eds.rlg 74 KB

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  1. Tue Feb 10 12:28:07 2004 run on Linux
  2. *** ^ redefined
  3. +++ depends redefined
  4. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  5. % Twisting type N solutions of GR %
  6. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  7. % The problem is to analyse an ansatz for a particular type of vacuum
  8. % solution to Einstein's equations for general relativity. The analysis was
  9. % described by Finley and Price (Proc Aspects of GR and Math Phys
  10. % (Plebanski Festschrift), Mexico City June 1993). The equations resulting
  11. % from the ansatz are:
  12. % F - F*gamma = 0
  13. % 3 3
  14. %
  15. % F *x + 2*F *x + x *F - x *Delta*F = 0
  16. % 2 2 1 2 1 2 1 2 2 1
  17. %
  18. % 2*F *x + 2*F *x + 2*F *x + 2*F *x + x *F = 0
  19. % 2 3 2 3 2 2 3 3 3 2 2 3 3 2 2 3 2 2 3 3
  20. %
  21. % Delta =0 Delta neq 0
  22. % 3 1
  23. %
  24. % gamma =0 gamma neq 0
  25. % 2 1
  26. % where the unknowns are {F,x,gamma,Delta} and the indices refer to
  27. % derivatives with respect to an anholonomic basis. The highest order is 4,
  28. % but the 4th order jet bundle is too large for practical computation, so
  29. % it is necessary to construct partial prolongations. There is a single
  30. % known solution, due to Hauser, which is verified at the end.
  31. on evallhseqp,edssloppy,edsverbose;
  32. off arbvars,edsdebug;
  33. pform {F,x,Delta,gamma,v,y,u}=0;
  34. pform v(i)=0,omega(i)=1;
  35. indexrange {i,j,k,l}={1,2,3};
  36. % Construct J1({v,y,u},{x}) and transform coordinates. Use ordering
  37. % statement to get v eliminated in favour of x where possible.
  38. % NB Coordinate change cc1 is invertible only when x(-1) neq 0.
  39. J1 := contact(1,{v,y,u},{x});
  40. j1 := EDS({d x - x *d u - x *d v - x *d y},d u^d v^d y)
  41. u v y
  42. korder x(-1),x(-2),v(-3);
  43. cc1 := {x(-v) = x(-1),
  44. x(-y) = x(-2),
  45. x(-u) = -x(-1)*v(-3)};
  46. cc1 := {x =x ,
  47. v 1
  48. x =x ,
  49. y 2
  50. x = - x *v }
  51. u 1 3
  52. J1 := restrict(pullback(J1,cc1),{x(-1) neq 0});
  53. j1 := EDS({d x + v *x *d u - x *d v - x *d y},d u^d v^d y)
  54. 3 1 1 2
  55. % Set up anholonomic cobasis
  56. bc1 := {omega(1) = d v - v(-3)*d u,
  57. omega(2) = d y,
  58. omega(3) = d u};
  59. 1 2 3
  60. bc1 := {omega = - v *d u + d v,omega =d y,omega =d u}
  61. 3
  62. J1 := transform(J1,bc1);
  63. 1 2 1 2 3
  64. j1 := EDS({d x - x *omega - x *omega },omega ^omega ^omega )
  65. 1 2
  66. % Prolong to J421: 4th order in x, 2nd in F and 1st in rest
  67. J2 := prolong J1$
  68. Prolongation using new equations:
  69. - x
  70. 2 3
  71. v =---------
  72. 3 2 x
  73. 1
  74. - x
  75. 1 3
  76. v =---------
  77. 3 1 x
  78. 1
  79. x =x
  80. 2 1 1 2
  81. x neq 0
  82. 1
  83. J20 := J2 cross {F}$
  84. J31 := prolong J20$
  85. Prolongation using new equations:
  86. 2*x *x - x *x
  87. 1 3 2 3 1 2 3 3
  88. v =-------------------------
  89. 3 3 2 2
  90. (x )
  91. 1
  92. 2
  93. - x *x + 2*(x )
  94. 1 3 3 1 1 3
  95. v =--------------------------
  96. 3 3 1 2
  97. (x )
  98. 1
  99. - x *x + x *x
  100. 1 2 2 3 1 2 2 3
  101. x =--------------------------
  102. 2 3 2 x
  103. 1
  104. x *x - x *x
  105. 1 2 3 1 1 2 1 3
  106. x =-----------------------
  107. 2 3 1 x
  108. 1
  109. x =x
  110. 2 2 1 1 2 2
  111. - x *x + x *x
  112. 1 1 2 3 1 2 3 1
  113. x =--------------------------
  114. 1 3 2 x
  115. 1
  116. x *x - x *x
  117. 1 1 3 1 1 1 1 3
  118. x =-----------------------
  119. 1 3 1 x
  120. 1
  121. x =x
  122. 1 2 1 1 1 2
  123. x neq 0
  124. 1
  125. J310 := J31 cross {Delta,gamma}$
  126. J421 := prolong J310$
  127. Prolongation using new equations:
  128. - f *x + f *x
  129. 1 2 3 2 3 1
  130. f =----------------------
  131. 3 2 x
  132. 1
  133. f *x - f *x
  134. 1 3 1 1 1 3
  135. f =-------------------
  136. 3 1 x
  137. 1
  138. f =f
  139. 2 1 1 2
  140. 2 2
  141. 3*x *x *x - 6*(x ) *x + 3*x *x *x - (x ) *x
  142. 1 3 3 1 2 3 1 3 2 3 1 3 1 2 3 3 1 2 3 3 3
  143. v =-----------------------------------------------------------------------
  144. 3 3 3 2 3
  145. (x )
  146. 1
  147. 2 3
  148. - x *(x ) + 6*x *x *x - 6*(x )
  149. 1 3 3 3 1 1 3 3 1 3 1 1 3
  150. v =--------------------------------------------------
  151. 3 3 3 1 3
  152. (x )
  153. 1
  154. x
  155. 2 3 3 2
  156. 2
  157. - 2*x *x *x + 2*x *x *x - x *x *x + (x ) *x
  158. 1 2 3 1 2 3 1 2 1 3 2 3 1 2 1 2 3 3 1 2 2 3 3
  159. =--------------------------------------------------------------------------
  160. 2
  161. (x )
  162. 1
  163. 2 2
  164. x *(x ) - 2*x *x *x - x *x *x + 2*x *(x )
  165. 1 2 3 3 1 1 2 3 1 3 1 1 2 1 3 3 1 1 2 1 3
  166. x =---------------------------------------------------------------------
  167. 2 3 3 1 2
  168. (x )
  169. 1
  170. - x *x + x *x
  171. 1 2 2 2 3 1 2 2 2 3
  172. x =------------------------------
  173. 2 2 3 2 x
  174. 1
  175. x *x - x *x
  176. 1 2 2 3 1 1 2 2 1 3
  177. x =---------------------------
  178. 2 2 3 1 x
  179. 1
  180. x =x
  181. 2 2 2 1 1 2 2 2
  182. x
  183. 1 3 3 2
  184. 2
  185. - 2*x *x *x + 2*x *x *x - x *x *x + x *(x )
  186. 1 1 3 1 2 3 1 1 1 3 2 3 1 1 1 2 3 3 1 2 3 3 1
  187. =--------------------------------------------------------------------------
  188. 2
  189. (x )
  190. 1
  191. 2 2
  192. x *(x ) - 2*x *x *x - x *x *x + 2*x *(x )
  193. 1 1 3 3 1 1 1 3 1 3 1 1 1 1 3 3 1 1 1 1 3
  194. x =---------------------------------------------------------------------
  195. 1 3 3 1 2
  196. (x )
  197. 1
  198. - x *x + x *x
  199. 1 1 2 2 3 1 2 2 3 1
  200. x =------------------------------
  201. 1 2 3 2 x
  202. 1
  203. x *x - x *x
  204. 1 1 2 3 1 1 1 2 1 3
  205. x =---------------------------
  206. 1 2 3 1 x
  207. 1
  208. x =x
  209. 1 2 2 1 1 1 2 2
  210. - x *x + x *x
  211. 1 1 1 2 3 1 1 2 3 1
  212. x =------------------------------
  213. 1 1 3 2 x
  214. 1
  215. x *x - x *x
  216. 1 1 1 3 1 1 1 1 1 3
  217. x =---------------------------
  218. 1 1 3 1 x
  219. 1
  220. x =x
  221. 1 1 2 1 1 1 1 2
  222. x neq 0
  223. 1
  224. cc4 := first pullback_maps;
  225. x *f - f *x
  226. 1 2 3 1 2 3
  227. cc4 := {f =-------------------,
  228. 3 2 x
  229. 1
  230. x *f - f *x
  231. 1 1 3 1 1 3
  232. f =-------------------,
  233. 3 1 x
  234. 1
  235. f =f ,
  236. 2 1 1 2
  237. 2
