737:
750:
1559:
2754:
1259:
2525:
2313:
3733:
2544:
2936:
1914:
1554:{\displaystyle {\mathcal {H}}(\phi _{1},\phi _{2},\ldots ,\pi _{1},\pi _{2},\ldots ,\mathbf {x} ,t)=\sum _{i}{\dot {\phi _{i}}}\pi _{i}-{\mathcal {L}}(\phi _{1},\phi _{2},\ldots \nabla \phi _{1},\nabla \phi _{2},\ldots ,\partial \phi _{1}/\partial t,\partial \phi _{2}/\partial t,\ldots ,\mathbf {x} ,t)\,.}
834:
and conjugate momenta, and possibly, time. For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field. Since each point mass has one or
2032:
1007:
3579:
is an index which takes values 0 (for the time coordinate), and 1, 2, 3 (for the spatial coordinates), so strictly only one derivative or coordinate would be present. In general, all the spatial and time derivatives will appear in the
Lagrangian density, for example in Cartesian coordinates, the
1177:
1040:. The Lagrangian density is a function of the fields in the system, their space and time derivatives, and possibly the space and time coordinates themselves. It is the field analogue to the Lagrangian function for a system of discrete particles described by generalized coordinates.
2319:
2107:
3585:
3328:
2749:{\displaystyle _{\phi ,\pi }=\int d^{3}x\sum _{i}\left({\frac {\delta {\mathcal {A}}}{\delta \phi _{i}}}{\frac {\delta {\mathcal {B}}}{\delta \pi _{i}}}-{\frac {\delta {\mathcal {B}}}{\delta \phi _{i}}}{\frac {\delta {\mathcal {A}}}{\delta \pi _{i}}}\right)\,,}
2806:
1656:
1781:
3244:
1929:
875:
1082:
3569:
3778:
Gotay, M., A multisymplectic framework for classical field theory and the calculus of variations. II. Space + time decomposition, in "Mechanics, Analysis and
Geometry: 200 Years after Lagrange" (North Holland,
3032:
2520:{\displaystyle B=\int d^{3}x{\mathcal {B}}\left(\phi _{1},\phi _{2},\ldots ,\pi _{1},\pi _{2},\ldots ,\nabla \phi _{1},\nabla \phi _{2},\ldots ,\nabla \pi _{1},\nabla \pi _{2},\ldots ,\mathbf {x} ,t\right)\,,}
2308:{\displaystyle A=\int d^{3}x{\mathcal {A}}\left(\phi _{1},\phi _{2},\ldots ,\pi _{1},\pi _{2},\ldots ,\nabla \phi _{1},\nabla \phi _{2},\ldots ,\nabla \pi _{1},\nabla \pi _{2},\ldots ,\mathbf {x} ,t\right)\,,}
3160:
3728:{\displaystyle {\mathcal {L}}\left(\phi ,{\frac {\partial \phi }{\partial x}},{\frac {\partial \phi }{\partial y}},{\frac {\partial \phi }{\partial z}},{\frac {\partial \phi }{\partial t}},x,y,z,t\right)}
91:
3746:
This is standard notation in this context, most of the literature does not explicitly mention it is a partial derivative. In general total and partial time derivatives of a function are not the same.
3255:
3055:
The following results are true if the
Lagrangian and Hamiltonian densities are explicitly time-independent (they can still have implicit time-dependence via the fields and their derivatives),
847:
The
Hamiltonian density is the continuous analogue for fields; it is a function of the fields, the conjugate "momentum" fields, and possibly the space and time coordinates themselves. For one
1719:
2795:
2931:{\displaystyle {\frac {\delta {\mathcal {F}}}{\delta f}}={\frac {\partial {\mathcal {F}}}{\partial f}}-\sum _{i}\nabla _{i}{\frac {\partial {\mathcal {F}}}{\partial (\nabla _{i}f)}}\,.}
1570:
115:
3109:
3085:
3442:
1909:{\displaystyle {\dot {\phi }}_{i}=+{\frac {\delta {\mathcal {H}}}{\delta \pi _{i}}}\,,\quad {\dot {\pi }}_{i}=-{\frac {\delta {\mathcal {H}}}{\delta \phi _{i}}}\,,}
3188:
2027:{\displaystyle {\frac {\delta }{\delta \phi _{i}}}={\frac {\partial }{\partial \phi _{i}}}-\nabla \cdot {\frac {\partial }{\partial (\nabla \phi _{i})}}\,,}
1002:{\displaystyle {\mathcal {H}}(\phi ,\pi ,\mathbf {x} ,t)={\dot {\phi }}\pi -{\mathcal {L}}(\phi ,\nabla \phi ,\partial \phi /\partial t,\mathbf {x} ,t)\,.}
17:
1172:{\displaystyle \pi ={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}}}\,,\quad {\dot {\phi }}\equiv {\frac {\partial \phi }{\partial t}}\,,}
3847:
Echeverria-Enriquez, A., Munos-Lecanda, M., Roman-Roy, N., Geometry of multisymplectic
Hamiltonian first-order field theories, J. Math. Phys.
781:
3512:
370:
2959:
836:
489:
3117:
462:
47:
3981:
3907:
3797:
3507:
It is a standard abuse of notation to abbreviate all the derivatives and coordinates in the
Lagrangian density as follows:
3323:{\displaystyle \mathbf {S} ={\frac {\partial {\mathcal {L}}}{\partial (\nabla \phi )}}{\frac {\partial \phi }{\partial t}}}
444:
1671:
3963:
774:
2762:
3864:-symplectic formalism) in classical field theory: Skinner-Rusk approach and the evolution operator, J. Math. Phys.
3478:
2941:
Under the same conditions of vanishing fields on the surface, the following result holds for the time evolution of
1043:
As in
Hamiltonian mechanics where every generalized coordinate has a corresponding generalized momentum, the field
736:
4010:
3397:. Covariant Hamiltonian field theory is developed in the Hamilton–De Donder, polysymplectic, multisymplectic and
110:
1076:, defined as the partial derivative of the Lagrangian density with respect to the time derivative of the field,
4020:
3937:
3884:
Variational
Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws -
365:
105:
4015:
4000:
4005:
767:
754:
515:
438:
360:
203:
1772:
for the fields are similar to the
Hamiltonian equations for discrete particles. For any number of fields:
1651:{\displaystyle \pi _{i}(\mathbf {x} ,t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}_{i}}}\,.}
3473:
434:
235:
2530:
and the fields are zero on the boundary of the volume the integrals are taken over, the field theoretic
213:
654:
543:
469:
329:
262:
3458:
3170:
Taking the partial time derivative of the definition of the
Hamiltonian density above, and using the
408:
3090:
3066:
3488:
3412:
3175:
684:
528:
3425:
3401:-symplectic variants. A phase space of covariant Hamiltonian field theory is a finite-dimensional
3483:
831:
811:
634:
398:
674:
3468:
3356:
2798:
1920:
807:
679:
496:
3823:
3789:
3364:
827:
803:
689:
664:
350:
168:
3737:
Here we write the same thing, but using ∇ to abbreviate all spatial derivatives as a vector.
