547:
202:
363:
588:
413:
474:
437:
612:
581:
507:
617:
574:
528:
128:
292:
607:
21:
25:
554:
444:
275:
119:
39:
518:
380:
59:
35:
374:
524:
503:
450:
115:
104:
558:
422:
495:
216:
100:
601:
499:
29:
215:} = 0. (In other words, they are pairwise in involution.) Here {–,–} is the
546:
16:
Theorem in symplectic geometry which generalizes
Darboux's theorem
476:
there is a symplectic chart such that one of its coordinates is
197:{\displaystyle df_{1}(p)\wedge \ldots \wedge df_{r}(p)\neq 0,}
358:{\displaystyle \omega =\sum _{i=1}^{n}df_{i}\wedge dg_{i}.}
517:
Libermann, P.; Marle, Charles-Michel (6 December 2012).
562:
373:
As a direct application we have the following. Given a
453:
425:
383:
295:
131:
468:
431:
407:
357:
196:
494:. Graduate Texts in Mathematics. Vol. 218.
419:is a symplectic manifold with symplectic form
582:
8:
520:Symplectic Geometry and Analytical Mechanics
589:
575:
452:
424:
382:
346:
330:
317:
306:
294:
170:
139:
130:
7:
543:
541:
561:. You can help Knowledge (XXG) by
14:
613:Theorems in differential geometry
62:with symplectic form ω. For
545:
492:Introduction to Smooth Manifolds
254:defined on an open neighborhood
282:, i.e., ω is expressed on
122:at each point, or equivalently
402:
384:
182:
176:
151:
145:
1:
408:{\displaystyle (M,\omega ,H)}
618:Differential geometry stubs
447:, around every point where
219:. Then there are functions
634:
540:
500:10.1007/978-1-4419-9982-5
469:{\displaystyle dH\neq 0}
432:{\displaystyle \omega }
557:-related article is a
470:
433:
409:
359:
322:
198:
555:differential geometry
490:Lee, John M. (2012).
471:
434:
410:
360:
302:
199:
451:
445:Hamiltonian function
423:
381:
293:
129:
120:linearly independent
608:Symplectic geometry
74: ≤
60:symplectic manifold
36:symplectic geometry
466:
429:
405:
375:Hamiltonian system
355:
194:
38:which generalizes
570:
569:
509:978-1-4419-9981-8
105:open neighborhood
40:Darboux's theorem
625:
591:
584:
577:
549:
542:
534:
513:
475:
473:
472:
467:
438:
436:
435:
430:
414:
412:
411:
406:
364:
362:
361:
356:
351:
350:
335:
334:
321:
316:
276:symplectic chart
203:
201:
200:
195:
175:
174:
144:
143:
101:smooth functions
34:is a theorem in
633:
632:
628:
627:
626:
624:
623:
622:
598:
597:
596:
595:
538:
531:
516:
510:
489:
486:
449:
448:
421:
420:
379:
378:
371:
342:
326:
291:
290:
273:
269:
253:
246:
239:
232:
225:
217:Poisson bracket
214:
210:
166:
135:
127:
126:
98:
91:
84:
48:
17:
12:
11:
5:
631:
629:
621:
620:
615:
610:
600:
599:
594:
593:
586:
579:
571:
568:
567:
550:
536:
535:
529:
514:
508:
485:
482:
465:
462:
459:
456:
428:
404:
401:
398:
395:
392:
389:
386:
370:
367:
366:
365:
354:
349:
345:
341:
338:
333:
329:
325:
320:
315:
312:
309:
305:
301:
298:
271:
267:
251:
244:
237:
230:
223:
212:
208:
205:
204:
193:
190:
187:
184:
181:
178:
173:
169:
165:
162:
159:
156:
153:
150:
147:
142:
138:
134:
103:defined on an
96:
89:
82:
47:
44:
15:
13:
10:
9:
6:
4:
3:
2:
630:
619:
616:
614:
611:
609:
606:
605:
603:
592:
587:
585:
580:
578:
573:
572:
566:
564:
560:
556:
551:
548:
544:
539:
532:
530:9789400938076
526:
522:
521:
515:
511:
505:
501:
497:
493:
488:
487:
483:
481:
479:
463:
460:
457:
454:
446:
442:
426:
418:
399:
396:
393:
390:
387:
376:
368:
352:
347:
343:
339:
336:
331:
327:
323:
318:
313:
310:
307:
303:
299:
296:
289:
288:
287:
285:
281:
277:
265:
261:
258: ⊂
257:
250:
243:
236:
229:
222:
218:
191:
188:
185:
179:
171:
167:
163:
160:
157:
154:
148:
140:
136:
132:
125:
124:
123:
121:
117:
116:differentials
113:
109:
106:
102:
95:
88:
81:
77:
73:
69:
66: ∈
65:
61:
58:-dimensional
57:
53:
45:
43:
41:
37:
33:
31:
27:
23:
563:expanding it
552:
537:
519:
491:
477:
440:
416:
372:
369:Applications
283:
279:
266:such that (f
263:
259:
255:
248:
241:
234:
227:
220:
206:
111:
107:
93:
86:
79:
75:
71:
67:
63:
55:
51:
49:
22:Carathéodory
20:
18:
602:Categories
484:References
461:≠
427:ω
394:ω
337:∧
304:∑
297:ω
186:≠
161:∧
158:…
155:∧
46:Statement
207:where {f
443:is the
274:) is a
247:, ...,
226:, ...,
92:, ...,
32:theorem
527:
506:
415:where
114:whose
78:, let
54:be a 2
26:Jacobi
553:This
559:stub
525:ISBN
504:ISBN
439:and
118:are
70:and
50:Let
19:The
496:doi
377:as
286:as
278:of
270:, g
262:of
224:r+1
211:, f
110:of
99:be
30:Lie
604::
523:.
502:.
480:.
240:,
233:,
85:,
42:.
590:e
583:t
576:v
565:.
533:.
512:.
498::
478:H
464:0
458:H
455:d
441:H
417:M
403:)
400:H
397:,
391:,
388:M
385:(
353:.
348:i
344:g
340:d
332:i
328:f
324:d
319:n
314:1
311:=
308:i
300:=
284:U
280:M
272:i
268:i
264:p
260:V
256:U
252:n
249:g
245:2
242:g
238:1
235:g
231:n
228:f
221:f
213:j
209:i
192:,
189:0
183:)
180:p
177:(
172:r
168:f
164:d
152:)
149:p
146:(
141:1
137:f
133:d
112:p
108:V
97:r
94:f
90:2
87:f
83:1
80:f
76:n
72:r
68:M
64:p
56:n
52:M
28:–
24:–
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