  238. v =( - (x ) *x + 3*x *x *x + 3*x *x *x
  239. 3 3 3 2 1 2 3 3 3 1 1 3 3 2 3 1 1 3 2 3 3
  240. 2 3
  241. - 6*(x ) *x )/(x ) ,
  242. 1 3 2 3 1
  243. 2 3
  244. - (x ) *x + 6*x *x *x - 6*(x )
  245. 1 1 3 3 3 1 1 3 3 1 3 1 3
  246. v =--------------------------------------------------,
  247. 3 3 3 1 3
  248. (x )
  249. 1
  250. 2
  251. x =((x ) *x - 2*x *x *x - x *x *x
  252. 2 3 3 2 1 2 2 3 3 1 1 2 3 2 3 1 1 2 2 3 3
  253. 2
  254. + 2*x *x *x )/(x ) ,
  255. 1 2 1 3 2 3 1
  256. x
  257. 2 3 3 1
  258. 2 2
  259. (x ) *x - 2*x *x *x - x *x *x + 2*x *(x )
  260. 1 1 2 3 3 1 1 2 3 1 3 1 1 2 1 3 3 1 2 1 3
  261. =---------------------------------------------------------------------,
  262. 2
  263. (x )
  264. 1
  265. x *x - x *x
  266. 1 2 2 2 3 1 2 2 2 3
  267. x =---------------------------,
  268. 2 2 3 2 x
  269. 1
  270. x *x - x *x
  271. 1 1 2 2 3 1 2 2 1 3
  272. x =---------------------------,
  273. 2 2 3 1 x
  274. 1
  275. x =x ,
  276. 2 2 2 1 1 2 2 2
  277. 2
  278. x =((x ) *x - 2*x *x *x - x *x *x
  279. 1 3 3 2 1 1 2 3 3 1 1 1 3 2 3 1 1 1 2 3 3
  280. 2
  281. + 2*x *x *x )/(x ) ,
  282. 1 1 1 3 2 3 1
  283. x
  284. 1 3 3 1
  285. 2 2
  286. (x ) *x - 2*x *x *x - x *x *x + 2*x *(x )
  287. 1 1 1 3 3 1 1 1 3 1 3 1 1 1 1 3 3 1 1 1 3
  288. =---------------------------------------------------------------------,
  289. 2
  290. (x )
  291. 1
  292. x *x - x *x
  293. 1 1 2 2 3 1 1 2 2 3
  294. x =---------------------------,
  295. 1 2 3 2 x
  296. 1
  297. x *x - x *x
  298. 1 1 1 2 3 1 1 2 1 3
  299. x =---------------------------,
  300. 1 2 3 1 x
  301. 1
  302. x =x ,
  303. 1 2 2 1 1 1 2 2
  304. x *x - x *x
  305. 1 1 1 2 3 1 1 1 2 3
  306. x =---------------------------,
  307. 1 1 3 2 x
  308. 1
  309. x *x - x *x
  310. 1 1 1 1 3 1 1 1 1 3
  311. x =---------------------------,
  312. 1 1 3 1 x
  313. 1
  314. x =x ,
  315. 1 1 2 1 1 1 1 2
  316. x neq 0}
  317. 1
  318. % Apply first order de and restrictions
  319. de1 := {Delta(-3) = 0,
  320. gamma(-2) = 0,
  321. Delta(-1) neq 0,
  322. gamma(-1) neq 0};
  323. de1 := {delta =0,
  324. 3
  325. gamma =0,
  326. 2
  327. delta neq 0,
  328. 1
  329. gamma neq 0}
  330. 1
  331. J421 := pullback(J421,de1)$
  332. % Main de in original coordinates
  333. de2 := {F(-3,-3) - gamma*F,
  334. x(-1)*F(-2,-2) + 2*x(-1,-2)*F(-2)
  335. + (x(-1,-2,-2) - x(-1)*Delta)*F,
  336. x(-2,-3)*(F(-2,-3)+F(-3,-2)) + x(-2,-2,-3)*F(-3)
  337. + x(-2,-3,-3)*F(-2) + (1/2)*x(-2,-2,-3,-3)*F};
  338. de2 := {f - f*gamma,
  339. 3 3
  340. f *x + 2*f *x + x *f - x *delta*f,
  341. 2 2 1 2 1 2 1 2 2 1
  342. 2*f *x + 2*f *x + 2*f *x + 2*f *x + x *f
  343. 2 3 2 3 2 2 3 3 3 2 2 3 3 2 2 3 2 2 3 3
  344. --------------------------------------------------------------------}
  345. 2
  346. % This is not expressed in terms of current coordinates.
  347. % Missing coordinates are seen from 1-form variables in following
  348. d de2 xmod cobasis J421;
  349. {d f *x }
  350. 3 2 2 3
  351. % The necessary equation is contained in the last prolongation
  352. pullback(d de2,cc4) xmod cobasis J421;
  353. {}
  354. % Apply main de
  355. pb1 := first solve(pullback(de2,cc4),{F(-3,-3),F(-2,-2),F(-2,-3)});
  356. pb1 := {f =f*gamma,
  357. 3 3
  358. - 2*f *x - x *f + x *delta*f
  359. 2 1 2 1 2 2 1
  360. f =--------------------------------------,
  361. 2 2 x
  362. 1
  363. 2
  364. 2*f *(x ) - 2*f *x *x - 2*f *x *x - x *x *f
  365. 1 2 3 2 1 2 3 3 3 1 2 2 3 1 2 2 3 3
  366. f =----------------------------------------------------------------}
  367. 2 3 4*x *x
  368. 1 2 3
  369. Y421 := pullback(J421,pb1)$
  370. % Check involution
  371. on ranpos;
  372. characters Y421;
  373. {15,7,0}
  374. dim_grassmann_variety Y421;
  375. 28
  376. % 15+2*7 = 29 > 28: Y421 not involutive, so prolong
  377. Y532 := prolong Y421$
  378. Prolongation using new equations:
  379. - gamma *x
  380. 1 2 3
  381. gamma =----------------
  382. 3 2 x
  383. 1
  384. gamma *x - gamma *x
  385. 1 3 1 1 1 3
  386. gamma =---------------------------
  387. 3 1 x
  388. 1
  389. gamma =0
  390. 1 2
  391. delta *x
  392. 1 2 3
  393. delta =-------------
  394. 2 3 x
  395. 1
  396. delta =delta
  397. 2 1 1 2
  398. delta *x
  399. 1 1 3
  400. delta =-------------
  401. 1 3 x
  402. 1
  403. 2 2
  404. f =(2*f *x *x + f *x *x - 2*f *(x ) + f *(x ) *gamma
  405. 1 3 3 1 3 1 3 1 1 1 3 3 1 1 1 3 1 1
  406. 2 2
  407. + gamma *(x ) *f)/(x )
  408. 1 1 1
  409. 3 2 2
  410. f =( - 2*f *x *(x ) + 4*f *x *x *(x ) - 2*f *(x ) *x *x
  411. 1 3 2 1 1 1 2 3 1 2 1 3 1 2 3 1 2 1 2 3 3 2 3
  412. 2 3 2
  413. - 2*f *(x ) *x *x - 2*f *x *(x ) + 2*f *x *x *(x )
  414. 1 3 1 2 2 3 2 3 1 1 1 2 3 1 1 2 3 1 2 3
  415. 2
  416. - 2*f *x *x *(x ) + 2*f *x *x *x *x
  417. 1 1 2 1 3 2 3 1 1 3 1 2 2 3 2 3
  418. 2 2
  419. - f *(x ) *x *x - 2*f *x *(x ) *x
  420. 1 1 2 2 3 3 2 3 2 1 2 3 3 1 2 3
  421. 2
  422. + 4*f *x *x *x *x + 2*f *x *(x ) *x
  423. 2 1 2 3 1 3 1 2 3 2 1 2 3 1 2 3 3
  424. 2
  425. + 2*f *x *x *x *x - 4*f *x *(x ) *x
  426. 2 1 2 1 3 3 1 2 3 2 1 2 1 3 2 3
  427. 2
  428. - 2*f *x *x *x *x - 2*f *x *(x ) *x
  429. 2 1 2 1 3 1 2 3 3 3 1 2 2 3 1 2 3
  430. 2
  431. + 2*f *x *x *x *x + 2*f *x *(x ) *x
  432. 3 1 2 2 1 3 1 2 3 3 1 2 3 1 2 2 3
  433. 2
  434. - 2*f *x *x *x *x + x *(x ) *x *f
  435. 3 1 2 1 3 1 2 2 3 1 2 3 1 2 2 3 3
  436. 2 2 2
  437. - x *x *x *x *f - (x ) *x *x *f)/(4*(x ) *(x ) )
  438. 1 2 1 3 1 2 2 3 3 1 2 2 3 3 1 2 3 1 2 3
  439. f *x - f *x
  440. 1 1 3 1 1 1 1 3
  441. f =-----------------------
  442. 1 3 1 x
  443. 1
  444. 3 2 2
  445. f =(2*f *x *(x ) + 4*f *x *x *(x ) - 2*f *(x ) *x *x
  446. 1 2 3 1 1 1 2 3 1 2 1 3 1 2 3 1 2 1 2 3 3 2 3
  447. 2 3 2
  448. - 2*f *(x ) *x *x - 2*f *x *(x ) + 2*f *x *x *(x )
  449. 1 3 1 2 2 3 2 3 1 1 1 2 3 1 1 2 3 1 2 3
  450. 2
  451. - 2*f *x *x *(x ) + 2*f *x *x *x *x
  452. 1 1 2 1 3 2 3 1 1 3 1 2 2 3 2 3
  453. 2 2
  454. - f *(x ) *x *x - 2*f *x *(x ) *x
  455. 1 1 2 2 3 3 2 3 2 1 2 3 3 1 2 3
  456. 2
  457. + 4*f *x *x *x *x + 2*f *x *(x ) *x
  458. 2 1 2 3 1 3 1 2 3 2 1 2 3 1 2 3 3
  459. 2
  460. + 2*f *x *x *x *x - 4*f *x *(x ) *x
  461. 2 1 2 1 3 3 1 2 3 2 1 2 1 3 2 3
  462. 2
  463. - 2*f *x *x *x *x - 2*f *x *(x ) *x
  464. 2 1 2 1 3 1 2 3 3 3 1 2 2 3 1 2 3
  465. 2
  466. + 2*f *x *x *x *x + 2*f *x *(x ) *x
  467. 3 1 2 2 1 3 1 2 3 3 1 2 3 1 2 2 3
  468. 2
  469. - 2*f *x *x *x *x + x *(x ) *x *f