3359:, that takes the form of the instantaneous Hamiltonian formalism on an infinite-dimensional
3063:
The Hamiltonian density is the total energy density, the sum of the kinetic energy density (
3887:
3453:
3368:
3345:
3044:
1725:
815:
709:
669:
577:
573:
565:
555:
345:
338:
94:
704:
8:
3406:
3402:
3239:{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial t}}+\nabla \cdot \mathbf {S} =0}
3179:
1769:
795:
484:
425:
403:
148:
143:
138:
38:
3891:
3952:
3913:
3831:
3352:
2042:
1183:
866:
614:
355:
230:
198:
158:
3367:
are field functions at some instant of time. This Hamiltonian formalism is applied to
1724:
The Hamiltonian density is the Hamiltonian per unit spatial volume. The corresponding
3977:
3959:
3933:
3925:
3917:
3903:
3793:
3394:
3390:
3376:
1017:
624:
581:
538:
533:
474:
250:
240:
133:
3895:
1196:
719:
699:
644:
639:
585:
560:
415:
273:
218:
193:
2531:
1662:
1027:
714:
659:
609:
604:
523:
3947:
3249:
in which the Hamiltonian density can be interpreted as the energy density, and
1665:
of the Hamiltonian density gives the Hamiltonian, in three spatial dimensions:
741:
649:
550:
267:
3899:
3994:
3416:
629:
456:
3385:
corresponds to derivatives of fields with respect to all world coordinates
3372:
1749:
1745:
1741:
848:
694:
619:
308:
188:
3463:
3360:
2072:
2038:
3333:
the energy flux, or flow of energy per unit time per unit surface area.
3835:
3171:
2535:
479:
153:
3932:(2nd ed.). San Francisco, CA: Addison Wesley. pp. 562–565.
3420:
501:
3564:{\displaystyle {\mathcal {L}}(\phi ,\partial _{\mu }\phi ,x_{\mu })}
1729:
420:
303:
278:
3027:{\displaystyle {\frac {dA}{dt}}=+{\frac {\partial A}{\partial t}}}
27:
Formalism in classical field theory based on Hamiltonian mechanics
2101:, their spatial derivatives, and the space and time coordinates,
1757:
1564:
where each conjugate field is defined with respect to its field,
393:
246:
163:
2075:, because fields have an infinite number of degrees of freedom.
839:, the field formulation has infinitely many degrees of freedom.
3155:{\displaystyle {\mathcal {H}}={\mathcal {T}}+{\mathcal {V}}\,.}
452:
298:
208:
3178:
and the definition of the conjugate momentum field, gives the
3826:, Covariant Hamiltonian equations for field theory, J. Phys.
3792:, "Advanced Classical Field Theory", World Scientific, 2009,
1753:
288:
283:
225:
1919:
where again the overdots are partial time derivatives, the
1037:
293:
256:
830:
for a system of discrete particles is a function of their
3860:
Rey, A., Roman-Roy, N. Saldago, M., Gunther's formalism (
3058:
86:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
3809:
Krupkova, O., Hamiltonian field theory, J. Geom. Phys.
3415:
is formulated as covariant Hamiltonian field theory on
1740:
The above equations and definitions can be extended to
3351:
Hamiltonian field theory usually means the symplectic
3928:(1980). "Chapter 12: Continuous Systems and Fields".
3588:
3515:
3428:
3389:. Covariant Hamilton equations are equivalent to the
3258:
3191:
3120:
3093:
3069:
3037:
which can be found from the total time derivative of
2962:
2809:
2765:
2547:
2322:
2110:
1932:
1784:
1674:
1573:
1262:
1085:
878:
50:
1253:the Hamiltonian density is a function of them all:
3951:
3727:
3563:
3436:
3322:
3238:
3154:
3103:
3079:
3026:
2930:
2789:
2748:
2519:
2307:
2026:
1908:
1713:
1650:
1553:
1171:
1001:
85:
3992:
3946:
3766:
1714:{\displaystyle H=\int {\mathcal {H}}\ d^{3}x\,.}
3974:Theoretical Mechanics of Particles and Continua
3881:
2790:{\displaystyle \delta {\mathcal {F}}/\delta f}
865:, the Hamiltonian density is defined from the
3971:
3050:
2083:For two functions which depend on the fields
802:is the field-theoretic analogue to classical
775:
3336:
2534:is defined as (not to be confused with the
1735:
3375:. In Covariant Hamiltonian field theory,
1661:In general, for any number of fields, the
782:
768:
3924:
3430:
3348:formulation of Hamiltonian field theory.
3148:
2924:
2742:
2513:
2301:
2020:
1902:
1841:
1707:
1644:
1547:
1165:
1125:
995:
3047:, and using the above Poisson bracket.
14:
3993:
3972:Fetter, A. L.; Walecka, J. D. (1980).
3580:Lagrangian density has the full form:
3165:
3059:Kinetic and potential energy densities
1763:
1752:. In physics, tensor fields describe
1212:
464:Newton's law of universal gravitation
3413:Hamiltonian non-autonomous mechanics
3087:) and the potential energy density (
842:
445:Mechanics of planar particle motion
53:
24:
3687:
3679:
3664:
3656:
3641:
3633:
3618:
3610:
3591:
3533:
3518:
3342:Covariant Hamiltonian field theory
3311:
3303:
3288:
3282:
3275:
3270:
3219:
3207:
3200:
3195:
3143:
3133:
3123:
3096:
3072:
3015:
3007:
2906:
2899:
2892:
2887:
2875:
2852:
2845:
2840:
2818:
2771:
2714:
2683:
2649:
2618:
2475:
2459:
2437:
2421:
2347:
2263:
2247:
2225:
2209:
2135:
2078:
2001:
1995:
1991:
1983:
1964:
1960:
1879:
1818:
1686:
1619:
1612:
1607:
1518:
1500:
1491:
1473:
1451:
1435:
1398:
1265:
1156:
1148:
1107:
1100:
1095:
972:
961:
952:
938:
881:
25:
18:Covariant Hamiltonian field theory
4032:
3882:Badin, G.; Crisciani, F. (2018).
