Example 6

Download the Jupyter Notebook here

In $\mathbb R^3$ take four masses $m_1 = m_2 = m_3 = m_4 = 1$ and consider the group $G$ with non-trivial $\mathrm{ker}\tau$ generated by $\kappa$ such that

\[ \rho(\kappa) = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1\end{pmatrix},\]

\[ \sigma(\kappa) = (1,2,3,4),\]

and such that $\bar G = G / \mathrm{ker}\tau$ is the cyclic group generated by the element $g$ such that

\[ \rho(g) = -\mathrm{Id}_3,\]

\[ \sigma(g) = ().\]

using SymOrb, GLMakie
P = initialize("non_trivial_kert.toml");

Orbit a)

starting_path = [0.7698922060880542, 0.8261311816638868, 0.843929914961185, 0.36576072609074084, 0.20188387034030897, 0.8423216631984697, 0.30402922184033176, 0.5462088209796815, 0.41957796014550175, 0.3140113556452936, 0.6169550758926127, 0.7481558347321142, 0.37084243949486984, 0.751776892139332, 0.06952200962611799, 0.2552852482940373, 0.7859092913405525, 0.14849915059451047, 0.7618732204301042, 0.020467379898463478, 0.2203162665839199, 0.6070686770569701, 0.9904310586606868, 0.209181284985119, 0.021463474983092312, 0.9122227853151593, 0.820848243941107, 0.7851204950881797, 0.0023365887877294655, 0.904285898748326, 0.008805805391491561, 0.38371835619644046, 0.11123754611254943, 0.6963439760502305, 0.8564050433086194, 0.1533799366182459, 0.4197857202785491, 0.44845817528955667, 0.9000057748259661, 0.3839166114313868, 0.1379539684028107, 0.8146833412677725, 0.8913536986316717, 0.28140825000572867, 0.7822173579758889, 0.6967057216017715, 0.27757391862976066, 0.08077520974880237, 0.9505907370399003, 0.8220294293947839, 0.8737756595309556, 0.08229992134873398, 0.6736570013639098, 0.23245539455652342, 0.39156117794874135, 0.9465326394900226, 0.7970650129306266, 0.7211245112150875, 0.3683934123105529, 0.36181135514830776, 0.19768598139930071, 0.2518069696645403, 0.7842838030727826, 0.6416423337737984, 0.11193007992882409, 0.23738219346773615, 0.7695273966805698, 0.6301972208802709, 0.027803513755643383, 0.9104734070952315, 0.4129096824393995, 0.2501734819780732, 0.5137607161076684, 0.33187512740676983, 0.5716339648815133, 0.09357253878705973, 0.0010888053776639817, 0.6941452570089673, 0.33275331687536336, 0.5544310474129187, 0.19090058436649937, 0.8131832002160323, 0.5682651864941121, 0.8301444650720248, 0.7465440102263661, 0.4672263501939524, 0.9590613890357232, 0.05582055177799028, 0.9152370016467682, 0.4889233061653593, 0.6756171936832217, 0.7755039842263829, 0.07073129514944443, 0.1587316764962292, 0.45691968598392674, 0.8686673724388403, 0.6324274645396873, 0.028833789579991942, 0.11031984565559083, 0.9081544104019114, 0.24436977049426034, 0.2191427001783821, 0.5848413494317043, 0.04184529118517011, 0.7469490964065992, 