762:
110:
4061:
3753:
3518:
705:
2104:
45:
36:
556:", treats one object as the immobile source of a force acting on the other. One then seeks to predict the motion of the single remaining mobile object. Such an approximation can give useful results when one object is much more massive than the other (as with a light planet orbiting a heavy star, where the star can be treated as essentially stationary).
3264:
1900:
4056:{\displaystyle {\begin{aligned}E_{1}&={\frac {\mu }{m_{1}}}E={\frac {1}{2}}m_{1}{\dot {\mathbf {x} }}_{1}^{2}+{\frac {\mu }{m_{1}}}U(\mathbf {r} )\\E_{2}&={\frac {\mu }{m_{2}}}E={\frac {1}{2}}m_{2}{\dot {\mathbf {x} }}_{2}^{2}+{\frac {\mu }{m_{2}}}U(\mathbf {r} )\\E_{\text{tot}}&=E_{1}+E_{2}\end{aligned}}}
3037:
632:
being the obvious physical example. In practice, such problems rarely arise. Except perhaps in experimental apparatus or other specialized equipment, we rarely encounter electrostatically interacting objects which are moving fast enough, and in such a direction, as to avoid colliding, and/or which
594:
The two-body problem is interesting in astronomy because pairs of astronomical objects are often moving rapidly in arbitrary directions (so their motions become interesting), widely separated from one another (so they will not collide) and even more widely separated from other objects (so outside
1608:
1760:
2265:
852:
2867:
3513:{\displaystyle E_{\text{tot}}={\frac {1}{2}}m_{1}{\dot {\mathbf {x} }}_{1}^{2}+{\frac {1}{2}}m_{2}{\dot {\mathbf {x} }}_{2}^{2}+U(\mathbf {r} )={\frac {1}{2}}(m_{1}+m_{2}){\dot {\mathbf {R} }}^{2}+{1 \over 2}\mu {\dot {\mathbf {r} }}^{2}+U(\mathbf {r} )}
2416:
2924:
2099:{\displaystyle {\ddot {\mathbf {r} }}={\ddot {\mathbf {x} }}_{1}-{\ddot {\mathbf {x} }}_{2}=\left({\frac {\mathbf {F} _{12}}{m_{1}}}-{\frac {\mathbf {F} _{21}}{m_{2}}}\right)=\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)\mathbf {F} _{12}}
1299:
1185:
1449:
3144:
2749:
2639:
602:, each member of a pair of such objects will orbit their mutual center of mass in an elliptical pattern, unless they are moving fast enough to escape one another entirely, in which case their paths will diverge along other planar
3598:
904:
1632:
4253:
678:
are necessary for any useful understanding of the electron's real behavior. Solving the classical two-body problem for an electron orbiting an atomic nucleus is misleading and does not produce many useful insights.
4151:
3726:
3670:
3758:
2169:
606:. If one object is very much heavier than the other, it will move far less than the other with reference to the shared center of mass. The mutual center of mass may even be inside the larger object.
2795:
2295:
1844:
1208:
1094:
2495:
2462:
559:
However, the one-body approximation is usually unnecessary except as a stepping stone. For many forces, including gravitational ones, the general version of the two-body problem can be
1799:
771:
3074:
2643:
2533:
754:
The complete two-body problem can be solved by re-formulating it as two one-body problems: a trivial one and one that involves solving for the motion of one particle in an external
953:
1436:
648:
Although the two-body model treats the objects as point particles, classical mechanics only apply to systems of macroscopic scale. Most behavior of subatomic particles
3529:
2285:
4187:
3032:{\displaystyle \mathbf {N} ={\frac {d\mathbf {L} }{dt}}={\dot {\mathbf {r} }}\times \mu {\dot {\mathbf {r} }}+\mathbf {r} \times \mu {\ddot {\mathbf {r} }}\ ,}
720:
688:
4094:
3674:
1603:{\displaystyle m_{1}{\ddot {\mathbf {x} }}_{1}+m_{2}{\ddot {\mathbf {x} }}_{2}=(m_{1}+m_{2}){\ddot {\mathbf {R} }}=\mathbf {F} _{12}+\mathbf {F} _{21}=0}
3621:
1380:
between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories
620:
In principle, the same solutions apply to macroscopic problems involving objects interacting not only through gravity, but through any other attractive
857:
526:
is to calculate and predict the motion of two massive bodies that are orbiting each other in space. The problem assumes that the two bodies are
1897:
Dividing both force equations by the respective masses, subtracting the second equation from the first, and rearranging gives the equation
549:. A two-point-particle model of such a system nearly always describes its behavior well enough to provide useful insights and predictions.
530:
that interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored.
1755:{\displaystyle {\ddot {\mathbf {R} }}\equiv {\frac {m_{1}{\ddot {\mathbf {x} }}_{1}+m_{2}{\ddot {\mathbf {x} }}_{2}}{m_{1}+m_{2}}}.}
77:
Two bodies with a "slight" difference in mass orbiting a common barycenter. Their sizes and this type of orbit are similar to the
505:
1083:
20:
1767:
277:
2260:{\displaystyle \mu {\ddot {\mathbf {r} }}=\mathbf {F} _{12}(\mathbf {x} _{1},\mathbf {x} _{2})=\mathbf {F} (\mathbf {r} )}
1338:
Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently.