2041:, must be used instead of simply
3822:Giachetta, G., Mangiarotti, L.,
3788:Giachetta, G., Mangiarotti, L.,
3479:Covariant classical field theory
3260:
3226:
2498:
2286:
1588:
1534:
1338:
982:
902:
749:
748:
735:
68:
1845:
1182:in which the overdot denotes a
1129:
3854:
3841:
3816:
3803:
3782:
3772:
3760:
3740:
3558:
3523:
3501:
3294:
3285:
3104:{\displaystyle {\mathcal {V}}}
3080:{\displaystyle {\mathcal {T}}}
2998:
2986:
2918:
2902:
2561:
2548:
2048:
2014:
1998:
1598:
1584:
1544:
1403:
1348:
1270:
992:
943:
912:
886:
814:. It also has applications in
13:
1:
3875:
3419:over the time axis, i.e. the
2071:form an infinite dimensional
1732:Joules per metre cubed, J m.
821:
371:Koopman–von Neumann mechanics
3767:Greiner & Reinhardt 1996
3753:
3437:{\displaystyle \mathbb {R} }
3393:in the case of hyperregular
1030:of some point in space, and
439:Non-inertial reference frame
7:
3976:. Dover. pp. 258–259.
3474:Hamiltonian fluid mechanics
3447:
1923:with respect to the fields
1776:Hamiltonian field equations
1756:and spinor fields describe
366:Appell's equation of motion
236:Inertial frame of reference
10:
4037:
3051:Explicit time-independence
3900:10.1007/978-3-319-59695-2
3886:. Springer. p. 218.
3337:Relativistic field theory
2538:from quantum mechanics).
1018:"del" or "nabla" operator
3950:; Reinhardt, J. (1996),
3494:
3489:Non-autonomous mechanics
3391:Euler–Lagrange equations
3176:implicit differentiation
1736:Tensor and spinor fields
1060:conjugate momentum field
800:Hamiltonian field theory
529:Rotating reference frame
361:Hamilton–Jacobi equation
3484:Polysymplectic manifold
832:generalized coordinates
812:Lagrangian field theory
806:. It is a formalism in
470:Newton's laws of motion
330:Newton's laws of motion
4011:Classical field theory
3729:
3565:
3469:Canonical quantization
3438:
3369:quantization of fields
3357:classical field theory
3324:
3240:
3156:
3105:
3081:
3028:
2932:
2799:variational derivative
2791:
2750:
2521:
2309:
2028:
1921:variational derivative
1910:
1715:
1652:
1555:
1173:
1003:
808:classical field theory
497:Simple harmonic motion
410:Euler's laws of motion
204:D'Alembert's principle
87:
4021:Differential geometry
3730:
3566:
3459:De Donder–Weyl theory
3439:
3365:canonical coordinates
3353:Hamiltonian formalism
3325:
3241:
3157:
3106:
3082:
3029:
2933:
2792:
2751:
2522:
2310:
2029:
1911:
1716:
1653:
1556:
1235:and their conjugates
1174:
1004:
804:Hamiltonian mechanics
351:Hamiltonian mechanics
169:Statistical mechanics
88:
4016:Quantum field theory
4001:Mathematical physics
3586:
3513:
3454:Analytical mechanics
3426:
3256:
3189:
3118:
3091:
3067:
3045:integration by parts
2960:
2807:
2763:
2545:
2320:
2108:
1930:
1782:
1672:
1571:
1260:
1083:
876:
816:quantum field theory
574:Angular acceleration
566:Rotational frequency
346:Lagrangian mechanics
339:Analytical mechanics
95:Second law of motion
48:
4006:Classical mechanics
3930:Classical Mechanics
3892:2018vffg.book.....B
3371:, e.g., in quantum
3180:continuity equation
3166:Continuity equation
2043:partial derivatives
1770:equations of motion
1764:Equations of motion
1744:and more generally
796:theoretical physics
426:Harmonic oscillator
404:Equations of motion
39:Classical mechanics
33:Part of a series on
3954:Field Quantization
3926:Goldstein, Herbert
3725:
3561:
3434:
3320:
3236:
3152:
3101:
3077:
3024:
2928:
2873:
2787:
2746:
2604:
2517:
2305:
2024:
1906:
1711:
1648:
1551:
1363:
1213:Many scalar fields
1169:
999:
867:Lagrangian density
837:degrees of freedom
742:Physics portal
356:Routhian mechanics
231:Frame of reference
83:
3983:978-0-486-43261-8
3909:978-3-319-59694-5
3824:Sardanashvily, G.
3798:978-981-283-895-7
3790:Sardanashvily, G.