0.40750839534677585, 0.34160841010512943, 0.3386088944595802, 0.5732908963997725, 0.31718574006948275, 0.09904795716430381, 0.6032631747818907, 0.0482535751199229, 0.4109079755747601, 0.9331915782449057, 0.7540719025643625, 0.9081594460697581, 0.9762992445683114, 0.6675747670400178, 0.37742410326771825, 0.917622813433172, 0.80104525898217, 0.19040188220421772, 0.45098219950909013, 0.6872262802395805, 0.4807735601934974, 0.657558593439856, 0.6968653128427043, 0.003825217086191657, 0.7157642841559342, 0.5909656280065237, 0.4047078102938362, 0.8097424013704053, 0.7541747508010884, 0.354863023951002, 0.4901194835302358, 0.6974391072381699, 0.0646296373546571, 0.01411565972830342, 0.26045087700089875, 0.46040964429903264, 0.5058597083161396, 0.8431535173816702, 0.13225763085140974, 0.37555487630034545, 0.083214372666646, 0.20063976071092604, 0.18472157675614365, 0.7363698206215898, 0.4510128298289774, 0.09596338390730808, 0.3805994357744582, 0.18922567036995308, 0.25026859536674684, 0.9734941318395355, 0.17889583752622207, 0.5938763818631198, 0.2669471509001584, 0.1671063330491156, 0.24382740336323738, 0.15139966308559683, 0.711839251698295, 0.15661886471005482, 0.9974662734791443, 0.6480320012790215, 0.6265959704017252, 0.26999995985234737, 0.9913341374889777, 0.5538535984384668, 0.08281239571517596, 0.807523036553398, 0.05773361586672432, 0.03491722379488882, 0.7105496155371365, 0.718586667162953, 0.21341890980113265, 0.6602100574133987, 0.013817737424785315, 0.02337080491697041, 0.6818871159207063, 0.6244662685164828, 0.6103757942233987, 0.23134108763362704, 0.7235587341185841, 0.3095031065988473, 0.6259626913624945, 0.9015564087509819, 0.4023503020573752, 0.3718681272918002, 0.43709046314636146, 0.5333215347185039, 0.5012706265702123, 0.15946238876518337, 0.1198605620097627, 0.8340599579339276, 0.9123277862260235, 0.22491183765004674, 0.5694930213308784, 0.8435765001776148, 0.3117934254125939, 0.8174646684049933, 0.9986014582207676, 0.3792268649308378, 0.346927350871037, 0.2856137284138939, 0.48924625347451, 0.9790329349854476, 0.15481509561173568, 0.03987809160598621, 0.4746562642686948, 0.858093827174272, 0.8320814346933844, 0.10874796327666258, 0.4049258999775056, 0.039165276339686805, 0.8310659372345686, 0.6476334280456096, 0.481512923973906, 0.5497953642366701, 0.4397395123217642, 0.9486956326980459, 0.06810245906821089, 0.5071764073219985, 0.4359728028177208, 0.6834491906683459, 0.6231913814736773, 0.006428761088890678, 0.5660434550337077, 0.3246419079317874, 0.3921044709734669, 0.9847419125689726, 0.7848292825270861, 0.7734409427542295, 0.3505164886323686]
orbits = find_orbits(P, OneMethod(BFGS(200)), starting_path=starting_path)
path_animation(P, orbits[1].fourier_coeff)

