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Carathéodory–Jacobi–Lie theorem

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558: 213: 374: 424: 485: 448: 599: 286: 17: 623: 139: 592: 303: 573: 518: 628: 585: 539: 618: 32: 529: 391: 36: 461: 565: 455: 433: 130: 50: 8: 70: 46: 385: 535: 514: 126: 115: 506: 227: 111: 569: 510: 612: 226:} = 0. (In other words, they are pairwise in involution.) Here {–,–} is the 40: 27:
Theorem in symplectic geometry which generalizes Darboux's theorem
487:
there is a symplectic chart such that one of its coordinates is
208:{\displaystyle df_{1}(p)\wedge \ldots \wedge df_{r}(p)\neq 0,} 557: 369:{\displaystyle \omega =\sum _{i=1}^{n}df_{i}\wedge dg_{i}.} 528:
Libermann, P.; Marle, Charles-Michel (6 December 2012).
384:
As a direct application we have the following. Given a
464: 436: 394: 306: 142: 479: 442: 418: 368: 207: 610: 505:. Graduate Texts in Mathematics. Vol. 218. 527: 430:is a symplectic manifold with symplectic form 593: 531:Symplectic Geometry and Analytical Mechanics 600: 586: 14: 611: 552: 500: 24: 25: 640: 624:Theorems in differential geometry 73:with symplectic form ω. For 556: 503:Introduction to Smooth Manifolds 265:defined on an open neighborhood 379: 293:, i.e., ω is expressed on 133:at each point, or equivalently 18:Carathéodory-Jacobi-Lie theorem 413: 395: 193: 187: 162: 156: 13: 1: 494: 419:{\displaystyle (M,\omega ,H)} 572:. You can help Knowledge by 56: 7: 629:Differential geometry stubs 458:, around every point where 230:. Then there are functions 10: 645: 551: 511:10.1007/978-1-4419-9982-5 480:{\displaystyle dH\neq 0} 443:{\displaystyle \omega } 568:-related article is a 481: 444: 420: 370: 333: 209: 566:differential geometry 501:Lee, John M. (2012). 482: 445: 421: 371: 313: 210: 462: 456:Hamiltonian function 434: 392: 304: 140: 131:linearly independent 619:Symplectic geometry 85: ≤  71:symplectic manifold 47:symplectic geometry 477: 440: 416: 386:Hamiltonian system 366: 205: 49:which generalizes 581: 580: 520:978-1-4419-9981-8 116:open neighborhood 51:Darboux's theorem 16:(Redirected from 636: 602: 595: 588: 560: 553: 545: 524: 486: 484: 483: 478: 449: 447: 446: 441: 425: 423: 422: 417: 375: 373: 372: 367: 362: 361: 346: 345: 332: 327: 287:symplectic chart 214: 212: 211: 206: 186: 185: 155: 154: 112:smooth functions 45:is a theorem in 21: 644: 643: 639: 638: 637: 635: 634: 633: 609: 608: 607: 606: 549: 542: 521: 497: 463: 460: 459: 435: 432: 431: 393: 390: 389: 382: 357: 353: 341: 337: 328: 317: 305: 302: 301: 284: 280: 264: 257: 250: 243: 236: 228:Poisson bracket 225: 221: 181: 177: 150: 146: 141: 138: 137: 109: 102: 95: 59: 28: 23: 22: 15: 12: 11: 5: 642: 632: 631: 626: 621: 605: 604: 597: 590: 582: 579: 578: 561: 547: 546: 540: 525: 519: 496: 493: 476: 473: 470: 467: 439: 415: 412: 409: 406: 403: 400: 397: 381: 378: 377: 376: 365: 360: 356: 352: 349: 344: 340: 336: 331: 326: 323: 320: 316: 312: 309: 282: 278: 262: 255: 248: 241: 234: 223: 219: 216: 215: 204: 201: 198: 195: 192: 189: 184: 180: 176: 173: 170: 167: 164: 161: 158: 153: 149: 145: 114:defined on an 107: 100: 93: 58: 55: 26: 9: 6: 4: 3: 2: 641: 630: 627: 625: 622: 620: 617: 616: 614: 603: 598: 596: 591: 589: 584: 583: 577: 575: 571: 567: 562: 559: 555: 554: 550: 543: 541:9789400938076 537: 533: 532: 526: 522: 516: 512: 508: 504: 499: 498: 492: 490: 474: 471: 468: 465: 457: 453: 437: 429: 410: 407: 404: 401: 398: 387: 363: 358: 354: 350: 347: 342: 338: 334: 329: 324: 321: 318: 314: 310: 307: 300: 299: 298: 296: 292: 288: 276: 272: 269: ⊂  268: 261: 254: 247: 240: 233: 229: 202: 199: 196: 190: 182: 178: 174: 171: 168: 165: 159: 151: 147: 143: 136: 135: 134: 132: 128: 127:differentials 124: 120: 117: 113: 106: 99: 92: 88: 84: 80: 77: ∈  76: 72: 69:-dimensional 68: 64: 54: 52: 48: 44: 42: 38: 34: 19: 574:expanding it 563: 548: 530: 502: 488: 451: 427: 383: 380:Applications 294: 290: 277:such that (f 274: 270: 266: 259: 252: 245: 238: 231: 217: 122: 118: 104: 97: 90: 86: 82: 78: 74: 66: 62: 60: 33:Carathéodory 31: 29: 613:Categories 495:References 472:≠ 438:ω 405:ω 348:∧ 315:∑ 308:ω 197:≠ 172:∧ 169:… 166:∧ 57:Statement 218:where {f 454:is the 285:) is a 258:, ..., 237:, ..., 103:, ..., 43:theorem 538:  517:  426:where 125:whose 89:, let 65:be a 2 37:Jacobi 564:This 570:stub 536:ISBN 515:ISBN 450:and 129:are 81:and 61:Let 30:The 507:doi 388:as 297:as 289:of 281:, g 273:of 235:r+1 222:, f 121:of 110:be 41:Lie 615:: 534:. 513:. 491:. 251:, 244:, 96:, 53:. 601:e 594:t 587:v 576:. 544:. 523:. 509:: 489:H 475:0 469:H 466:d 452:H 428:M 414:) 411:H 408:, 402:, 399:M 396:( 364:. 359:i 355:g 351:d 343:i 339:f 335:d 330:n 325:1 322:= 319:i 311:= 295:U 291:M 283:i 279:i 275:p 271:V 267:U 263:n 260:g 256:2 253:g 249:1 246:g 242:n 239:f 232:f 224:j 220:i 203:, 200:0 194:) 191:p 188:( 183:r 179:f 175:d 163:) 160:p 157:( 152:1 148:f 144:d 123:p 119:V 108:r 105:f 101:2 98:f 94:1 91:f 87:n 83:r 79:M 75:p 67:n 63:M 39:– 35:– 20:)

Index

Carathéodory-Jacobi-Lie theorem
Carathéodory
Jacobi
Lie
symplectic geometry
Darboux's theorem
symplectic manifold
smooth functions
open neighborhood
differentials
linearly independent
Poisson bracket
symplectic chart
Hamiltonian system
Hamiltonian function
doi
10.1007/978-1-4419-9982-5
ISBN
978-1-4419-9981-8
Symplectic Geometry and Analytical Mechanics
ISBN
9789400938076
Stub icon
differential geometry
stub
expanding it
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