  470. 3 1 2 1 3 1 2 2 3 1 2 3 1 2 2 3 3
  471. 2 2 2
  472. - x *x *x *x *f - (x ) *x *x *f)/(4*(x ) *(x ) )
  473. 1 2 1 3 1 2 2 3 3 1 2 2 3 3 1 2 3 1 2 3
  474. 2 2
  475. f =(delta *(x ) *f - 2*f *x *x - f *x *x + f *(x ) *delta
  476. 1 2 2 1 1 1 2 1 2 1 1 1 2 2 1 1 1
  477. - 2*f *x *x + 2*f *x *x - x *x *f + x *x *f)/
  478. 2 1 1 2 1 2 1 1 1 2 1 1 2 2 1 1 1 1 2 2
  479. 2
  480. (x )
  481. 1
  482. f =f
  483. 1 2 1 1 1 2
  484. 2
  485. v =(4*x *(x ) *x - 24*x *x *x *x
  486. 3 3 3 3 2 1 3 3 3 1 2 3 1 3 3 1 3 1 2 3
  487. 2 3 2
  488. + 6*x *(x ) *x + 24*(x ) *x - 12*(x ) *x *x
  489. 1 3 3 1 2 3 3 1 3 2 3 1 3 1 2 3 3
  490. 2 3 4
  491. + 4*x *(x ) *x - (x ) *x )/(x )
  492. 1 3 1 2 3 3 3 1 2 3 3 3 3 1
  493. 3 2 2 2
  494. v =( - x *(x ) + 8*x *x *(x ) + 6*(x ) *(x )
  495. 3 3 3 3 1 1 3 3 3 3 1 1 3 3 3 1 3 1 1 3 3 1
  496. 2 4 4
  497. - 36*x *(x ) *x + 24*(x ) )/(x )
  498. 1 3 3 1 3 1 1 3 1
  499. 2 3 3
  500. x =( - 12*f *(x ) *(x ) + 12*f *x *x *(x )
  501. 2 3 3 3 2 1 3 1 2 3 1 1 3 1 2 3
  502. 2 2 3
  503. - 6*f *(x ) *x *(x ) - 4*f *(x ) *x *x
  504. 1 1 2 3 3 2 3 2 1 2 3 3 3 2 3
  505. 3 2 3 2
  506. + 6*f *(x ) *(x ) - 8*f *(x ) *(x ) *gamma
  507. 2 1 2 3 3 2 1 2 3
  508. 3 3
  509. - 6*f *(x ) *x *x + 6*f *(x ) *x *x
  510. 3 1 2 2 3 3 2 3 3 1 2 2 3 2 3 3
  511. 2 2 2
  512. - 6*x *(x ) *(x ) *f + 12*x *x *x *(x ) *f
  513. 1 2 3 3 1 2 3 1 2 3 1 3 1 2 3
  514. 2 2
  515. - 6*x *(x ) *x *x *f + 6*x *x *x *(x ) *f
  516. 1 2 3 1 2 3 3 2 3 1 2 1 3 3 1 2 3
  517. 2 2
  518. - 12*x *(x ) *(x ) *f + 6*x *x *x *x *x *f
  519. 1 2 1 3 2 3 1 2 1 3 1 2 3 3 2 3
  520. 2 3
  521. - 2*x *(x ) *x *x *f + 3*(x ) *x *x *f
  522. 1 2 1 2 3 3 3 2 3 1 2 2 3 3 2 3 3
  523. 3 3
  524. - 4*(x ) *x *x *f*gamma)/(2*(x ) *x *f)
  525. 1 2 2 3 2 3 1 2 3
  526. 3 2 2
  527. x =(x *(x ) - 3*x *x *(x ) - 3*x *x *(x )
  528. 2 3 3 3 1 1 2 3 3 3 1 1 2 3 3 1 3 1 1 2 3 1 3 3 1
  529. 2 2
  530. + 6*x *(x ) *x - x *x *(x ) + 6*x *x *x *x
  531. 1 2 3 1 3 1 1 2 1 3 3 3 1 1 2 1 3 3 1 3 1
  532. 3 3
  533. - 6*x *(x ) )/(x )
  534. 1 2 1 3 1
  535. 3 3 2
  536. x =( - 12*f *x *(x ) + 12*f *x *(x ) - 6*f *x *x *(x )
  537. 2 2 3 3 3 1 3 1 2 3 1 1 3 2 3 1 1 2 3 3 2 3
  538. 2 2 2
  539. - 4*f *(x ) *x *x + 6*f *(x ) *(x )
  540. 2 1 2 3 3 3 2 3 2 1 2 3 3
  541. 2 2 2
  542. - 8*f *(x ) *(x ) *gamma - 6*f *(x ) *x *x
  543. 2 1 2 3 3 1 2 2 3 3 2 3
  544. 2 2
  545. + 6*f *(x ) *x *x + 3*(x ) *x *x *f
  546. 3 1 2 2 3 2 3 3 1 2 2 3 3 2 3 3
  547. 2 2
  548. - 4*(x ) *x *x *f*gamma)/(2*(x ) *x *f)
  549. 1 2 2 3 2 3 1 2 3
  550. 3 2
  551. x =(12*f *x *(x ) + 6*f *x *x *(x )
  552. 2 2 3 3 2 1 2 1 2 3 1 1 2 2 3 2 3
  553. 2 2
  554. + 24*f *x *x *(x ) - 24*f *x *x *(x )
  555. 2 1 2 3 1 2 3 2 1 2 1 3 2 3
  556. 2 2
  557. - 6*f *(x ) *x *x + 6*f *(x ) *x *x
  558. 2 1 2 2 3 3 2 3 2 1 2 2 3 2 3 3
  559. 2
  560. + 12*f *x *x *(x ) - 12*f *x *x *x *x
  561. 3 1 2 2 1 2 3 3 1 2 1 2 2 3 2 3
  562. 2 2 2
  563. - 4*f *(x ) *x *x + 6*f *(x ) *(x )
  564. 3 1 2 2 2 3 2 3 3 1 2 2 3
  565. 2 2 2
  566. - 8*f *(x ) *(x ) *delta + 8*x *x *(x ) *f
  567. 3 1 2 3 1 2 2 3 1 2 3
  568. 2
  569. - 8*x *x *(x ) *f + 4*x *x *x *x *f
  570. 1 2 2 1 3 2 3 1 2 2 1 2 3 3 2 3
  571. 2
  572. - 6*x *x *x *x *f + 3*(x ) *x *x *f
  573. 1 2 1 2 2 3 3 2 3 1 2 2 3 3 2 2 3
  574. 2 2
  575. - 4*(x ) *x *x *delta*f)/(2*(x ) *x *f)
  576. 1 2 3 3 2 3 1 2 3
  577. 3 2
  578. x =(12*f *x *(x ) + 6*f *x *x *(x )
  579. 2 2 2 3 3 1 2 1 2 3 1 1 2 2 3 2 3
  580. 2 2
  581. + 24*f *x *x *(x ) - 24*f *x *x *(x )
  582. 2 1 2 3 1 2 3 2 1 2 1 3 2 3
  583. 2 2
  584. - 6*f *(x ) *x *x + 6*f *(x ) *x *x
  585. 2 1 2 2 3 3 2 3 2 1 2 2 3 2 3 3
  586. 2
  587. + 12*f *x *x *(x ) - 12*f *x *x *x *x
  588. 3 1 2 2 1 2 3 3 1 2 1 2 2 3 2 3
  589. 2 2 2
  590. - 4*f *(x ) *x *x + 6*f *(x ) *(x )
  591. 3 1 2 2 2 3 2 3 3 1 2 2 3
  592. 2 2 2
  593. - 8*f *(x ) *(x ) *delta + 12*x *x *(x ) *f
  594. 3 1 2 3 1 2 2 3 1 2 3
  595. 2
  596. - 12*x *x *(x ) *f + 6*x *x *x *x *f
  597. 1 2 2 1 3 2 3 1 2 2 1 2 3 3 2 3
  598. 2
  599. - 6*x *x *x *x *f + 3*(x ) *x *x *f
  600. 1 2 1 2 2 3 3 2 3 1 2 2 3 3 2 2 3
  601. 2 2
  602. - 4*(x ) *x *x *delta*f)/(2*(x ) *x *f)
  603. 1 2 3 3 2 3 1 2 3
  604. - x *x + x *x
  605. 1 2 2 2 2 3 1 2 2 2 2 3
  606. x =----------------------------------
  607. 2 2 2 3 2 x
  608. 1
  609. x *x - x *x
  610. 1 2 2 2 3 1 1 2 2 2 1 3
  611. x =-------------------------------
  612. 2 2 2 3 1 x
  613. 1
  614. x =x
  615. 2 2 2 2 1 1 2 2 2 2
  616. 2
  617. x =( - 3*x *(x ) *x + 6*x *x *x *x
  618. 1 3 3 3 2 1 1 3 3 1 2 3 1 1 3 1 3 1 2 3
  619. 2
  620. - 3*x *(x ) *x + 3*x *x *x *x
  621. 1 1 3 1 2 3 3 1 1 1 3 3 1 2 3
  622. 2 2
  623. - 6*x *(x ) *x + 3*x *x *x *x - x *(x ) *x
  624. 1 1 1 3 2 3 1 1 1 3 1 2 3 3 1 1 1 2 3 3 3
  625. 3 3
  626. + x *(x ) )/(x )
  627. 1 2 3 3 3 1 1
  628. 3 2 2
  629. x =(x *(x ) - 3*x *x *(x ) - 3*x *x *(x )
  630. 1 3 3 3 1 1 1 3 3 3 1 1 1 3 3 1 3 1 1 1 3 1 3 3 1
  631. 2 2
  632. + 6*x *(x ) *x - x *x *(x ) + 6*x *x *x *x
  633. 1 1 3 1 3 1 1 1 1 3 3 3 1 1 1 1 3 3 1 3 1
  634. 3 3
  635. - 6*x *(x ) )/(x )
  636. 1 1 1 3 1
  637. x =( - 2*x *x *x + 2*x *x *x - x *x *x
  638. 1 2 3 3 2 1 1 2 3 1 2 3 1 1 2 1 3 2 3 1 1 2 1 2 3 3
  639. 2
  640. + 2*x *x *x + x *x *x - 2*x *(x )
  641. 1 2 2 3 1 3 1 1 2 2 1 3 3 1 1 2 2 1 3
  642. 2 2
  643. + (x ) *x )/(x )
  644. 1 2 2 3 3 1 1
  645. x
  646. 1 2 3 3 1
  647. 2 2
  648. x *(x ) - 2*x *x *x - x *x *x + 2*x *(x )
  649. 1 1 2 3 3 1 1 1 2 3 1 3 1 1 1 2 1 3 3 1 1 1 2 1 3
  650. =-----------------------------------------------------------------------------
  651. 2
  652. (x )
  653. 1
  654. 2
  655. x =(2*x *x *x + x *x *x - 2*x *(x )
  656. 1 2 2 3 3 1 2 2 3 1 3 1 1 2 2 1 3 3 1 1 2 2 1 3
  657. 2 2
  658. + (x ) *x )/(x )
  659. 1 2 2 3 3 1 1
  660. - x *x + x *x
  661. 1 1 2 2 2 3 1 2 2 2 3 1
  662. x =----------------------------------
  663. 1 2 2 3 2 x
  664. 1
  665. x *x - x *x
  666. 1 1 2 2 3 1 1 1 2 2 1 3
  667. x =-------------------------------
  668. 1 2 2 3 1 x
  669. 1
  670. x =x
  671. 1 2 2 2 1 1 1 2 2 2
  672. x =( - 2*x *x *x + 2*x *x *x - x *x *x
  673. 1 1 3 3 2 1 1 1 3 1 2 3 1 1 1 1 3 2 3 1 1 1 1 2 3 3
  674. 2 2
  675. + x *(x ) )/(x )
  676. 1 1 2 3 3 1 1
  677. x
  678. 1 1 3 3 1
  679. 2 2
  680. x *(x ) - 2*x *x *x - x *x *x + 2*x *(x )
  681. 1 1 1 3 3 1 1 1 1 3 1 3 1 1 1 1 1 3 3 1 1 1 1 1 3