3694:
3671:
3648:
3625:
3377:canonical momenta
3318:
3298:
3214:
3022:
2981:
2922:
2864:
2859:
2832:
2735:
2704:
2670:
2639:
2595:
2018:
1978:
1953:
1900:
1856:
1839:
1795:
1693:
1642:
1632:
1380:
1354:
1163:
1139:
1123:
1119:
927:
792:
791:
539:Centrifugal force
534:Centripetal force
490:Euler's equations
475:Relative velocity
251:Moment of inertia
81:
55:
16:(Redirected from
4028:
3987:
3968:
3957:
3943:
3921:
3869:
3858:
3852:
3845:
3839:
3820:
3814:
3807:
3801:
3786:
3780:
3776:
3770:
3764:
3747:
3744:
3738:
3734:
3732:
3731:
3726:
3724:
3720:
3695:
3693:
3685:
3677:
3672:
3670:
3662:
3654:
3649:
3647:
3639:
3631:
3626:
3624:
3616:
3608:
3595:
3594:
3578:
3570:
3568:
3567:
3562:
3557:
3556:
3541:
3540:
3522:
3521:
3505:
3443:
3441:
3440:
3435:
3433:
3355:when applied to
3329:
3327:
3326:
3321:
3319:
3317:
3309:
3301:
3299:
3297:
3280:
3279:
3278:
3268:
3263:
3245:
3243:
3242:
3237:
3229:
3215:
3213:
3205:
3204:
3203:
3193:
3161:
3159:
3158:
3153:
3147:
3146:
3137:
3136:
3127:
3126:
3110:
3108:
3107:
3102:
3100:
3099:
3086:
3084:
3083:
3078:
3076:
3075:
3042:
3033:
3031:
3030:
3025:
3023:
3021:
3013:
3005:
2982:
2980:
2972:
2964:
2952:
2946:
2937:
2935:
2934:
2929:
2923:
2921:
2914:
2913:
2897:
2896:
2895:
2885:
2883:
2882:
2872:
2860:
2858:
2850:
2849:
2848:
2838:
2833:
2831:
2823:
2822:
2821:
2811:
2796:
2794:
2793:
2788:
2780:
2775:
2774:
2755:
2753:
2752:
2747:
2741:
2737:
2736:
2734:
2733:
2732:
2719:
2718:
2717:
2707:
2705:
2703:
2702:
2701:
2688:
2687:
2686:
2676:
2671:
2669:
2668:
2667:
2654:
2653:
2652:
2642:
2640:
2638:
2637:
2636:
2623:
2622:
2621:
2611:
2603:
2591:
2590:
2575:
2574:
2526:
2524:
2523:
2518:
2512:
2508:
2501:
2487:
2486:
2471:
2470:
2449:
2448:
2433:
2432:
2411:
2410:
2398:
2397:
2379:
2378:
2366:
2365:
2351:
2350:
2341:
2340:
2314:
2312:
2311:
2306:
2300:
2296:
2289:
2275:
2274:
2259:
2258:
2237:
2236:
2221:
2220:
2199:
2198:
2186:
2185:
2167:
2166:
2154:
2153:
2139:
2138:
2129:
2128:
2100:
2091:
2070:
2061:
2033:
2031:
2030:
2025:
2019:
2017:
2013:
2012:
1990:
1979:
1977:
1976:
1975:
1959:
1954:
1952:
1951:
1950:
1934:
1915:
1913:
1912:
1907:
1901:
1899:
1898:
1897:
1884:
1883:
1882:
1872:
1864:
1863:
1858:
1857:
1849:
1840:
1838:
1837:
1836:
1823:
1822:
1821:
1811:
1803:
1802:
1797:
1796:
1788:
1720:
1718:
1717:
1712:
1703:
1702:
1691:
1690:
1689:
1657:
1655:
1654:
1649:
1643:
1641:
1640:
1639:
1634:
1633:
1625:
1617:
1616:
1615:
1605:
1591:
1583:
1582:
1560:
1558:
1557:
1552:
1537:
1517:
1512:
1511:
1490:
1485:
1484:
1463:
1462:
1447:
1446:
1428:
1427:
1415:
1414:
1402:
1401:
1392:
1391:
1382:
1381:
1376:
1375:
1366:
1362:
1341:
1327:
1326:
1314:
1313:
1295:
1294:
1282:
1281:
1269:
1268:
1252:
1234:
1217:For many fields
1208:
1199:time derivative
1194:
1188:time derivative
1178:
1176:
1175:
1170:
1164:
1162:
1154:
1146:
1141:
1140:
1132:
1124:
1122:
1121:
1120:
1112:
1105:
1104:
1103:
1093:
1075:
1057:
1035:
1025:
1015:
1008:
1006:
1005:
1000:
985:
971:
942:
941:
929:
928:
920:
905:
885:
884:
864:
843:One scalar field
784:
777:
770:
757:
752:
751:
744:
740:
739:
645:Johann Bernoulli
640:Daniel Bernoulli
561:Tangential speed
465:
441:
416:Fictitious force
411:
263:Mechanical power
253:
194:Angular momentum
92:
90:
89:
84:
82:
80:
72:
71:
62:
57:
56:
30:
29:
21:
4036:
4035:
4031:
4030:
4029:
4027:
4026:
4025:
3991:
3990:
3984:
3966:
3940:
3910:
3878:
3873:
3872:
3859:
3855:
3846:
3842:
3821:
3817:
3808:
3804:
3787:
3783:
3777:
3773:
3765:
3761:
3756:
3751:
3750:
3745:
3741:
3686:
3678:
3676:
3663:
3655:
3653:
3640:
3632:
3630:
3617:
3609:
3607:
3600:
3596:
3590:
3589:
3587:
3584:
3583:
3574:
3552:
3548:
3536:
3532:
3517:
3516:
3514:
3511:
3510:
3506:
3502:
3497:
3450:
3429:
3427:
3424:
3423:
3407:multisymplectic
3383:
3339:
3310:
3302:
3300:
3281:
3274:
3273:
3269:
3267:
3259:
3257:
3254:
3253:
3225:
3206:
3199:
3198:
3194:
3192:
3190:
3187:
3186:
3168:
3142:
3141:
3132:
3131:
3122:
3121:
3119:
3116:
3115:
3095:
3094:
3092:
3089:
3088:
3071:
3070:
3068:
3065:
3064:
3061:
3053:
3038:
3014:
3006:
3004:
2973:
2965:
2963:
2961:
2958:
2957:
2948:
2947:(similarly for
2942:
2909:
2905:
2898:
2891:
2890:
2886:
2884:
2878:
2874:
2868:
2851:
2844:
2843:
2839:
2837:
2824:
2817:
2816:
2812:
2810:
2808:
2805:
2804:
2776:
2770:
2769:
2764:
2761:
2760:
2728:
2724:
2720:
2713:
2712:
2708:
2706:
2697:
2693:
2689:
2682:
2681:
2677:
2675:
2663:
2659:
2655:
2648:
2647:
2643:
2641:
2632:
2628:
2624:
2617:
2616:
2612:
2610:
2609:
2605:
2599:
2586:
2582:
