaBAABaBa
F51(1,0.8,0.6) AbAbaAbAbaABaBaABaBa
F52(1,0.8,0.6) AAAbAbbbbbBBBBBaBaaaA
F53(1,0.8,0.6) AbaBabaBabAaBAbABAbABa
F54(1,0.8,0.6) AbAbaaabAbAaBaBAAABaBa
F55(1,0.8,0.6) AbABBBBAbAbBaBabbbbaBa
F56(1,0.8,0.6) AbaBAABaBaaAAbAbaabABa
F57(1,0.8,0.6) AbABBBBBAbAaBabbbbbaBa
F58(1,0.8,0.6) AAAbAbabAbAaBaBABaBaaa
F59(1,0.8,0.6) AbAbaaaabAbAaBaBAAAABaBa
F60(1,0.8,0.6) AbABBBBAbAbbBBaBabbbbaBa
F61(1,0.8,0.6) AbABBBBBBAbAaBabbbbbbaBa
F62(1,0.8,0.6) AbABABaBABAbAaBababAbabaBa
F63(1,0.8,0.6) AbABBBBBBBAbAaBabbbbbbbaBa
F64(1,0.8,0.6) AbABBBBBBBBAbAaBabbbbbbbbaBa
F65(1,0.8,0.6) AbaBAbaBabABabAaBAbaBAbABabABa
F66(1,0.8,0.6) AbABBBBBBBBBAbAaBabbbbbbbbbaBa
F67(1,0.8,0.6) AAbAbabAbabAbaBbABaBABaBABaBaa
F68(1,0.8,0.6) ABaBabaBabbbaBaBbAbABBBAbABAbAba
F69(1,0.8,0.6) AbaBabABabABAbABBBAbAaBabbbaBabaBAbaBAbABa
F1(1,0.8,0.4) AbAbBaBa
F2(1,0.8,0.4) AbABbaBa
F3(1,0.8,0.4) AbAbAaBaBa
F4(1,0.8,0.4) AbAbbBBaBa
F5(1,0.8,0.4) AAbAbBaBaa
F6(1,0.8,0.4) BaBBBbbbAb
F7(1,0.8,0.4) AbAbAAaaBaBa
F8(1,0.8,0.4) AAbAbbBBaBaa
F9(1,0.8,0.4) AbAbbbBBBaBa
F10(1,0.8,0.4) BaBaaaAAAbAb
F11(1,0.8,0.4) AAbAbaABaBaa
F12(1,0.8,0.4) AAbAbbbBBBaBaa
F13(1,0.8,0.4) AbAbbaBbABBaBa
F14(1,0.8,0.4) AbAbbbbBBBBaBa
F15(1,0.8,0.4) AABaBaaAAbAbaa
F16(1,0.8,0.4) AAAbAbbBBaBaaa
F17(1,0.8,0.4) AAbAbAAaaBaBaa
F18(1,0.8,0.4) AAAbAbAaBaBaaa
F19(1,0.8,0.4) BaBaaaaAAAAbAb
F20(1,0.8,0.4) AAAbAbaABaBaaa
F21(1,0.8,0.4) AAbAbbbbBBBBaBaa
F22(1,0.8,0.4) AbAbAAAAaaaaBaBa
F23(1,0.8,0.4) AAAbAbbbBBBaBaaa
F24(1,0.8,0.4) AbAbbbbbBBBBBaBa
F25(1,0.8,0.4) AAABaBaaAAbAbaaa
F26(1,0.8,0.4) BaBaaaaaAAAAAbAb
F27(1,0.8,0.4) AbAbabAbAaBaBABaBa
F28(1,0.8,0.4) AbAbABAbAaBabaBaBa
F29(1,0.8,0.4) AbAbAAAAAaaaaaBaBa
F30(1,0.8,0.4) AAAbAbbbbBBBBaBaaa
F31(1,0.8,0.4) AAbAbbbbbBBBBBaBaa
F32(1,0.8,0.4) AAAbAbaaaAAABaBaaa