  682. =-----------------------------------------------------------------------------
  683. 2
  684. (x )
  685. 1
  686. - x *x + x *x
  687. 1 1 1 2 2 3 1 1 2 2 3 1
  688. x =----------------------------------
  689. 1 1 2 3 2 x
  690. 1
  691. x *x - x *x
  692. 1 1 1 2 3 1 1 1 1 2 1 3
  693. x =-------------------------------
  694. 1 1 2 3 1 x
  695. 1
  696. x =x
  697. 1 1 2 2 1 1 1 1 2 2
  698. - x *x + x *x
  699. 1 1 1 1 2 3 1 1 1 2 3 1
  700. x =----------------------------------
  701. 1 1 1 3 2 x
  702. 1
  703. x *x - x *x
  704. 1 1 1 1 3 1 1 1 1 1 1 3
  705. x =-------------------------------
  706. 1 1 1 3 1 x
  707. 1
  708. x =x
  709. 1 1 1 2 1 1 1 1 1 2
  710. x neq 0
  711. 1
  712. x neq 0
  713. 2 3
  714. f neq 0
  715. characters Y532;
  716. {22,6,0}
  717. dim_grassmann_variety Y532;
  718. 34
  719. % 22+2*6 = 34: just need to check for integrability conditions
  720. torsion Y532;
  721. {}
  722. % Y532 involutive. Dimensions?
  723. dim Y532;
  724. 79
  725. length one_forms Y532;
  726. 48
  727. % The following puts in part of Hauser's solution and ends up with an ODE
  728. % system (all characters 0), so no more solutions, as described by Finley
  729. % at MG6.
  730. hauser := {x=-v+(1/2)*(y+u)**2,delta=3/(8x),gamma=3/(8v)};
  731. 2 2
  732. u + 2*u*y - 2*v + y
  733. hauser := {x=-----------------------,
  734. 2
  735. 3
  736. delta=-----,
  737. 8*x
  738. 3
  739. gamma=-----}
  740. 8*v
  741. H532 := pullback(Y532,hauser)$
  742. New 0-form conditions detected
  743. 2
  744. - 8*gamma *v - 3*v
  745. 3 3
  746. -----------------------
  747. 2
  748. 8*v
  749. 2
  750. - 8*gamma *v - 3
  751. 1
  752. --------------------
  753. 2
  754. 8*v
  755. 3*(v - u - y)
  756. 3
  757. ----------------------------------------------------------------------------
  758. 4 3 2 2 2 3 2 2 4
  759. 2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
  760. 4 3 2 2 2
  761. ( - 2*delta *u - 8*delta *u *y + 8*delta *u *v - 12*delta *u *y
  762. 2 2 2 2
  763. 3 2 2 4
  764. + 16*delta *u*v*y - 8*delta *u*y - 8*delta *v + 8*delta *v*y - 2*delta *y
  765. 2 2 2 2 2
  766. 4 3 2 2 2 3 2
  767. - 3*u - 3*y)/(2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v
  768. 2 4
  769. - 4*v*y + y ))
  770. 4 3 2 2 2
  771. ( - 2*delta *u - 8*delta *u *y + 8*delta *u *v - 12*delta *u *y
  772. 1 1 1 1
  773. 3 2 2 4
  774. + 16*delta *u*v*y - 8*delta *u*y - 8*delta *v + 8*delta *v*y - 2*delta *y
  775. 1 1 1 1 1
  776. 4 3 2 2 2 3 2 2
  777. + 3)/(2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y
  778. 4
  779. + y ))
  780. - v + u + y
  781. 3
  782. - x + u + y
  783. 2
  784. - (x + 1)
  785. 1
  786. lift ws;
  787. Solving 0-forms
  788. New equations:
  789. - 3*(u + y)
  790. gamma =--------------
  791. 3 2
  792. 8*v
  793. - 3
  794. gamma =------
  795. 1 2
  796. 8*v
  797. delta
  798. 2
  799. - 3*(u + y)
  800. =----------------------------------------------------------------------------
  801. 4 3 2 2 2 3 2 2 4
  802. 2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
  803. delta
  804. 1
  805. 3
  806. =----------------------------------------------------------------------------
  807. 4 3 2 2 2 3 2 2 4
  808. 2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
  809. v =u + y
  810. 3
  811. x =u + y
  812. 2
  813. x =-1
  814. 1
  815. New 0-form conditions detected
  816. 3 2 2
  817. - 8*gamma *v + 6*u + 12*u*y - 3*v + 6*y
  818. 3 3
  819. -----------------------------------------------
  820. 3
  821. 8*v
  822. 3*(x - 1)
  823. 2 3
  824. --------------
  825. 2
  826. 8*v
  827. 3
  828. - 8*gamma *v + 3*x *v + 6*u + 6*y
  829. 1 3 1 3
  830. -----------------------------------------
  831. 3
  832. 8*v
  833. 3
  834. - 4*gamma *v + 3*u + 3*y
  835. 1 3
  836. ------------------------------
  837. 3
  838. 4*v
  839. 3
  840. - 4*gamma *v + 3
  841. 1 1
  842. ----------------------
  843. 3
  844. 4*v
  845. 3*(x - 1)
  846. 2 3
  847. ----------------------------------------------------------------------------
  848. 4 3 2 2 2 3 2 2 4
  849. 2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
  850. 8 7 6 6 2
  851. ( - 2*delta *u - 16*delta *u *y + 16*delta *u *v - 56*delta *u *y
  852. 2 2 2 2 2 2 2 2
  853. 5 5 3 4 2
  854. + 96*delta *u *v*y - 112*delta *u *y - 48*delta *u *v
  855. 2 2 2 2 2 2
  856. 4 2 4 4 3 2
  857. + 240*delta *u *v*y - 140*delta *u *y - 192*delta *u *v *y
  858. 2 2 2 2 2 2
  859. 3 3 3 5 2 3
  860. + 320*delta *u *v*y - 112*delta *u *y + 64*delta *u *v
  861. 2 2 2 2 2 2
  862. 2 2 2 2 4 2 6
  863. - 288*delta *u *v *y + 240*delta *u *v*y - 56*delta *u *y
  864. 2 2 2 2 2 2
  865. 3 2 3 5
  866. + 128*delta *u*v *y - 192*delta *u*v *y + 96*delta *u*v*y
  867. 2 2 2 2 2 2
  868. 7 4 3 2 2 4
  869. - 16*delta *u*y - 32*delta *v + 64*delta *v *y - 48*delta *v *y
  870. 2 2 2 2 2 2 2 2
  871. 6 8 4 3 2 2 2
  872. + 16*delta *v*y - 2*delta *y + 9*u + 36*u *y - 12*u *v + 54*u *y
  873. 2 2 2 2
  874. 3 2 2 4 8 7 6
  875. - 24*u*v*y + 36*u*y - 12*v - 12*v*y + 9*y )/(2*(u + 8*u *y - 8*u *v
  876. 6 2 5 5 3 4 2 4 2 4 4
  877. + 28*u *y - 48*u *v*y + 56*u *y + 24*u *v - 120*u *v*y + 70*u *y
  878. 3 2 3 3 3 5 2 3 2 2 2
  879. + 96*u *v *y - 160*u *v*y + 56*u *y - 32*u *v + 144*u *v *y
  880. 2 4 2 6 3 2 3 5 7
  881. - 120*u *v*y + 28*u *y - 64*u*v *y + 96*u*v *y - 48*u*v*y + 8*u*y
  882. 4 3 2 2 4 6 8
  883. + 16*v - 32*v *y + 24*v *y - 8*v*y + y ))
  884. 8 7 6 6 2
  885. ( - delta *u - 8*delta *u *y + 8*delta *u *v - 28*delta *u *y
  886. 1 2 1 2 1 2 1 2
  887. 5 5 3 4 2
  888. + 48*delta *u *v*y - 56*delta *u *y - 24*delta *u *v
  889. 1 2 1 2 1 2
  890. 4 2 4 4 3 2
  891. + 120*delta *u *v*y - 70*delta *u *y - 96*delta *u *v *y
  892. 1 2 1 2 1 2
  893. 3 3 3 5 2 3
  894. + 160*delta *u *v*y - 56*delta *u *y + 32*delta *u *v
  895. 1 2 1 2 1 2
  896. 2 2 2 2 4 2 6
  897. - 144*delta *u *v *y + 120*delta *u *v*y - 28*delta *u *y
  898. 1 2 1 2 1 2
  899. 3 2 3 5
  900. + 64*delta *u*v *y - 96*delta *u*v *y + 48*delta *u*v*y
  901. 1 2 1 2 1 2
  902. 7 4 3 2 2 4
  903. - 8*delta *u*y - 16*delta *v + 32*delta *v *y - 24*delta *v *y
  904. 1 2 1 2 1 2 1 2
  905. 6 8 3 2 2
  906. + 8*delta *v*y - delta *y - 6*u - 18*u *y + 12*u*v - 18*u*y + 12*v*y
  907. 1 2 1 2
  908. 3 8 7 6 6 2 5 5 3 4 2
  909. - 6*y )/(u + 8*u *y - 8*u *v + 28*u *y - 48*u *v*y + 56*u *y + 24*u *v
  910. 4 2 4 4 3 2 3 3 3 5