Orbit b)

starting_path = [0.719833237888818, 0.3156903228446277, 0.2377497546149483, 0.938446795415135, 0.6525171109484798, 0.25215069290113135, 0.20383580090779319, 0.2062022104888851, 0.15057200601456822, 0.47263701595933516, 0.2986610410137528, 0.3942430628941105, 0.9820528105047532, 0.9944523374986898, 0.34486702072340214, 0.6201286589772129, 0.14378412727598278, 0.14000911525089044, 0.8779634052567217, 0.25785630995294084, 0.9497963368794856, 0.5243047519069751, 0.7351271989930398, 0.5826221439836307, 0.1496226756489092, 0.6581262353740662, 0.5839685363840956, 0.6447317430784045, 0.772178216111542, 0.5265861662258833, 0.006368152574415253, 0.6250007936366905, 0.04403846347456175, 0.47086331164766637, 0.5105225246808532, 0.7948042274282243, 0.19167063299153053, 0.013993214214713445, 0.7462687784442099, 0.867819180202428, 0.1500891778933635, 0.5885967020215034, 0.3923389449920942, 0.6637551953013537, 0.18445083394503337, 0.1687286628539001, 0.5343344492154674, 0.33472987075271177, 0.03052457393230623, 0.9714353643658519, 0.9632765212996902, 0.24381811515193097, 0.8875048030583736, 0.577330408889619, 0.5675908154758456, 0.30349471185916277, 0.24009421182215995, 0.08375206388932599, 0.07115797081341657, 0.9740367058684156, 0.4614223522785008, 0.08460750828684982, 0.26665124981373767, 0.05790366730082663, 0.8558867176238222, 0.9580764648625946, 0.4479586537501934, 0.37154378037421765, 0.52484056737673, 0.1378406361424942, 0.1888291908740981, 0.96506391885087, 0.024496410564979088, 0.7866288713719368, 0.4382553513497811, 0.8563684561104867, 0.6393410677389947, 0.253267961682072, 0.07575623176470891, 0.9127847605044965, 0.05238296093957451, 0.9327161395609761, 0.05900968709415233, 0.7763728992138513, 0.9034655336437066, 0.1151570257333906, 0.5868289991110244, 0.7596704028104854, 0.8856800606955205, 0.9712357922183503, 0.2045030763238146, 0.6954816079610194, 0.6791816747181336, 0.41010943567223423, 0.7038784264354696, 0.01997881697238113, 0.41565773490916225, 0.2203657665796187, 0.40729884761004354, 0.09856865982919416, 0.09375193883687849, 0.805157434419815, 0.23707692599572117, 0.8270202348404997, 0.38379698818736374, 0.36552867528552346, 0.1042090708873018, 0.08601959008658688, 0.3058208732475861, 0.2347130445156279, 0.5289828167049193, 0.9955528991255179, 0.20665258822864419, 0.6494520575375806, 0.20066508938500727, 0.4896125928319669, 0.5132675901172689, 0.9544645780787352, 0.9335645154522366, 0.4533900967029645, 0.8540075285452333, 0.8468249714570377, 0.3259605515620291, 0.30496968400973545, 0.40632345886638077, 0.4507054300820552, 0.11550057478165865, 0.9848848560031659, 0.18601316850550131, 0.962937643595487, 0.8421668515380124, 0.2073461398117643, 0.03604076527035216, 0.9267731339205013, 0.14424783739029934, 0.679438423090717, 0.838832619601901, 0.03653905613364672, 0.9025148082720176, 0.7642854343120009, 0.4710090882991337, 0.8520392961174306, 0.07163921337514711, 0.9781496321774426, 0.6761712860460642, 0.035522053309652524, 0.09910273140573067, 0.5699607040645247, 0.8247775140168656, 0.038920443952717765, 0.3236968182705583, 0.7406636468105057, 0.8961729171696902, 0.9177922674048761, 0.7079304358668049, 0.1388388568562846, 0.7939786068853174, 0.5965037542459015, 0.1409604406392022, 0.6090336518088901, 0.6055642857913955, 0.9660992330695456, 0.9800090259759423, 0.9681549560749664, 0.03911096711215156, 0.03289647878151236, 0.6362296992371407, 0.8401102140352676, 0.6015467499972412, 0.2716157713178846, 0.6854830515579006, 0.6052546945545445, 0.7571904036716196, 0.06055483079124846, 0.08413921081002829, 0.2578220556458959, 0.5181415484642048, 0.5915399144642763, 0.6321303850557604, 0.6570777917258923, 0.19889194729144533, 0.4897452880516825, 0.609658224281927, 0.4951357304014775, 0.02941873437638498, 0.2869350040310219, 0.1558823394568697, 0.7704223257593007, 0.795085544587958, 0.8149881729401831, 0.15513784134080733, 0.49419941934984624, 0.4585734680358543, 0.9155873726269382, 0.8998747901456302, 0.7440766389713159, 0.5315016866413438, 0.7741055305161327, 0.13352248508960507, 0.37015407370154463, 0.789160166717228, 0.032008394966727916, 0.40212596929349054, 0.4861663082261759, 0.7211301674656069, 0.09305086187653877, 0.00208298851112243, 0.13355731805364823, 0.5622995770718108, 0.2282087375005435, 0.2938512260291851, 0.7814529692072368, 0.8770515785873737, 0.1025913161998735, 0.47151581556330513, 0.18585847779408482, 0.23237678127910222, 0.7426761632674381, 0.20669084985596564, 0.1074022724768341, 0.9818145914466081, 0.24183831244175058, 0.0708145016973507, 0.14129520872225665, 0.24517383777236945, 0.4804932544680891, 0.8036847799205397, 0.7148805376992884, 0.9932180520118964, 0.09449734814266542, 0.41310978455268654, 0.6482688864690396, 0.8915343966076775, 0.38320943818685516]
orbits = find_orbits(P, OneMethod(Newton(200)), starting_path=starting_path)
path_animation(P, orbits[1].fourier_coeff)

Orbit c)