346:
2431:
is the key to the two-body problem. The solution depends on the specific force between the bodies, which is defined by
2166:, and the laws of physics would have to change from place to place. The subtracted equation can therefore be written:
2138:
The force between the two objects, which originates in the two objects, should only be a function of their separation
4454:
4426:
4383:
3189:
1804:
4441:
4071:
847:{\displaystyle {\boldsymbol {R}}={\frac {m_{1}}{M}}{\boldsymbol {x}}_{1}+{\frac {m_{2}}{M}}{\boldsymbol {x}}_{2}}
610:
553:
2862:{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {r} \times \mu {\frac {d\mathbf {r} }{dt}},}
2467:
2434:
167:
4282:
758:. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved.
2411:{\displaystyle \mu ={\frac {1}{{\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}}}={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}.}
498:
431:
4494:
3171:
1611:
637:
1335:
position vectors denote their second derivative with respect to time, or their acceleration vectors.
909:
24:
19:
This article is about the two-body problem in classical mechanics. For the relativistic version, see
3174:) that the force between two particles acts along the line between their positions, it follows that
1294:{\displaystyle \mathbf {F} _{21}(\mathbf {x} _{1},\mathbf {x} _{2})=m_{2}{\ddot {\mathbf {x} }}_{2}}
1180:{\displaystyle \mathbf {F} _{12}(\mathbf {x} _{1},\mathbf {x} _{2})=m_{1}{\ddot {\mathbf {x} }}_{1}}
652:
be predicted under the classical assumptions underlying this article or using the mathematics here.
1875:
563:, allowing it to be solved completely, and giving a solution simple enough to be used effectively.
426:
341:
3523:
1419:
730:
533:
The most prominent example of the classical two-body problem is the gravitational case (see also
297:
1889:
of the center of mass can be determined at all times from the initial positions and velocities.
4470:
4277:
3139:{\displaystyle \mathbf {N} \ =\ {\frac {d\mathbf {L} }{dt}}=\mathbf {r} \times \mathbf {F} \ ,}
2744:{\displaystyle \mathbf {x} _{2}(t)=\mathbf {R} (t)-{\frac {m_{1}}{m_{1}+m_{2}}}\mathbf {r} (t)}
2634:{\displaystyle \mathbf {x} _{1}(t)=\mathbf {R} (t)+{\frac {m_{2}}{m_{1}+m_{2}}}\mathbf {r} (t)}
2163:
491:
214:
2132:
2107:
726:
537:), arising in astronomy for predicting the orbits (or escapes from orbit) of objects such as
399:
234:
142:
4342:
3040:
2768:
272:
229:
219:
147:
2270:
636:
The dynamical system of a two-body system under the influence of torque turns out to be a
8:
519:
314:
152:
4346:
4372:
4305:
3240:
2901:(with these written taking the center of mass as the origin, and thus both parallel to
2498:
1331:
is the force on mass 2 due to its interactions with mass 1. The two dots on top of the
765:
625:
581: ≥ 3) cannot be solved in terms of first integrals, except in special cases.
567:
387:
262:
4450:
4436:
4422:
4379:
3258:
675:
302:
239:
118:
1360:
equation (2) from equation (1) results in an equation that describes how the vector
4350:
3244:
2784:
2767:
The motion of two bodies with respect to each other always lies in a plane (in the
629:
621:
472:
421:
130:
4489:
4406:
2775:
734:
714:
689:
Classical central-force problem § Relation to the classical two-body problem
477:
382:
292:
267:
674:"). However, electrons don't actually orbit nuclei in any meaningful sense, and
4446:
4418:
4355:
4330:
4310:
4297:
4292:
2502:
1439:
1349:
694:
671:
659:
614:
571:
534:
527:
449:
365:
359:
282:
207:
201:
196:
65:
4411:
1846:
of the center of mass is constant, from which follows that the total momentum
761:
4483:
4088:
3593:{\displaystyle E={\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}+U(\mathbf {r} )}
3211:
899:{\displaystyle {\boldsymbol {r}}={\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{2}}
603:
454:
287:
244:
733:
to the section by replacing the section with a link and a summary or by
4474:
4287:
4272:
2876:
2289:
172:
162:
157:
69:
721:
Classical central-force problem#Relation to the classical two-body problem
4248:{\displaystyle \mu {\ddot {\mathbf {r} }}={F}(r){\hat {\mathbf {r} }}\ ,}
4181:
336:
78:
4402:
3170:
Introducing the assumption (true of most physical forces, as they obey
1443:
1353:
667:
663:
416:
372:
331:
109:
61:
755:
538:
81:(in which the barycenter is external to both bodies), as well as the
4146:{\displaystyle \mathbf {F} (\mathbf {r} )=F(r){\hat {\mathbf {r} }}}
2792:
of the system, with respect to the center of mass, by the equations
1892:
1446:) of the system. Addition of the force equations (1) and (2) yields
560:
3721:{\displaystyle \mathbf {x} _{2}=-{\frac {\mu }{m_{2}}}\mathbf {r} }
655:
643:
3665:{\displaystyle \mathbf {x} _{1}={\frac {\mu }{m_{1}}}\mathbf {r} }
682:
599:
2530:
have been determined, the original trajectories may be obtained
1324:
is the force on mass 1 due to its interactions with mass 2, and
589:
89:
system (in which the barycenter is internal to the larger body).