  911. - 120*u *v*y + 70*u *y + 96*u *v *y - 160*u *v*y + 56*u *y
  912. 2 3 2 2 2 2 4 2 6 3
  913. - 32*u *v + 144*u *v *y - 120*u *v*y + 28*u *y - 64*u*v *y
  914. 2 3 5 7 4 3 2 2 4
  915. + 96*u*v *y - 48*u*v*y + 8*u*y + 16*v - 32*v *y + 24*v *y
  916. 6 8
  917. - 8*v*y + y )
  918. 3*x
  919. 1 3
  920. ----------------------------------------------------------------------------
  921. 4 3 2 2 2 3 2 2 4
  922. 2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
  923. 6 5 4 4 2
  924. ( - delta *u - 6*delta *u *y + 6*delta *u *v - 15*delta *u *y
  925. 1 2 1 2 1 2 1 2
  926. 3 3 3 2 2
  927. + 24*delta *u *v*y - 20*delta *u *y - 12*delta *u *v
  928. 1 2 1 2 1 2
  929. 2 2 2 4 2
  930. + 36*delta *u *v*y - 15*delta *u *y - 24*delta *u*v *y
  931. 1 2 1 2 1 2
  932. 3 5 3 2 2
  933. + 24*delta *u*v*y - 6*delta *u*y + 8*delta *v - 12*delta *v *y
  934. 1 2 1 2 1 2 1 2
  935. 4 6 6 5 4 4 2
  936. + 6*delta *v*y - delta *y - 6*u - 6*y)/(u + 6*u *y - 6*u *v + 15*u *y
  937. 1 2 1 2
  938. 3 3 3 2 2 2 2 2 4 2
  939. - 24*u *v*y + 20*u *y + 12*u *v - 36*u *v*y + 15*u *y + 24*u*v *y
  940. 3 5 3 2 2 4 6
  941. - 24*u*v*y + 6*u*y - 8*v + 12*v *y - 6*v*y + y )
  942. 6 5 4 4 2
  943. ( - delta *u - 6*delta *u *y + 6*delta *u *v - 15*delta *u *y
  944. 1 1 1 1 1 1 1 1
  945. 3 3 3 2 2
  946. + 24*delta *u *v*y - 20*delta *u *y - 12*delta *u *v
  947. 1 1 1 1 1 1
  948. 2 2 2 4 2
  949. + 36*delta *u *v*y - 15*delta *u *y - 24*delta *u*v *y
  950. 1 1 1 1 1 1
  951. 3 5 3 2 2
  952. + 24*delta *u*v*y - 6*delta *u*y + 8*delta *v - 12*delta *v *y
  953. 1 1 1 1 1 1 1 1
  954. 4 6 6 5 4 4 2
  955. + 6*delta *v*y - delta *y + 6)/(u + 6*u *y - 6*u *v + 15*u *y
  956. 1 1 1 1
  957. 3 3 3 2 2 2 2 2 4 2
  958. - 24*u *v*y + 20*u *y + 12*u *v - 36*u *v*y + 15*u *y + 24*u*v *y
  959. 3 5 3 2 2 4 6
  960. - 24*u*v*y + 6*u*y - 8*v + 12*v *y - 6*v*y + y )
  961. - v + 1
  962. 3 3
  963. - x + 1
  964. 2 3
  965. - x + 1
  966. 2 2
  967. - x
  968. 1 3
  969. - x
  970. 1 2
  971. - x
  972. 1 1
  973. Solving 0-forms
  974. New equations:
  975. 2 2
  976. 3*(2*u + 4*u*y - v + 2*y )
  977. gamma =-----------------------------
  978. 3 3 3
  979. 8*v
  980. 3*(u + y)
  981. gamma =-----------
  982. 1 3 3
  983. 4*v
  984. 3
  985. gamma =------
  986. 1 1 3
  987. 4*v
  988. 4 3 2 2 2 3 2
  989. delta =(3*(3*u + 12*u *y - 4*u *v + 18*u *y - 8*u*v*y + 12*u*y - 4*v
  990. 2 2
  991. 2 4 8 7 6 6 2 5
  992. - 4*v*y + 3*y ))/(2*(u + 8*u *y - 8*u *v + 28*u *y - 48*u *v*y
  993. 5 3 4 2 4 2 4 4 3 2
  994. + 56*u *y + 24*u *v - 120*u *v*y + 70*u *y + 96*u *v *y
  995. 3 3 3 5 2 3 2 2 2 2 4
  996. - 160*u *v*y + 56*u *y - 32*u *v + 144*u *v *y - 120*u *v*y
  997. 2 6 3 2 3 5 7 4
  998. + 28*u *y - 64*u*v *y + 96*u*v *y - 48*u*v*y + 8*u*y + 16*v
  999. 3 2 2 4 6 8
  1000. - 32*v *y + 24*v *y - 8*v*y + y ))
  1001. 3 2 2 3 8 7
  1002. delta =(6*( - u - 3*u *y + 2*u*v - 3*u*y + 2*v*y - y ))/(u + 8*u *y
  1003. 1 2
  1004. 6 6 2 5 5 3 4 2 4 2
  1005. - 8*u *v + 28*u *y - 48*u *v*y + 56*u *y + 24*u *v - 120*u *v*y
  1006. 4 4 3 2 3 3 3 5 2 3
  1007. + 70*u *y + 96*u *v *y - 160*u *v*y + 56*u *y - 32*u *v
  1008. 2 2 2 2 4 2 6 3 2 3
  1009. + 144*u *v *y - 120*u *v*y + 28*u *y - 64*u*v *y + 96*u*v *y
  1010. 5 7 4 3 2 2 4 6 8
  1011. - 48*u*v*y + 8*u*y + 16*v - 32*v *y + 24*v *y - 8*v*y + y )
  1012. 6 5 4 4 2 3 3 3 2 2
  1013. delta =6/(u + 6*u *y - 6*u *v + 15*u *y - 24*u *v*y + 20*u *y + 12*u *v
  1014. 1 1
  1015. 2 2 2 4 2 3 5 3
  1016. - 36*u *v*y + 15*u *y + 24*u*v *y - 24*u*v*y + 6*u*y - 8*v
  1017. 2 2 4 6
  1018. + 12*v *y - 6*v*y + y )
  1019. v =1
  1020. 3 3
  1021. x =1
  1022. 2 3
  1023. x =1
  1024. 2 2
  1025. x =0
  1026. 1 3
  1027. x =0
  1028. 1 2
  1029. x =0
  1030. 1 1
  1031. New 0-form conditions detected
  1032. - v
  1033. 3 3 3
  1034. - x
  1035. 2 3 3
  1036. - x
  1037. 2 2 3
  1038. - x
  1039. 2 2 2
  1040. - x
  1041. 1 3 3
  1042. - x
  1043. 1 2 3
  1044. - x
  1045. 1 2 2
  1046. - x
  1047. 1 1 3
  1048. - x
  1049. 1 1 2
  1050. - x
  1051. 1 1 1
  1052. Solving 0-forms
  1053. New equations:
  1054. v =0
  1055. 3 3 3
  1056. x =0
  1057. 2 3 3
  1058. x =0
  1059. 2 2 3
  1060. x =0
  1061. 2 2 2
  1062. x =0
  1063. 1 3 3
  1064. x =0
  1065. 1 2 3
  1066. x =0
  1067. 1 2 2
  1068. x =0
  1069. 1 1 3
  1070. x =0
  1071. 1 1 2
  1072. x =0
  1073. 1 1 1
  1074. New 0-form conditions detected
  1075. - v
  1076. 3 3 3 3
  1077. - x
  1078. 2 3 3 3
  1079. - x
  1080. 2 2 3 3
  1081. - x
  1082. 2 2 2 3
  1083. - x
  1084. 2 2 2 2
  1085. - x
  1086. 1 3 3 3
  1087. - x
  1088. 1 2 3 3
  1089. - x
  1090. 1 2 2 3
  1091. - x
  1092. 1 2 2 2
  1093. - x
  1094. 1 1 3 3
  1095. - x
  1096. 1 1 2 3
  1097. - x
  1098. 1 1 2 2
  1099. - x
  1100. 1 1 1 3
  1101. - x
  1102. 1 1 1 2
  1103. - x
  1104. 1 1 1 1
  1105. Solving 0-forms
  1106. New equations:
  1107. v =0
  1108. 3 3 3 3
  1109. x =0
  1110. 2 3 3 3
  1111. x =0
  1112. 2 2 3 3
  1113. x =0
  1114. 2 2 2 3
  1115. x =0
  1116. 2 2 2 2
  1117. x =0
  1118. 1 3 3 3
  1119. x =0
  1120. 1 2 3 3
  1121. x =0
  1122. 1 2 2 3
  1123. x =0
  1124. 1 2 2 2
  1125. x =0
  1126. 1 1 3 3
  1127. x =0
  1128. 1 1 2 3
  1129. x =0
  1130. 1 1 2 2
  1131. x =0
  1132. 1 1 1 3
  1133. x =0
  1134. 1 1 1 2
  1135. x =0
  1136. 1 1 1 1
  1137. New 0-form conditions detected
  1138. - v
  1139. 3 3 3 3 3
  1140. - x
  1141. 2 3 3 3 3
  1142. 3*( - 4*f *v + f )
  1143. 1 3 2
  1144. ----------------------
  1145. 2*f*v
  1146. 2 2
  1147. 3*(2*f *u + 4*f *u*y - 4*f *v + 2*f *y + f )
  1148. 1 2 1 2 1 2 1 2 3
  1149. --------------------------------------------------------
  1150. 2 2
  1151. f*(u + 2*u*y - 2*v + y )
  1152. - x
  1153. 2 2 2 2 3
  1154. - x
  1155. 2 2 2 2 2
  1156. - x
  1157. 1 3 3 3 3
  1158. - x
  1159. 1 2 3 3 3
  1160. - x
  1161. 2 2 3 3 1
  1162. - x
  1163. 1 2 2 2 3
  1164. - x
  1165. 1 2 2 2 2
  1166. - x
  1167. 1 1 3 3 3
  1168. - x
  1169. 1 1 2 3 3
  1170. - x
  1171. 1 1 2 2 3
  1172. - x
  1173. 1 1 2 2 2
  1174. - x
  1175. 1 1 1 3 3
  1176. - x
  1177. 1 1 1 2 3
  1178. - x
  1179. 1 1 1 2 2
  1180. - x
  1181. 1 1 1 1 3
  1182. - x
  1183. 1 1 1 1 2
  1184. - x
  1185. 1 1 1 1 1
  1186. Solving 0-forms
  1187. New equations:
  1188. v =0
  1189. 3 3 3 3 3
  1190. x =0
  1191. 2 3 3 3 3
  1192. x =0
  1193. 2 2 3 3 1
  1194. x =0
  1195. 2 2 2 2 3
  1196. x =0
  1197. 2 2 2 2 2
  1198. x =0