starting_path = [0.44710121599437924, 0.6821366846155972, 0.9804509321580339, 0.010221858329031552, 0.7416747443234546, 0.646487345986261, 0.6245524680454806, 0.49244827509372324, 0.7585904320013361, 0.25480025675330087, 0.2565201887016135, 0.531691139632819, 0.9853005635812985, 0.38634617589602605, 0.6575823929372141, 0.012033815479751242, 0.7845095896274416, 0.8457368655066106, 0.2803008970153159, 0.4474499767109814, 0.8221627344465734, 0.868524320181091, 0.3609062488223983, 0.24697931815614416, 0.6011428222470878, 0.527094664779945, 0.024959922489210196, 0.992192969724247, 0.5941593257118112, 0.8265321329754883, 0.16897818313946222, 0.2025409893766399, 0.528367958396191, 0.035136150781698516, 0.11449900409236147, 0.6971167543173222, 0.20689533304694485, 0.5264211177551429, 0.1812223822799528, 0.23459738582485534, 0.7121576781984669, 0.5061275041125114, 0.06773734238696061, 0.8141777818760009, 0.8092521232285382, 0.5986289357272743, 0.6407032058670715, 0.8302671690422174, 0.25063810706664547, 0.9569358869368843, 0.19142216510712529, 0.33610908781392046, 0.2228814593537859, 0.992243006998407, 0.43135496725980693, 0.38545274719063427, 0.6271080785805446, 0.10023060772513626, 0.1429484843312715, 0.901775255920774, 0.8970362349103289, 0.2655812114807313, 0.8497415537899505, 0.45392296182524405, 0.02922247042890458, 0.43322216358083465, 0.5650096239352702, 0.6067742203052751, 0.9775088355790904, 0.6320874509895578, 0.08255982473847101, 0.8784627711709652, 0.7897345319436058, 0.9418825990597539, 0.398041550189819, 0.5766860069849383, 0.5657624009070927, 0.28967743737764906, 0.6032676874325386, 0.1350188981630286, 0.9498830567932406, 0.4565938370121676, 0.9531998540490156, 0.9270778239453008, 0.6704886906761254, 0.12336802290802906, 0.5390709056654192, 0.8031446268323589, 0.18987828626340286, 0.640809383818778, 0.6497014254519508, 0.8617473922422615, 0.8317295913479116, 0.9816126260014372, 0.9469019645053846, 0.71773074149578, 0.24284436897273798, 0.8602277164539495, 0.4466967220472533, 0.22396775867898266, 0.038310885364845726, 0.6698109475984154, 0.3153221974152707, 0.2967210596486016, 0.7259100007236916, 0.9494154667387082, 0.26875877724992703, 0.6664377022712673, 0.9472566823035877, 0.6044118018063156, 0.25516144772177973, 0.299281982418372, 0.769482372853951, 0.08975160035392149, 0.372786908879965, 0.9321054266124297, 0.280049429631278, 0.7837284491364603, 0.011498009487897032, 0.7799972654213857, 0.8025708724409881, 0.9050295587017683, 0.8327209342072183, 0.25209353402810997, 0.27679411533892395, 0.7018358146767577, 0.015685772852260582, 0.5390048973343222, 0.09374430719968818, 0.9688616064882504, 0.2143715817570273, 0.34036360842605484, 0.5914173562201498, 0.6993332554447593, 0.997453108687559, 0.25509654253527636, 0.6838087883780875, 0.7070970949589298, 0.2527047210240132, 0.3173156888251528, 0.6284891004181542, 0.6925357298725451, 0.0838304204087017, 0.032000662238337196, 0.8890845312041781, 0.7948470190429583, 0.3661065409379143, 0.3614027732994777, 0.15648986755559058, 0.5107342961463519, 0.5990092688263975, 0.9847969745592996, 0.3042787873471855, 0.144633643531336, 0.7309076841763511, 0.03795832991757975, 0.47945813341314103, 0.2543676947239326, 0.6018026601362, 0.10448029086559663, 0.939902022664921, 0.274727170706648, 4.6083734122959186e-5, 0.25634077300354186, 0.5572969685513811, 0.4763824240358612, 0.1862737287840177, 0.07641457891356918, 0.6623069818469199, 0.17957603432348435, 0.8395278237978477, 0.5154914648847507, 0.1491128637658985, 0.7134669662122607, 0.6621461313208266, 0.14813188462835236, 0.2254747972517548, 0.18857182173886122, 0.2383880601480023, 0.5500502638494053, 0.6297380085055924, 0.13434960346330427, 0.6506674554343473, 0.22318462084277468, 0.7036890469905456, 0.1827101395017947, 0.7878717768201781, 0.6388236460362299, 0.8087628986975401, 0.8183780788974814, 0.9634077042123816, 0.2196235300077436, 0.32151735228662737, 0.43240977830402005, 0.608384739414471, 0.7601276220672626, 0.9425177531898248, 0.36457756877739667, 0.453207753567065, 0.23873831753495034, 0.7599032650927172, 0.9877142548024612, 0.7111567198690604, 0.3110424477680047, 0.4928728340891356, 0.351650009126539, 0.02634358903598122, 0.03836113938719943, 0.2514505654776795, 0.24458529409034158, 0.9462080841129379, 0.3998174119441793, 0.2618604703478977, 0.6529452684423926, 0.1353054820376851, 0.7120748430014272, 0.42789901163416, 0.8875514554908029, 0.9180508426830545, 0.612925068466186, 0.11362450755070908, 0.6654323641010322, 0.002960668016975365, 0.032060154629373505, 0.457898435599015, 0.18710247675162972, 0.21393302157953475, 0.2528534110956995, 0.20227845742070505, 0.6287963551763216, 0.1657990633994746, 0.7498471983885785, 0.05220953845048859, 0.789135857039381]
orbits = find_orbits(P, OneMethod(Newton(200)), starting_path=starting_path)
path_animation(P, orbits[1].fourier_coeff)