2564:
2560:
2546:
2543:
2542:
2532:Poisson bracket
2497:
2482:
2478:
2466:
2462:
2444:
2440:
2428:
2424:
2406:
2402:
2393:
2389:
2374:
2370:
2361:
2357:
2356:
2352:
2346:
2345:
2336:
2332:
2321:
2318:
2317:
2285:
2270:
2266:
2254:
2250:
2232:
2228:
2216:
2212:
2194:
2190:
2181:
2177:
2162:
2158:
2149:
2145:
2144:
2140:
2134:
2133:
2124:
2120:
2109:
2106:
2105:
2098:
2093:
2089:
2084:
2081:
2079:Poisson bracket
2068:
2063:
2062:and conjugates
2059:
2054:
2051:
2008:
2004:
1994:
1989:
1971:
1967:
1963:
1958:
1946:
1942:
1938:
1933:
1931:
1928:
1927:
1917:
1893:
1889:
1885:
1878:
1877:
1873:
1871:
1859:
1848:
1847:
1846:
1832:
1828:
1824:
1817:
1816:
1812:
1810:
1798:
1787:
1786:
1785:
1783:
1780:
1779:
1766:
1738:
1698:
1694:
1685:
1684:
1673:
1670:
1669:
1663:volume integral
1635:
1624:
1623:
1622:
1618:
1611:
1610:
1606:
1604:
1587:
1578:
1574:
1572:
1569:
1568:
1533:
1513:
1507:
1503:
1486:
1480:
1476:
1458:
1454:
1442:
1438:
1423:
1419:
1410:
1406:
1397:
1396:
1387:
1383:
1371:
1367:
1365:
1364:
1358:
1337:
1322:
1318:
1309:
1305:
1290:
1286:
1277:
1273:
1264:
1263:
1261:
1258:
1257:
1241:
1236:
1223:
1218:
1215:
1200:
1189:
1155:
1147:
1145:
1131:
1130:
1111:
1110:
1106:
1099:
1098:
1094:
1092:
1084:
1081:
1080:
1062:
1044:
1031:
1028:position vector
1021:
1013:
981:
967:
937:
936:
919:
918:
901:
880:
879:
877:
874:
873:
851:
845:
824:
788:
747:
734:
733:
726:
725:
724:
599:
591:
590:
570:
524:Circular motion
518:
508:
507:
506:
463:
433:
430:
409:
388:
380:
379:
376:
375:
333:
323:
315:
314:
313:
272:
268:Mechanical work
261:
245:
183:
175:
174:
173:
128:
120:
97:
73:
67:
63:
61:
52:
51:
49:
46:
45:
28:
23:
22:
15:
12:
11:
5:
4034:
4024:
4023:
4018:
4013:
4008:
4003:
3989:
3988:
3982:
3969:
3964:
3944:
3938:
3922:
3908:
3877:
3874:
3871:
3870:
3868:(2005) 052901.
3853:
3840:
3836:hep-th/9904062
3815:
3802:
3781:
3771:
3758:
3757:
3755:
3752:
3749:
3748:
3739:
3736:
3735:
3723:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3692:
3689:
3684:
3681:
3675:
3669:
3666:
3661:
3658:
3652:
3646:
3643:
3638:
3635:
3629:
3623:
3620:
3615:
3612:
3606:
3603:
3599:
3593:
3572:
3571:
3560:
3555:
3551:
3547:
3544:
3539:
3535:
3531:
3528:
3525:
3520:
3499:
3498:
3496:
3493:
3492:
3491:
3486:
3481:
3476:
3471:
3466:
3461:
3456:
3449:
3446:
3432:
3403:polysymplectic
3381:
3338:
3335:
3331:
3330:
3316:
3313:
3308:
3305:
3296:
3293:
3290:
3287:
3284:
3277:
3272:
3266:
3262:
3247:
3246:
3235:
3232:
3228:
3224:
3221:
3218:
3212:
3209:
3202:
3197:
3167:
3164:
3163:
3162:
3151:
3145:
3140:
3135:
3130:
3125:
3098:
3074:
3060:
3057:
3052:
3049:
3035:
3034:
3020:
3017:
3012:
3009:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2979:
2976:
2971:
2968:
2939:
2938:
2927:
2920:
2917:
2912:
2908:
2904:
2901:
2894:
2889:
2881:
2877:
2871:
2867:
2863:
2857:
2854:
2847:
2842:
2836:
2830:
2827:
2820:
2815:
2786:
2783:
2779:
2773:
2768:
2757:
2756:
2745:
2740:
2731:
2727:
2723:
2716:
2711:
2700:
2696:
2692:
2685:
2680:
2674:
2666:
2662:
2658:
2651:
2646:
2635:
2631:
2627:
2620:
2615:
2608:
2602:
2598:
2594:
2589:
2585:
2581:
2578:
2573:
2570:
2567:
2563:
2559:
2556:
2553:
2550:
2528:
2527:
2516:
2511:
2507:
2504:
2500:
2496:
2493:
2490:
2485:
2481:
2477:
2474:
2469:
2465:
2461:
2458:
2455:
2452:
2447:
2443:
2439:
2436:
2431:
2427:
2423:
2420:
2417:
2414:
2409:
2405:
2401:
2396:
2392:
2388:
2385:
2382:
2377:
2373:
2369:
2364:
2360:
2355:
2349:
2344:
2339:
2335:
2331:
2328:
2325:
2315:
2304:
2299:
2295:
2292:
2288:
2284:
2281:
2278:
2273:
2269:
2265:
2262:
2257:
2253:
2249:
2246:
2243:
2240:
2235:
2231:
2227:
2224:
2219:
2215:
2211:
2208:
2205:
2202:
2197:
2193:
2189:
2184:
2180:
2176:
2173:
2170:
2165:
2161:
2157:
2152:
2148:
2143:
2137:
2132:
2127:
2123:
2119:
2116:
2113:
2096:
2087:
2080:
2077:
2066:
2057:
2050:
2047:
2035:
2034:
2023:
2016:
2011:
2007:
2003:
2000:
1997:
1993:
1988:
1985:
1982:
1974:
1970:
1966:
1962:
1957:
1949:
1945:
1941:
1937:
1905:
1896:
1892:
1888:
1881:
1876:
1870:
1867:
1862:
1855:
1852:
1844:
1835:
1831:
1827:
1820:
1815:
1809:
1806:
1801:
1794:
1791:
1774:
1765:
1762:
1737:
1734:
1722:
1721:
1710:
1706:
1701:
1697:
1688:
1683:
1680:
1677:
1659:
1658:
1647:
1638:
1631:
1628:
1621:
1614:
1609:
1603:
1600:
1597:
1594:
1590:
1586:
1581:
1577:
1562:
1561:
1550:
1546:
1543:
1540:
1536:
1532:
1529:
1526:
1523:
1520:
1516:
1510:
1506:
1502:
1499:
1496:
1493:
1489:
1483:
1479:
1475:
1472:
1469:
1466:
1461:
1457:
1453:
1450:
1445:
1441:
1437:
1434:
1431:
1426:
1422:
1418:
1413:
1409:
1405:
1400:
1395:
1390:
1386:
1379:
1374:
1370:
1361:
1357:
1353:
1350:
1347:
1344:
1340:
1336:
1333:
1330:
1325:
1321:
1317:
1312:
1308:
1304:
1301:
1298:
1293:
1289:
1285:
1280:
1276:
1272:
1267:
1239:
1221:
1214:
1211:
1180:
1179:
1168:
1161:
1158:
1153:
1150:
1144:
1138:
1135:
1128:
1118:
1115:
1109:
1102:
1097:
1091:
1088:
1010:
1009:
998:
994:
991:
988:
984:
980:
977:
974:
970:
966:
963:
960:
957:
954:
951:
948:
945:
940:
935:
932:
926:
923:
917:
914:
911:
908:
904:
900:
897:
894:
891:
888:
883:
844:
841:
823:
820:
790:
789:
787:
786:
779:
772:
764:
761:
760:
759:
758:
745:
728:
727:
723:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
617:
612:
607:
601:
600:
597:
596:
593:
592:
589:
588:
569:
568:
563:
558:
553:
551:Coriolis force
548:
547:
546:
536:
531:
526:
520:
519:
514:
513:
510:
509:
505:
504:
499:
494:
493:
492:
487:
477:
472:
467:
460:
449:
448:
447:
442:
429:
428:
423:
418:
413:
406:
401:
396:
390:
389:
386:
385:
382:
381:
378:
377:
374:
373:
368:
363:
358:
353:
348:
342:
336:
334:
327:
324:
321:
320:
317:
316:
312:
311:
306:
301:
296:
291:
286:
281:
276:
270:
265:
259:
254:
243:
238:
233:
228:
223:
222:
221:
216:
206:
201:
196:
191:
185:
184:
181:
180:
177:
176:
172:
171:
166:
161:
156:
151:
146:
141:
136:
130:
129:
126:
125:
122:
121:
119:
118:
113:
108:
102:
99:
98:
93:
79:
76:
70:
66:
60:
42:
41:
35:
34:
26:
9:
6:
4:
3:
2:
4033:
4022:
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3998:
3996:
3985:
3979:
3975:
3970:
3967:
3965:3-540-59179-6
3961:
3956:
3955:
3949:
3945:
3941:
3935:
3931:
3927:
3923:
3919:
3915:
3911:
3905:
3901:
3897:
3893:
3889:
3885:
3880:
3879:
3867:
3863:
3857:
3850:
3844:
3837:
3833:
3830:(1999) 6629;
3829:
3825:
3819:
3812:
3806:
3799:
3795:
3791:
3785:
3775:
3768:
3763:
3759:
3743:
3721:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3690:
3682:
3673:
3667:
3659:
3650:
3644:
3636:
3627:
3621:
3613:
3604:
3601:
3597:
3582:
3581:
3577:
3553:
3549:
3545:
3542:
3537:
3529:
3526:
3509:
3508:
3504:
3500:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3451:
3445:
3422:
3418:
3417:fiber bundles
3414:
3410:
3408:
3404:
3400:
3396:
3392:
3388:
3384:
3378:
3374:
3370:
3366:
3362:
3358:
3354:
3349:
3347:
3343:
3334:
3314:
3306:
3291:
3264:
3252:
3251:
3250:
3233:
3230:
3222:
3216:
3210:
3185:
3184:
3183:
3181:
3177:
3173:
3149:
3138:
3128:
3114:
3113:
3112:
3056:
3048:
3046:
3041:
3018:
3010:
3001:
2995:
2992:
2989:
2983:
2977:
2974:
2969:
2966:
2956:
2955:
2954:
2951:
2945:
2925:
2915:
2910:
2879:
2869:
2865:
2861:
2855:
2834:
2828:
2825:
2813:
2803:
2802:
2801:
2800:
2784:
2781:
2777:
2766:
2743:
2738:
2729:
2725:
2721:
2709:
2698:
2694:
2690:
2678:
2672:
2664:
2660:
2656:
2644:
2633:
2629:
2625:
2613:
2606:
2600:
2596:
2592:
2587:
2583:
2579:
2576:
2571:
2568:
2565:
2557:
2554:
2551:
2541:
2540:
2539:
2537:
2533:
2514:
2509:
2505:
2502:
2494:
2491:
2488:
2483:
2479:
2472:
2467:
2463:
2456:
2453:
2450:
2445:
2441:
2434:
2429:
2425:
2418:
2415:
2412:
2407:
2403:
2399:
2394:
2390:
2386:
2383:
2380:
2375:
2371:
2367:
2362:
2358:
2353:
2342:
2337:
2333:
2329:
2326:
2323:
2316:
2302:
2297:
2293:
2290:
2282:
2279:
2276:
2271:
2267:
2260:
2255:
2251:
2244:
2241:
2238:
2233:
2229:
2222:
2217:
2213:
2206:
2203:
2200:
2195:
2191:
2187:
2182:
2178:
2174:
2171:
2168:
2163:
2159:
2155:
2150:
2146:
2141:
2130:
2125:
2121:
2117:
2114:
2111:
2104:
2103:
2102:
2099:
2090:
2076:
2074:
2069:
2060:
2046:
2044:
2040:
2021:
2009:
2005:
1986:
1980:
1972:
1968:
1955:
1947:
1943:
1939:
1935:
1926:
1925:
1924:
1922:
1916:
1903:
1894:
1890:
1886:
1874:
1868:
1865:
1860:
1853:
1850:
1842:
1833:
1829:
1825:
1813:
1807:
1804:
1799:
1792:
1789:
1777:
1773:
1771:
1761:
1759:
1755:
1751:
1750:spinor fields
1747:
1746:tensor fields
1743:
1742:vector fields
1733:
1731:
1727:
1708:
1704:
1699:
1695:
1681:
1678:
1675:
1668:
1667:
1666:
1664:
1645:
1636:
1629:
1626:
1601:
1595:
1592:
1579:
1575:
1567:
1566:
1565:
1548:
1541:
1538:
1530:
1527:
1524:
1521:
1514:
1508:
1504:
1497:
1494:
1487:
1481:
1477:
1470:
1467:
1464:
1459:
1455:
1448:
1443:
1439:
1432:
1429:
1424:
1420:
1416:
1411:
1407:
1393:
1388:
1384:
1377:
1372:
1368:
1359:
1355:
1351:
1345:
1342:
1334:
1331:
1328:
1323:
1319:
1315:
1310:
1306:
1302:
1299:
1296:
1291:
1287:
1283:
1278:
1274:
1256:
1255:
1254:
1250:
1246:
1242:
1232:
1228:
1224:
1210:
1207:
1203:
1198:
1193:
1187:
1186:
1166:
1159:
1151:
1142:
1136:
1133:
1126:
1116:
1113:
1089:
1086:
1079:
1078:
1077:
1073:
1069:
1065:
1061:
1055:
1051:
1047:
1041:
1039:
1034:
1029:
1024:
1019:
996:
989:
986:
978:
975:
968:
964:
958:
955:
949:
946:
933:
930:
924:
921:
915:
909:
906:
898:
895:
892:
889:
872:
871:
870:
868:
862:
858:
854:
850:
840:
838:
833:
829:
819:
817:
813:
809:
805:
801:
797:
785:
780:
778:
773:
771:
766:
765:
763:
762:
756:
746:
743:
738:
732:
731:
730:
729:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
602:
595:
594:
587:
583:
579:
575:
572:
571:
567:
564:
562:
559:
557:
554:
552:
549:
545:
542:
541:
540:
537:
535:
532:
530:
527:
525:
522:
521:
517:
512:
511:
503:
500:
498:
495:
491:
488:
486:
483:
482:
481:
478:
476:
473:
471:
468:
466:
461:
458:
454:
451:
450:
446:
443:
440:
436:
432:
431:
427:
424:
422:
419:
417:
414:
412:
407:
405:
402:
400:
397:
395:
392:
391:
384:
383:
372:
369:
367:
364:
362:
359:
357:
354:
352:
349:
347:
344:
343:
341:
340:
335:
332:
331:
326:
325:
319:
318:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
282:
280:
277:
275:
271:
269:
266:
264:
260:
258:
255:
252:
248:
244:
242:
239:
237:
234:
232:
229:
227:
224:
220:
217:
215:
212:
211:
210:
207:
205:
202:
200:
197:
195:
192:
190:
187:
186:
179:
178:
170:
167:
165:
162:
160:
157:
155:
152:
150:
147:
145:
142:
140:
137:
135:
132:
131:
124:
123:
117:
114:
112:
109:
107:
104:
103:
101:
100:
96:
77:
74:
64:
58:
44:
43:
40:
37:
36:
32:
31:
19:
3973:
3958:, Springer,
3953:
3929:
3883:
3865:
3861:
3856:
3851:(2002) 7402.
3848:
3843:
3827:
3818:
3810:
3805:
3784:
3774:
3762:
3742:
3575:
3503:
3411:
3398:
3386:
3379:
3373:gauge theory
3363:, and where
3350:
3346:relativistic
3341:
3340:
3332:
3248:
3169:
3062:
3054:
3039:
3036:
2949:
2943:
2940:
2758:
2529:
2094:
2085:
2082:
2064:
2055:
2052:
2036:
1918:
1778:
1775:
1767:
1739:
1723:
1660:
1563:
1248:
1244:
1237:
1230:
1226:
1219:
1216:
1205:
1201:
1191:
1184:
1181:
1071:
1067:
1063:
1059:
1053:
1049:
1045:
1042:
1032:
1022:
1011:
860:
856:
852:
849:scalar field
846:
825:
799:
793:
584: /
580: /
578:displacement
576: /
437: /
399:Displacement
337:
328:
322:Formulations
309:Virtual work
249: /
189:Acceleration
182:Fundamentals
3948:Greiner, W.
3769:, Chapter 2
3464:Four-vector
3395:Lagrangians
3361:phase space
2073:phase space
2053:The fields
2049:Phase space
2039:dot product
2037:with · the
828:Hamiltonian
720:von Neumann
387:Core topics
3995:Categories
3939:0201029189
3876:References
3813:(2002) 93.
3409:manifold.
3172:chain rule
2536:commutator
822:Definition
810:alongside
655:d'Alembert
635:Maupertuis
598:Scientists
480:Rigid body
154:Kinematics
3918:125902566
3754:Citations
3688:∂
3683:ϕ
3680:∂
3665:∂
3660:ϕ
3657:∂
3642:∂
3637:ϕ
3634:∂
3619:∂
3614:ϕ
3611:∂
3602:ϕ
3554:μ
3543:ϕ
3538:μ
3534:∂
3527:ϕ
3421:real line
3312:∂
3307:ϕ
3304:∂
3292:ϕ
3289:∇
3283:∂
3271:∂
3223:⋅
3220:∇
3208:∂
3196:∂
3016:∂
3008:∂
2907:∇
2900:∂
2888:∂
2876:∇
2866:∑
2862:−
2853:∂
2841:∂
2826:δ
2814:δ
2782:δ
2767:δ
2726:π
2722:δ
2710:δ
2695:ϕ
2691:δ
2679:δ
2673:−
2661:π
2657:δ
2645:δ
2630:ϕ
2626:δ
2614:δ
2597:∑
2580:∫
2572:π
2566:ϕ
2492:…
2480:π
2476:∇
2464:π
2460:∇
2454:…
2442:ϕ
2438:∇
2426:ϕ
2422:∇
2416:…
2404:π
2391:π
2384:…
2372:ϕ
2359:ϕ
2330:∫
2280:…
2268:π
2264:∇
2252:π
2248:∇
2242:…
2230:ϕ
2226:∇
2214:ϕ
2210:∇
2204:…
2192:π
2179:π
2172:…
2160:ϕ
2147:ϕ
2118:∫
2006:ϕ
2002:∇
1996:∂
1992:∂
1987:⋅
1984:∇
1981:−
1969:ϕ
1965:∂
1961:∂
1944:ϕ
1940:δ
1936:δ
1891:ϕ
1887:δ
1875:δ
1869:−
1854:˙
1851:π
1830:π
1826:δ
1814:δ
1793:˙
1790:ϕ
1726:dimension
1682:∫
1630:˙
1627:ϕ
1620:∂
1608:∂
1576:π
1528:…
1519:∂
1505:ϕ
1501:∂
1492:∂
1478:ϕ
1474:∂
1468:…
1456:ϕ
1452:∇
1440:ϕ
1436:∇
1433:…
1421:ϕ
1408:ϕ
1394:−
1385:π
1378:˙
1369:ϕ
1356:∑
1332:…
1320:π
1307:π
1300:…
1288:ϕ
1275:ϕ
1157:∂
1152:ϕ
1149:∂
1143:≡
1137:˙
1134:ϕ
1117:˙
1114:ϕ
1108:∂
1096:∂
1087:π
973:∂
965:ϕ
962:∂
956:ϕ
953:∇
947:ϕ
934:−
931:π
925:˙
922:ϕ
896:π
890:ϕ
700:Liouville
582:frequency
502:Vibration
219:potential
144:Continuum
139:Celestial
116:Textbooks
3448:See also
1758:fermions
1730:SI units
1728:is , in
1195:, not a
755:Category
680:Hamilton
665:Lagrange
660:Clairaut
625:Horrocks
586:velocity
556:Pendulum
544:reactive
516:Rotation
485:dynamics
435:Inertial
421:Friction
304:Velocity
279:Momentum
159:Kinetics
149:Dynamics
127:Branches
111:Timeline
3888:Bibcode
3344:is the
2797:is the
1185:partial
1026:is the
715:Koopman
675:Poisson
670:Laplace
615:Huygens
610:Galileo
455: (
394:Damping
247:Inertia
241:Impulse
214:kinetic
164:Statics
134:Applied
106:History
3980:
3962:
3936:
3916:
3906:
3796:
3779:1991).