  1199. 1 3 3 3 3
  1200. x =0
  1201. 1 2 3 3 3
  1202. x =0
  1203. 1 2 2 2 3
  1204. x =0
  1205. 1 2 2 2 2
  1206. x =0
  1207. 1 1 3 3 3
  1208. x =0
  1209. 1 1 2 3 3
  1210. x =0
  1211. 1 1 2 2 3
  1212. x =0
  1213. 1 1 2 2 2
  1214. x =0
  1215. 1 1 1 3 3
  1216. x =0
  1217. 1 1 1 2 3
  1218. x =0
  1219. 1 1 1 2 2
  1220. x =0
  1221. 1 1 1 1 3
  1222. x =0
  1223. 1 1 1 1 2
  1224. x =0
  1225. 1 1 1 1 1
  1226. f
  1227. 2
  1228. f =-----
  1229. 1 3 4*v
  1230. - f
  1231. 3
  1232. f =---------------------------
  1233. 1 2 2 2
  1234. 2*(u + 2*u*y - 2*v + y )
  1235. New 0-form conditions detected
  1236. - 4*f *v - 2*f *u - 2*f *y + 3*f
  1237. 1 2 2
  1238. -----------------------------------
  1239. 2
  1240. 8*v
  1241. 2 2 2
  1242. - 8*f *u *v - 16*f *u*v*y + 16*f *v - 8*f *v*y + 3*f
  1243. 1 1 1 1 1 1 1 1
  1244. -----------------------------------------------------------------
  1245. 2 2
  1246. 16*v*(u + 2*u*y - 2*v + y )
  1247. 2 2 2 3 2 2 2
  1248. ( - 8*f *u *v - 16*f *u*v *y + 16*f *v - 8*f *v *y - 2*f *u
  1249. 1 1 3 1 1 3 1 1 3 1 1 3 2
  1250. 2 2 2 2
  1251. - 4*f *u*y + 4*f *v - 2*f *y - f *v)/(8*v *(u + 2*u*y - 2*v + y ))
  1252. 2 2 2 3
  1253. 2 2 2
  1254. 8*f *u *v + 16*f *u*v*y - 16*f *v + 8*f *v*y - 3*f
  1255. 1 1 1 1 1 1 1 1
  1256. --------------------------------------------------------------
  1257. 2 2
  1258. 16*v*(u + 2*u*y - 2*v + y )
  1259. 2 2
  1260. - 2*f *u - 4*f *u*y + 4*f *v - 2*f *y + 2*f *u + 2*f *y - 3*f
  1261. 1 1 1 1 3 3
  1262. ----------------------------------------------------------------------------
  1263. 4 3 2 2 2 3 2 2 4
  1264. 2*(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
  1265. 4 3 2 2 2 2
  1266. ( - 8*f *u *v - 32*f *u *v*y + 32*f *u *v - 48*f *u *v*y
  1267. 1 1 2 1 1 2 1 1 2 1 1 2
  1268. 2 3 3 2 2
  1269. + 64*f *u*v *y - 32*f *u*v*y - 32*f *v + 32*f *v *y
  1270. 1 1 2 1 1 2 1 1 2 1 1 2
  1271. 4 2 2
  1272. - 8*f *v*y - f *u - 2*f *u*y + 2*f *v - f *y - 8*f *v)/(8*v
  1273. 1 1 2 2 2 2 2 3
  1274. 4 3 2 2 2 3 2 2 4
  1275. *(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y ))
  1276. Solving 0-forms
  1277. New equations:
  1278. 2 2 2 2
  1279. f =(3*(2*f *u *v + 4*f *u*v*y - 4*f *v + 2*f *v*y - 2*f*u - 4*f*u*y
  1280. 1 1 3 1 1 1 1
  1281. 2 2
  1282. + 3*f*v - 2*f*y ))/(16*v
  1283. 3 2 2 3
  1284. *(u + 3*u *y - 2*u*v + 3*u*y - 2*v*y + y ))
  1285. 2 2 2 2
  1286. f =(3*( - 4*f *u *v - 8*f *u*v*y + 8*f *v - 4*f *v*y - f*u - 2*f*u*y
  1287. 1 1 2 1 1 1 1
  1288. 2 5 4 3 3 2 2
  1289. - 6*f*v - f*y ))/(16*v*(u + 5*u *y - 4*u *v + 10*u *y - 12*u *v*y
  1290. 2 3 2 2 4 2 3 5
  1291. + 10*u *y + 4*u*v - 12*u*v*y + 5*u*y + 4*v *y - 4*v*y + y ))
  1292. 3*f
  1293. f =-----------------------------
  1294. 1 1 2 2
  1295. 8*v*(u + 2*u*y - 2*v + y )
  1296. 2 2
  1297. 2*f *u + 4*f *u*y - 4*f *v + 2*f *y + 3*f
  1298. 1 1 1 1
  1299. f =---------------------------------------------
  1300. 3 2*(u + y)
  1301. - 4*f *v + 3*f
  1302. 1
  1303. f =-----------------
  1304. 2 2*(u + y)
  1305. New 0-form conditions detected
  1306. 4 2 3 2 2 3 2 2 2
  1307. ( - 8*f *u *v - 32*f *u *v *y + 32*f *u *v - 48*f *u *v *y
  1308. 1 1 1 1 1 1 1 1 1 1 1 1
  1309. 3 2 3 4 3 2
  1310. + 64*f *u*v *y - 32*f *u*v *y - 32*f *v + 32*f *v *y
  1311. 1 1 1 1 1 1 1 1 1 1 1 1
  1312. 2 4 2 2 2 2
  1313. - 8*f *v *y + 3*f *u *v + 6*f *u*v*y - 6*f *v + 3*f *v*y - 3*f*u
  1314. 1 1 1 1 1 1 1
  1315. 2 2
  1316. - 6*f*u*y + 12*f*v - 3*f*y )/(8*v
  1317. 4 3 2 2 2 3 2 2 4
  1318. *(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y ))
  1319. Solving 0-forms
  1320. New equations:
  1321. f
  1322. 1 1 1
  1323. 2 2 2 2 2
  1324. 3*(f *u *v + 2*f *u*v*y - 2*f *v + f *v*y - f*u - 2*f*u*y + 4*f*v - f*y )
  1325. 1 1 1 1
  1326. =-------------------------------------------------------------------------------
  1327. 2 4 3 2 2 2 3 2 2 4
  1328. 8*v *(u + 4*u *y - 4*u *v + 6*u *y - 8*u*v*y + 4*u*y + 4*v - 4*v*y + y )
  1329. 4*f *v - 3*f
  1330. 1 1 2
  1331. EDS({d f - f *omega + --------------*omega
  1332. 1 2*(u + y)
  1333. 2 2
  1334. - 2*f *u - 4*f *u*y + 4*f *v - 2*f *y - 3*f
  1335. 1 1 1 1 3
  1336. + ------------------------------------------------*omega ,
  1337. 2*(u + y)
  1338. 3*f 1
  1339. d f - -----------------------------*omega
  1340. 1 2 2
  1341. 8*v*(u + 2*u*y - 2*v + y )
  1342. 2 2
  1343. 2*f *u + 4*f *u*y - 4*f *v + 2*f *y + 3*f
  1344. 1 1 1 1 2
  1345. + -----------------------------------------------*omega
  1346. 3 2 2 3
  1347. 4*(u + 3*u *y - 2*u*v + 3*u*y - 2*v*y + y )
  1348. 4*f *v - 3*f
  1349. 1 3 1 2 3
  1350. + --------------*omega },omega ^omega ^omega )
  1351. 8*v*(u + y)
  1352. characters ws;
  1353. {0,0,0}
  1354. clear v(i),omega(i);
  1355. clear F,x,Delta,gamma,v,y,u,omega;
  1356. off ranpos;
  1357. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  1358. % Isometric embeddings of Ricci-flat R(4) in ISO(10) %
  1359. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  1360. % Determine the Cartan characters of a Ricci-flat embedding of R(4) into
  1361. % the orthonormal frame bundle ISO(10) over flat R(6). Reference:
  1362. % Estabrook & Wahlquist, Class Quant Grav 10(1993)1851
  1363. % Indices
  1364. indexrange {p,q,r,s}={1,2,3,4,5,6,7,8,9,10},
  1365. {i,j,k,l}={1,2,3,4},{a,b,c,d}={5,6,7,8,9,10};
  1366. % Metric for R10
  1367. pform g(p,q)=0;
  1368. g(p,q) := 0$
  1369. g(-p,-q) := 0$
  1370. g(-p,-p) := g(p,p) := 1$
  1371. % Hodge map for R4
  1372. pform epsilon(i,j,k,l)=0;
  1373. index_symmetries epsilon(i,j,k,l):antisymmetric;
  1374. epsilon(1,2,3,4) := 1;
  1375. 1 2 3 4
  1376. epsilon := 1
  1377. % Coframe for ISO(10)
  1378. % NB index_symmetries must come after o(p,-q) := ... (EXCALC bug)
  1379. pform e(r)=1,o(r,s)=1;
  1380. korder index_expand {e(r)};
  1381. e(-p) := g(-p,-q)*e(q)$
  1382. o(p,-q) := o(p,r)*g(-r,-q)$
  1383. index_symmetries o(p,q):antisymmetric;
  1384. % Structure equations
  1385. flat_no_torsion := {d e(p) => -o(p,-q)^e(q),
  1386. d o(p,q) => -o(p,-r)^o(r,q)};
  1387. p p q
  1388. flat_no_torsion := {d e => - o ^e ,
  1389. q
  1390. p q p r q
  1391. d o => - o ^o }
  1392. r
  1393. % Coframing structure
  1394. ISO := coframing({e(p),o(p,q)},flat_no_torsion)$
  1395. dim ISO;
  1396. 55
  1397. % 4d curvature 2-forms
  1398. pform F(i,j)=2;
  1399. index_symmetries F(i,j):antisymmetric;