F33(1,0.8,0.4) AAAAbAbbbBBBaBaaaa
F34(1,0.8,0.4) AbAbbbbbbBBBBBBaBa
F35(1,0.8,0.4) AABaBaaaaAAAAbAbaa
F36(1,0.8,0.4) AbAbaabAbAaBaBAABaBa
F37(1,0.8,0.4) AbABAbAbAAaaBaBabaBa
F38(1,0.8,0.4) AAAbAbbbbbBBBBBaBaaa
F39(1,0.8,0.4) AAbAbbbbbbBBBBBBaBaa
F40(1,0.8,0.4) AAAAbAbbbbbBBBBBaBaaaa
F41(1,0.8,0.4) AbAbaaaabAbAaBaBAAAABaBa
F42(1,0.8,0.4) AAAAbAbbbbbbBBBBBBaBaaaa
F43(1,0.8,0.4) AbaBaBaBaBabAaBAbAbAbAbABa
F44(1,0.8,0.4) AbAbAbAbAbAbAbAaBaBaBaBaBaBaBa
F1(1,0.8,0.2) BaAb
F2(1,0.8,0.2) AbAaBa
F3(1,0.8,0.2) BabBAb
F4(1,0.8,0.2) AbAbBaBa
F5(1,0.8,0.2) BabbBBAb
F6(1,0.8,0.2) AbAbAaBaBa
F7(1,0.8,0.2) BaBaaAAbAb
F8(1,0.8,0.2) AbAbbBBaBa
F9(1,0.8,0.2) AbABaAbaBa
F10(1,0.8,0.2) BaBaaaAAAbAb
F11(1,0.8,0.2) AbAbbbBBBaBa
F12(1,0.8,0.2) AAbAbAaBaBaa
F13(1,0.8,0.2) AAbAbbBBaBaa
F14(1,0.8,0.2) BabbbbBBBBAb
F15(1,0.8,0.2) AbAbbbbBBBBaBa
F16(1,0.8,0.2) AAbAbAAaaBaBaa
F17(1,0.8,0.2) AAAbAbAaBaBaaa
F18(1,0.8,0.2) BabbbbbBBBBBAb
F19(1,0.8,0.2) AbAbbaBbABBaBa
F20(1,0.8,0.2) AAAbAbbBBaBaaa
F21(1,0.8,0.2) AbAbAbAbBaBaBaBa
F22(1,0.8,0.2) AbAbAAAAaaaaBaBa
F23(1,0.8,0.2) AAAbAbAAaaBaBaaa
F24(1,0.8,0.2) AAAbAbbbBBBaBaaa
F25(1,0.8,0.2) AAbAbAAAAaaaaBaBaa
F26(1,0.8,0.2) AbAbAAAAAaaaaaBaBa
F27(1,0.8,0.2) AAAAbAbbbbBBBBaBaaaa
F28(1,0.8,0.2) AbAbBaBaabAaBAAbAbBaBa
F29(1,0.8,0.2) AAAAAbAbbbbBBBBaBaaaaa
F30(1,0.8,0.2) AbAbBaBAbABaAbaBabAbBaBa
F31(1,0.8,0.2) AbAbBaBaBaBaAbAbAbAbBaBa
F32(1,0.8,0.2) AAAAAbAbbbbbBBBBBaBaaaaa
F1(0.6,0.8,1) BaAb
F2(0.6,0.8,1) AbAaBa
F3(0.6,0.8,1) BabBAb
F4(0.6,0.8,1) AbAbBaBa
F5(0.6,0.8,1) BabbBBAb
F6(0.6,0.8,1) AbAbAaBaBa
F7(0.6,0.8,1) BaBaaAAbAb
F8(0.6,0.8,1) AbAbbBBaBa
F9(0.6,0.8,1) AbABaAbaBa
F10(0.6,0.8,1) BaBaaaAAAbAb
F11(0.6,0.8,1) AbAbbbBBBaBa
F12(0.6,0.8,1) AAbAbAaBaBaa
F13(0.6,0.8,1) AAbAbbBBaBaa
F14(0.6,0.8,1) BabbbbBBBBAb