2759:where
1754:bosons
1692:
1058:has a
753:
705:Appell
690:Cauchy
685:Jacobi
630:Halley
620:Newton
605:Kepler
457:linear
453:Motion
299:Torque
274:Moment
209:Energy
199:Couple
3914:S2CID
3832:arXiv
3495:Notes
1197:total
1012:with
835:more
710:Gibbs
695:Routh
650:Euler
289:Speed
284:Space
226:Force
3978:ISBN
3960:ISBN
3934:ISBN
3904:ISBN
3794:ISBN
3573:The
3174:for
2092:and
1768:The
1748:and
1038:time
1016:the
826:The
294:Time
257:Mass
3896:doi
3828:A32
3405:or
3111:),
2953:):
1190:∂/∂
1036:is
869:by
794:In
3997::
3912:.
3902:.
3894:.
3866:46
3849:41
3811:43
3444:.
3182::
3043:,
2045:.
1760:.
1247:,
1229:,
1209:.
1206:dt
1070:,
1052:,
1020:,
859:,
818:.
798:,
3986:.
3942:.
3920:.
3898::
3890::
3862:k
3838:.
3834::
3800:.
3722:)
3718:t
3715:,
3712:z
3709:,
3706:y
3703:,
3700:x
3697:,
3691:t
3674:,
3668:z
3651:,
3645:y
3628:,
3622:x
3605:,
3598:(
3592:L
3576:μ
3559:)
3550:x
3546:,
3530:,
3524:(
3519:L
3431:R
3399:k
3387:x
3382:i
3380:p
3315:t
3295:)
3286:(
3276:L
3265:=
3261:S
3234:0
3231:=
3227:S
3217:+
3211:t
3201:H
3150:.
3144:V
3139:+
3134:T
3129:=
3124:H
3097:V
3073:T
3040:A
3019:t
3011:A
3002:+
2999:]
2996:H
2993:,
2990:A
2987:[
2984:=
2978:t
2975:d
2970:A
2967:d
2950:B
2944:A
2926:.
2919:)
2916:f
2911:i
2903:(
2893:F
2880:i
2870:i
2856:f
2846:F
2835:=
2829:f
2819:F
2785:f
2778:/
2772:F
2744:,
2739:)
2730:i
2715:A
2699:i
2684:B
2665:i
2650:B
2634:i
2619:A
2607:(
2601:i
2593:x
2588:3
2584:d
2577:=
2569:,
2562:]
2558:B
2555:,
2552:A
2549:[
2515:,
2510:)
2506:t
2503:,
2499:x
2495:,
2489:,
2484:2
2473:,
2468:1
2457:,
2451:,
2446:2
2435:,
2430:1
2419:,
2413:,
2408:2
2400:,
2395:1
2387:,
2381:,
2376:2
2368:,
2363:1
2354:(
2348:B
2343:x
2338:3
2334:d
2327:=
2324:B
2303:,
2298:)
2294:t
2291:,
2287:x
2283:,
2277:,
2272:2
2261:,
2256:1
2245:,
2239:,
2234:2
2223:,
2218:1
2207:,
2201:,
2196:2
2188:,
2183:1
2175:,
2169:,
2164:2
2156:,
2151:1
2142:(
2136:A
2131:x
2126:3
2122:d
2115:=
2112:A
2097:i
2095:π
2088:i
2086:φ
2067:i
2065:π
2058:i
2056:φ
2022:,
2015:)
2010:i
1999:(
1973:i
1956:=
1948:i
1904:,
1895:i
1880:H
1866:=
1861:i
1843:,
1834:i
1819:H
1808:+
1805:=
1800:i
1709:.
1705:x
1700:3
1696:d
1687:H
1679:=
1676:H
1646:.
1637:i
1613:L
1602:=
1599:)
1596:t
1593:,
1589:x
1585:(
1580:i
1549:.
1545:)
1542:t
1539:,
1535:x
1531:,
1525:,
1522:t
1515:/
1509:2
1498:,
1495:t
1488:/
1482:1
1471:,
1465:,
1460:2
1449:,
1444:1
1430:,
1425:2
1417:,
1412:1
1404:(
1399:L
1389:i
1373:i
1360:i
1352:=
1349:)
1346:t
1343:,
1339:x
1335:,
1329:,
1324:2
1316:,
1311:1
1303:,
1297:,
1292:2
1284:,
1279:1
1271:(
1266:H
1251:)
1249:t
1245:x
1243:(
1240:i
1238:π
1233:)
1231:t
1227:x
1225:(
1222:i
1220:φ
1204:/
1202:d
1192:t
1167:,
1160:t
1127:,
1101:L
1090:=
1074:)
1072:t
1068:x
1066:(
1064:π
1056:)
1054:t
1050:x
1048:(
1046:φ
1033:t
1023:x
1014:∇
997:.
993:)
990:t
987:,
983:x
979:,
976:t
969:/
959:,
950:,
944:(
939:L
916:=
913:)
910:t
907:,
903:x
899:,
893:,
887:(
882:H
863:)
861:t
857:x
855:(
853:φ
783:e
776:t
769:v
459:)
78:t
75:d
69:p
65:d
59:=
54:F
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.