  1400. F(-i,-j) := -g(-i,-k)*o(k,-a)^o(a,-j);
  1401. 1 10 2 10 1 5 2 5 1 6 2 6 1 7 2 7 1 8 2 8 1 9 2 9
  1402. f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
  1403. 1 2
  1404. 1 10 3 10 1 5 3 5 1 6 3 6 1 7 3 7 1 8 3 8 1 9 3 9
  1405. f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
  1406. 1 3
  1407. 2 10 3 10 2 5 3 5 2 6 3 6 2 7 3 7 2 8 3 8 2 9 3 9
  1408. f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
  1409. 2 3
  1410. 1 10 4 10 1 5 4 5 1 6 4 6 1 7 4 7 1 8 4 8 1 9 4 9
  1411. f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
  1412. 1 4
  1413. 2 10 4 10 2 5 4 5 2 6 4 6 2 7 4 7 2 8 4 8 2 9 4 9
  1414. f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
  1415. 2 4
  1416. 3 10 4 10 3 5 4 5 3 6 4 6 3 7 4 7 3 8 4 8 3 9 4 9
  1417. f := o ^o + o ^o + o ^o + o ^o + o ^o + o ^o
  1418. 3 4
  1419. % EDS for vacuum GR (Ricci-flat) in 4d
  1420. GR0 := eds({e(a),epsilon(i,j,k,l)*F(-j,-k)^e(-l)},
  1421. {e(i)},
  1422. ISO)$
  1423. % Find an integral element, and linearise
  1424. Z := integral_element GR0$
  1425. 45 free variables
  1426. 39 free variables
  1427. 29 free variables
  1428. 21 free variables
  1429. GRZ := linearise(GR0,Z)$
  1430. % This actually tells us the characters already:
  1431. % {45-39,39-29,29-21,21} = {6,10,8,21}
  1432. % Get the characters and dimension at Z
  1433. characters GRZ;
  1434. Cauchy characteristics detected from characters
  1435. {6,10,8,21}
  1436. dim_grassmann_variety GRZ;
  1437. 134
  1438. % 6+2*10+3*8+4*21 = 134, so involutive
  1439. clear e(r),o(r,s),g(p,q),epsilon(i,j,k,l),F(i,j);
  1440. clear e,o,g,epsilon,F,Z;
  1441. indexrange 0;
  1442. %%%%%%%%%%%%%%%%%%%%%%%%%%
  1443. % Janet's PDE system %
  1444. %%%%%%%%%%%%%%%%%%%%%%%%%%
  1445. % This is something of a standard test problem in analysing integrability
  1446. % conditions. Although it looks very innocent, it must be prolonged five
  1447. % times from the second jet bundle before reaching involution. The initial
  1448. % equations are just
  1449. %
  1450. % u =w, u =u *y + v
  1451. % y y z z x x
  1452. load sets;
  1453. off varopt;
  1454. pform {x,y,z,u,v,w}=0$
  1455. janet := contact(2,{x,y,z},{u,v,w})$
  1456. janet := pullback(janet,{u(-y,-y)=w,u(-z,-z)=y*u(-x,-x)+v})$
  1457. % Prolong to involution
  1458. involutive janet;
  1459. 0
  1460. involution janet;
  1461. Prolongation using new equations:
  1462. u =u *y + u + v
  1463. y z z x x y x x y
  1464. u =w
  1465. y y z z
  1466. u =u *y + v
  1467. x z z x x x x
  1468. u =w
  1469. x y y x
  1470. Reduction using new equations:
  1471. - v - w *y + w
  1472. y y x x z z
  1473. u =-------------------------
  1474. x x y 2
  1475. Reduction using new equations:
  1476. w =v + w *y + 3*w
  1477. y z z y y y x x y x x
  1478. Prolongation using new equations:
  1479. w =v + w *y + 3*w
  1480. y z z z y y y z x x y z x x z
  1481. w =v + w *y + 4*w
  1482. y y z z y y y y x x y y x x y
  1483. w =v + w *y + 3*w
  1484. x y z z x y y y x x x y x x x
  1485. 2
  1486. 2*u - v *y + 2*v - w *y + w *y
  1487. x x x x y y x y x x x x z z
  1488. u =-----------------------------------------------------
  1489. x y z z 2
  1490. u =w
  1491. x y y z x z
  1492. u =u *y + v
  1493. x x z z x x x x x x
  1494. - v - w *y + w
  1495. y y z x x z z z z
  1496. u =-------------------------------
  1497. x x y z 2
  1498. - v - w *y + w
  1499. x y y x x x x z z
  1500. u =-------------------------------
  1501. x x x y 2
  1502. Reduction using new equations:
  1503. w
  1504. z z z z
  1505. 2
  1506. =2*u - v *y + 2*v + v - w *y + 2*w *y
  1507. x x x x x x y y x x y y y z z x x x x x x z z
  1508. EDS({d u - u *d x - u *d y - u *d z,
  1509. x y z
  1510. d v - v *d x - v *d y - v *d z,
  1511. x y z
  1512. d w - w *d x - w *d y - w *d z,
  1513. x y z
  1514. d u - u *d x - u *d y - u *d z,
  1515. x x x x y x z
  1516. d u - u *d x - w*d y - u *d z,
  1517. y x y y z
  1518. d u - u *d x - u *d y - (u *y + v)*d z,
  1519. z x z y z x x
  1520. d v - v *d x - v *d y - v *d z,
  1521. x x x x y x z
  1522. d v - v *d x - v *d y - v *d z,
  1523. y x y y y y z
  1524. d v - v *d x - v *d y - v *d z,
  1525. z x z y z z z
  1526. d w - w *d x - w *d y - w *d z,
  1527. x x x x y x z
  1528. d w - w *d x - w *d y - w *d z,
  1529. y x y y y y z
  1530. d w - w *d x - w *d y - w *d z,
  1531. z x z y z z z
  1532. v + w *y - w
  1533. y y x x z z
  1534. d u - u *d x + ----------------------*d y - u *d z,
  1535. x x x x x 2 x x z
  1536. v + w *y - w
  1537. y y x x z z
  1538. d u + ----------------------*d x - w *d y - u *d z,
  1539. x y 2 x x y z
  1540. d u - u *d x - u *d y - (u *y + v )*d z,
  1541. x z x x z x y z x x x x
  1542. d u - u *d x - w *d y
  1543. y z x y z z
  1544. 2
  1545. - 2*u + v *y - 2*v + w *y - w *y
  1546. x x y y y x x z z
  1547. + ----------------------------------------------*d z,
  1548. 2
  1549. d v - v *d x - v *d y - v *d z,
  1550. x x x x x x x y x x z
  1551. d v - v *d x - v *d y - v *d z,
  1552. x y x x y x y y x y z
  1553. d v - v *d x - v *d y - v *d z,
  1554. x z x x z x y z x z z
  1555. d v - v *d x - v *d y - v *d z,
  1556. y y x y y y y y y y z
  1557. d v - v *d x - v *d y - v *d z,
  1558. y z x y z y y z y z z
  1559. d v - v *d x - v *d y - v *d z,
  1560. z z x z z y z z z z z
  1561. d w - w *d x - w *d y - w *d z,
  1562. x x x x x x x y x x z
  1563. d w - w *d x - w *d y - w *d z,
  1564. x y x x y x y y x y z
  1565. d w - w *d x - w *d y - w *d z,
  1566. x z x x z x y z x z z
  1567. d w - w *d x - w *d y - w *d z,
  1568. y y x y y y y y y y z
  1569. d w - w *d x - w *d y + ( - v - w *y - 3*w )*d z,
  1570. y z x y z y y z y y y x x y x x
  1571. d w - w *d x + ( - v - w *y - 3*w )*d y - w *d z,
  1572. z z x z z y y y x x y x x z z z
  1573. v + w *y - w
  1574. x y y x x x x z z
  1575. d u - u *d x + ----------------------------*d y - u *d z,
  1576. x x x x x x x 2 x x x z
  1577. v + w *y - w
  1578. y y z x x z z z z
  1579. d u - u *d x + ----------------------------*d y
  1580. x x z x x x z 2
  1581. - (u *y + v )*d z,
  1582. x x x x x x
  1583. v + w *y - w
  1584. y y z x x z z z z
  1585. d u + ----------------------------*d x - w *d y
  1586. x y z 2 x z
  1587. 2
  1588. - 2*u + v *y - 2*v + w *y - w *y
  1589. x x x x y y x y x x x x z z
  1590. + --------------------------------------------------------*d z,