F15(0.6,0.8,1) AbAbbbbBBBBaBa
F16(0.6,0.8,1) AAbAbAAaaBaBaa
F17(0.6,0.8,1) AAAbAbAaBaBaaa
F18(0.6,0.8,1) BabbbbbBBBBBAb
F19(0.6,0.8,1) AbAbbaBbABBaBa
F20(0.6,0.8,1) AAAbAbbBBaBaaa
F21(0.6,0.8,1) AbAbAbAbBaBaBaBa
F22(0.6,0.8,1) AbAbAAAAaaaaBaBa
F23(0.6,0.8,1) AAAbAbAAaaBaBaaa
F24(0.6,0.8,1) AAAbAbbbBBBaBaaa
F25(0.6,0.8,1) AAbAbAAAAaaaaBaBaa
F26(0.6,0.8,1) AbAbAAAAAaaaaaBaBa
F27(0.6,0.8,1) AAAAbAbbbbBBBBaBaaaa
F28(0.6,0.8,1) AbAbBaBaabAaBAAbAbBaBa
F29(0.6,0.8,1) AAAAAbAbbbbBBBBaBaaaaa
F30(0.6,0.8,1) AbAbBaBAbABaAbaBabAbBaBa
F1(1,0.412,0.267) BaBaAbAb
F2(1,0.412,0.267) ABaBbAba
F3(1,0.412,0.267) BaBaBbAbAb
F4(1,0.412,0.267) BaBaaAAbAb
F5(1,0.412,0.267) AbAbAaBaBa
F6(1,0.412,0.267) AABaBbAbaa
F7(1,0.412,0.267) AbAbAAaaBaBa
F8(1,0.412,0.267) AAABaBbAbaaa
F9(1,0.412,0.267) AAABaBAabAbaaa
F10(1,0.412,0.267) AAAABaBbAbaaaa
F11(1,0.412,0.267) AbAbAAAaaaBaBa
F12(1,0.412,0.267) BaBAbbAbBaBBabAb
F13(1,0.412,0.267) AAAAABaBbAbaaaaa
F14(1,0.412,0.267) BaBaBaBaBaBbAbAbAbAbAb
F15(1,0.412,0.267) AbAbAbAbAbAbAbaBbABaBaBaBaBaBaBa
F1(0.393,0.288,0.662) AbaabAaBAABa
F2(0.393,0.288,0.662) AbabaBbABABa
F3(0.393,0.288,0.662) AbaabaBbABAABa
F4(0.393,0.288,0.662) AbabbaBbABBABa
F5(0.393,0.288,0.662) AbaabAAaaBAABa
F6(0.393,0.288,0.662) AbababAaBABABa
F7(0.393,0.288,0.662) AAbabaBbABABaa
F8(0.393,0.288,0.662) AbabaBBbbABABa
F9(0.393,0.288,0.662) AbaaabaBbABAAABa
F10(0.393,0.288,0.662) AAbaabaBbABAABaa
F11(0.393,0.288,0.662) AbaabaBBbbABAABa
F12(0.393,0.288,0.662) AbaabAAAaaaBAABa
F13(0.393,0.288,0.662) AbababAAaaBABABa
F14(0.393,0.288,0.662) AbabbaBBbbABBABa
F15(0.393,0.288,0.662) AbaaaabaBbABAAAABa
F16(0.393,0.288,0.662) AbaaabaBBbbABAAABa
F17(0.393,0.288,0.662) AbaabaBBBbbbABAABa
F18(0.393,0.288,0.662) AbabaBABAababABABa