  1591. 2
  1592. d v - v *d x - v *d y - v *d z,
  1593. x x x x x x x x x x y x x x z
  1594. d v - v *d x - v *d y - v *d z,
  1595. x x y x x x y x x y y x x y z
  1596. d v - v *d x - v *d y - v *d z,
  1597. x x z x x x z x x y z x x z z
  1598. d v - v *d x - v *d y - v *d z,
  1599. x y y x x y y x y y y x y y z
  1600. d v - v *d x - v *d y - v *d z,
  1601. x y z x x y z x y y z x y z z
  1602. d v - v *d x - v *d y - v *d z,
  1603. x z z x x z z x y z z x z z z
  1604. d v - v *d x - v *d y - v *d z,
  1605. y y y x y y y y y y y y y y z
  1606. d v - v *d x - v *d y - v *d z,
  1607. y y z x y y z y y y z y y z z
  1608. d v - v *d x - v *d y - v *d z,
  1609. y z z x y z z y y z z y z z z
  1610. d v - v *d x - v *d y - v *d z,
  1611. z z z x z z z y z z z z z z z
  1612. d w - w *d x - w *d y - w *d z,
  1613. x x x x x x x x x x y x x x z
  1614. d w - w *d x - w *d y - w *d z,
  1615. x x y x x x y x x y y x x y z
  1616. d w - w *d x - w *d y - w *d z,
  1617. x x z x x x z x x y z x x z z
  1618. d w - w *d x - w *d y - w *d z,
  1619. x y y x x y y x y y y x y y z
  1620. d w - w *d x - w *d y
  1621. x y z x x y z x y y z
  1622. + ( - v - w *y - 3*w )*d z,
  1623. x y y y x x x y x x x
  1624. d w - w *d x + ( - v - w *y - 3*w )*d y
  1625. x z z x x z z x y y y x x x y x x x
  1626. - w *d z,
  1627. x z z z
  1628. d w - w *d x - w *d y - w *d z,
  1629. y y y x y y y y y y y y y y z
  1630. d w - w *d x - w *d y
  1631. y y z x y y z y y y z
  1632. + ( - v - w *y - 4*w )*d z,
  1633. y y y y x x y y x x y
  1634. d w - w *d x + ( - v - w *y - 3*w )*d y + (
  1635. z z z x z z z y y y z x x y z x x z
  1636. 2
  1637. - 2*u + v *y - 2*v - v + w *y
  1638. x x x x x x y y x x y y y z z x x x x
  1639. - 2*w *y)*d z,
  1640. x x z z
  1641. d u ^d x + d u ^d z
  1642. x x x x x x x z
  1643. - v - w *y + w
  1644. x x y y x x x x x x z z
  1645. + -------------------------------------*d x^d y
  1646. 2
  1647. v + w *y - w
  1648. x y y z x x x z x z z z
  1649. + ----------------------------------*d y^d z,
  1650. 2
  1651. 1
  1652. d u ^d z + ---*d u ^d x
  1653. x x x x y x x x z
  1654. - v - w *y + w v
  1655. x y y z x x x z x z z z x x x
  1656. + -------------------------------------*d x^d y + --------*d x^d z
  1657. 2*y y
  1658. v + w *y - w
  1659. x x y y x x x x x x z z
  1660. + ----------------------------------*d y^d z,
  1661. 2
  1662. y 1
  1663. d u ^d z - ---*d v ^d z + ---*d v ^d y
  1664. x x x x 2 x x y y 2 y y y z
  1665. 2
  1666. 1 y y
  1667. + ---*d v ^d z - ----*d w ^d z + ---*d w ^d y
  1668. 2 y y z z 2 x x x x 2 x x y z
  1669. 3*w
  1670. 1 x x x z
  1671. + y*d w ^d z + ---*d w ^d x + ------------*d x^d y
  1672. x x z z 2 x z z z 2
  1673. v - 2*w *y - w
  1674. x x y y x x x x x x z z
  1675. + v *d x^d z + ------------------------------------*d y^d z,
  1676. x x x y 2
  1677. d v ^d x + d v ^d y + d v ^d z,
  1678. x x x x x x x y x x x z
  1679. d v ^d x + d v ^d y + d v ^d z,
  1680. x x x y x x y y x x y z
  1681. d v ^d x + d v ^d y + d v ^d z,
  1682. x x x z x x y z x x z z
  1683. d v ^d x + d v ^d y + d v ^d z,
  1684. x x y y x y y y x y y z
  1685. d v ^d x + d v ^d y + d v ^d z,
  1686. x x y z x y y z x y z z
  1687. d v ^d x + d v ^d y + d v ^d z,
  1688. x x z z x y z z x z z z
  1689. d v ^d x + d v ^d y + d v ^d z,
  1690. x y y y y y y y y y y z
  1691. d v ^d y + y*d w ^d y + d w ^d x + d w ^d z
  1692. x y y y x x x y x x z z x z z z
  1693. + 3*w *d x^d y - 3*w *d y^d z,
  1694. x x x x x x x z
  1695. d v ^d z + y*d w ^d z + d w ^d x + d w ^d y
  1696. x y y y x x x y x x y z x y y z
  1697. + 3*w *d x^d z + 4*w *d y^d z,
  1698. x x x x x x x y
  1699. d v ^d x + d v ^d y + d v ^d z,
  1700. x y y z y y y z y y z z
  1701. d v ^d x + d v ^d y + d v ^d z,
  1702. x y z z y y z z y z z z
  1703. d v ^d x + d v ^d y + d v ^d z,
  1704. x z z z y z z z z z z z
  1705. d v ^d z + y*d w ^d z + d w ^d x + d w ^d y
  1706. y y y y x x y y x y y z y y y z
  1707. + 4*w *d x^d z + 5*w *d y^d z,
  1708. x x x y x x y y
  1709. d w ^d x + d w ^d y + d w ^d z,
  1710. x x x x x x x y x x x z
  1711. d w ^d x + d w ^d y + d w ^d z,
  1712. x x x y x x y y x x y z
  1713. d w ^d x + d w ^d y + d w ^d z,
  1714. x x x z x x y z x x z z
  1715. d w ^d x + d w ^d y + d w ^d z,
  1716. x x y y x y y y x y y z
  1717. d w ^d x + d w ^d y + d w ^d z},d x^d y^d z)
  1718. x y y y y y y y y y y z
  1719. involutive ws;
  1720. 1
  1721. % Solve the homogeneous system, for which the
  1722. % involutive prolongation is completely integrable
  1723. fdomain u=u(x,y,z),v=v(x,y,z),w=w(x,y,z);
  1724. janet := {@(u,y,y)=0,@(u,z,z)=y*@(u,x,x)};
  1725. janet := {@ u=0,@ u=@ u*y}
  1726. y y z z x x
  1727. janet := involution pde2eds janet$
  1728. Prolongation using new equations:
  1729. u =u *y + u
  1730. y z z x x y x x
  1731. u =0
  1732. y y z
  1733. u =u *y
  1734. x z z x x x
  1735. u =0
  1736. x y y
  1737. Reduction using new equations:
  1738. u =0
  1739. x x y
  1740. Prolongation using new equations:
  1741. u =u
  1742. x y z z x x x
  1743. u =0
  1744. x y y z
  1745. u =u *y
  1746. x x z z x x x x
  1747. u =0
  1748. x x y z
  1749. u =0
  1750. x x x y
  1751. Reduction using new equations:
  1752. u =0
  1753. x x x x
  1754. Prolongation using new equations:
  1755. u =0
  1756. x x x z z
  1757. u =0
  1758. x x x y z
  1759. u =0
  1760. x x x x z
  1761. % Check if completely integrable
  1762. if frobenius janet then write "yes" else write "no";
  1763. yes
  1764. length one_forms janet;
  1765. 12
  1766. % So there are 12 constants in the solution: there should be 12 invariants
  1767. length(C := invariants janet);
  1768. 12
  1769. solve(for i:=1:length C collect
  1770. part(C,i) = mkid(k,i),coordinates janet \ {x,y,z})$
  1771. S := select(lhs ~q = u,first ws);
  1772. 3 2 3 3
  1773. s := {u=(k1*x + 3*k1*x*y*z - 6*k10*y*z - 6*k11 - 6*k12*z - k2*x *z - k2*x*y*z
  1774. 2 3 2
  1775. - 6*k3*x*y*z - 6*k4*x*y - 3*k5*x *z - k5*y*z - 6*k6*x*z - 3*k7*x
  1776. 2
  1777. - 3*k7*y*z - 6*k8*x - 6*k9*y)/6}
  1778. % Check solution
  1779. mkdepend dependencies;
  1780. sub(S,{@(u,y,y),@(u,z,z)-y*@(u,x,x)});
  1781. {0,0}
  1782. clear u(i,j),v(i,j),w(i,j),u(i),v(i),w(i);
  1783. clear x,y,z,u,v,w,C,S;
  1784. end;
  1785. Time for test: 5850 ms, plus GC time: 170 ms