F19(0.393,0.288,0.662) AbabaBABABababAaBABAbababABABa
F1(1,0.1,0.01) BaBbAb
F2(1,0.1,0.01) ABaBbAba
F3(1,0.1,0.01) BaBaAbAb
F4(1,0.1,0.01) BaBBbbAb
F5(1,0.1,0.01) AbaBAabABa
F6(1,0.1,0.01) AbAbaABaBa
F7(1,0.1,0.01) BaBBBbbbAb
F8(1,0.1,0.01) AbAbaaAABaBa
F9(1,0.1,0.01) AABaBAabAbaa
F10(1,0.1,0.01) AbaBAAaabABa
F11(1,0.1,0.01) BaBBaBbAbbAb
F12(1,0.1,0.01) AAABaBAabAbaaa
F13(1,0.1,0.01) AbaBAAAaaabABa
F14(1,0.1,0.01) AABaBAAaabAbaa
F15(1,0.1,0.01) BaBBaBBbbAbbAb
F16(1,0.1,0.01) AAABaBAAaabAbaaa
F17(1,0.1,0.01) AbAbaaaaAAAABaBa
F18(1,0.1,0.01) AAbaBAAAaaabABaa
F19(1,0.1,0.01) AAbAbaaaAAABaBaa
F20(1,0.1,0.01) AbaBAAAAaaaabABa
F21(1,0.1,0.01) BaBBBaBBbbAbbbAb
F22(1,0.1,0.01) AbAbabAbAaBaBABaBa
F23(1,0.1,0.01) AAAABaBAAaabAbaaaa
F24(1,0.1,0.01) AAABaBAAAaaabAbaaa
F25(1,0.1,0.01) AAbaBAAAAaaaabABaa
F26(1,0.1,0.01) AbaBAAAAAaaaaabABa
F27(1,0.1,0.01) BaBaBBaBaAbAbbAbAb
F28(1,0.1,0.01) BaBBabbAbBaBBAbbAb
F29(1,0.1,0.01) BaBBaBaBBbbAbAbbAb
F30(1,0.1,0.01) BaBBaBBaBbAbbAbbAb
F31(1,0.1,0.01) BaBabbAbAbBaBaBBAbAb
F32(1,0.1,0.01) BaBabbAbAAaaBaBBAbAb
F33(1,0.1,0.01) BaBaBbAbAbBaBaBbAbAb
F34(1,0.1,0.01) AAbAbbbAbAbBaBaBBBaBaa
F35(1,0.1,0.01) AAAAABaBAAAaaabAbaaaaa
F36(1,0.1,0.01) AAAABaBAAAAaaaabAbaaaa
F37(1,0.1,0.01) AAAbaBAAAAAaaaaabABaaa
F38(1,0.1,0.01) BaBBaBaBaBBbbAbAbAbbAb
F39(1,0.1,0.01) BaBBaBBaBaBBbbAbAbbAbbAb
F40(1,0.1,0.01) BaBBaBBaBBaBbAbbAbbAbbAb
F41(1,0.1,0.01) BaBBaBBBBaBBBbbbAbbbbAbbAb
F42(1,0.1,0.01) BaBAbABaBabAbaABaBAbAbaBabAb
F43(1,0.1,0.01) BaBBaBBaBaBaBBbbAbAbAbbAbbAb
F44(1,0.1,0.01) BaBAbAbaBaBaBaBBbbAbAbAbABaBabAb
F45(1,0.1,0.01) BaBBaBBaBBaBBaBBaAbbAbbAbbAbbAbbAb
F46(1,0.1,0.01) BaBabAbbAbbAbbAbbAaBBaBBaBBaBBaBAbAb
F47(1,0.1,0.01) BaBBaBBaBBaBBaBBaBbAbbAbbAbbAbbAbbAb
F48(1,0.1,0.01) BaBBaBBaBaBBaBaBBaBBbbAbbAbAbbAbAbbAbbAb