2914:
1411:
542:
546:
137:
82:
23:. For the career management problem of working couples, see
991:
be their masses. The goal is to determine the trajectories
86:
57:
3750:
that separately contain the kinetic energy of each body:
44:
35:
609:
For the derivation of the solutions to the problem, see
595:
influences will be small enough to be ignored safely).
2751:
as may be verified by substituting the definitions of
4190:
4097:
3756:
3677:
3624:
3532:
3267:
3077:
2927:
2798:
2646:
2536:
2470:
2437:
2298:
2273:
2172:
1903:
1807:
1770:
1635:
1452:
1422:
1211:
1097:
912:
860:
774:
658:
in an atom are sometimes described as "orbiting" its
4401:
4410:
4371:
4263:) is negative in the case of an attractive force.
4247:
4145:
4055:
3720:
3664:
3592:
3512:
3138:
3031:
2861:
2759:into the right-hand sides of these two equations.
2743:
2633:
2489:
2456:
2410:
2279:
2259:
2098:
1838:
1793:
1754:
1602:
1430:
1293:
1179:
947:
898:
846:
1893:Displacement vector motion (2nd one-body problem)
4481:
4331:"The Sturm-Liouville problem of two-body system"
3223:
3198:(conserved). Therefore, the displacement vector
644:Inapplicability to atoms and subatomic particles
977:be the vector positions of the two bodies, and
683:Reduction to two independent, one-body problems
4369:
584:
2907:) the rate of change of the angular momentum
1839:{\displaystyle \mathbf {v} ={\frac {dR}{dt}}}
768:for two-body problem; Jacobi coordinates are
633:are isolated enough from their surroundings.
590:Gravitation and other inverse-square examples
499:
2762:
1412:Center of mass motion (1st one-body problem)
4435:
3526:is the lowest and the total energy becomes
2490:{\displaystyle \mathbf {F} (\mathbf {r} )}
2457:{\displaystyle \mathbf {F} (\mathbf {r} )}
506:
492:
4354:
2135:from mass 2 to mass 1, as defined above.
1794:{\displaystyle {\ddot {\mathbf {R} }}=0}
1348:) results in an equation describing the
760:
4363:
886:
871:
862:
834:
802:
776:
4482:
4076:For many physical problems, the force
561:reduced to a pair of one-body problems
21:Two-body problem in general relativity
4378:. Springer. p. 58; Figure 2.15.
16:Motion problem in classical mechanics
2144:and not of their absolute positions
1202:
1088:
698:
4328:
3172:Newton's strong third law of motion
13:
14:
4506:
4475:Eric Weisstein's World of Physics
4464:
4335:Journal of Physics Communications
4065:
670:(this is the source of the term "
552:A simpler "one body" model, the "
278:Kepler's laws of planetary motion
4229:
4198:
4133:
4107:
4099:
3998:
3950:
3874:
3826:
3728:and in a similar way the energy
3714:
3680:
3658:
3627:
3583:
3557:
3522:In the center of mass frame the
3503:
3477:
3440:
3386:
3355:
3306:
3126:
3118:
3099:
3079:
3069:pointing in the same direction,
3013:
2999:
2985:
2965:
2943:
2929:
2841:
2824:
2816:
2808:
2800:
2728:
2672:
2649:
2618:
2562:
2539:
2480:
2472:
2447:
2439:
2250:
2242:
2225:
2210:
2195:
2180:
2162:; otherwise, there would not be
2086:
2007:
1978:
1950:
1926:
1908:
1809:
1775:
1705:
1671:
1640:
1584:
1569:
1554:
1502:
1468:
1424:
1275:
1244:
1229:
1214:
1161:
1130:
1115:
1100:
1082:When applied to the two masses,
703:
108:
43:
34:
4395:
4072:Classical central-force problem
948:{\displaystyle M=m_{1}+m_{2}\ }
611:Classical central-force problem
4322:
4233:
4222:
4216:
4137:
4126:
4120:
4111:
4103:
4002:
3994:
3878:
3870:
3587:
3579:
3507:
3499:
3432:
3406:
3390:
3382:
3039:and using the property of the
2738:
2732:
2682:
2676:
2665:
2659:
2628:
2622:
2572:
2566:
2555:
2549:
2484:
2476:
2451:
2443:
2254:
2246:
2235:
2205:
1547:
1521:
1254:
1224:
1140:
1110:
1023:, given the initial positions
64:external to both bodies, with
1:
4316:
3224:Energy of the two-body system
1431:{\displaystyle \mathbf {R} }
7:
4445:(2nd. ed.). New York:
4417:(3rd. ed.). New York:
4329:Luo, Siwei (22 June 2020).
4266:
3732:is related to the energies
1344:
1307:
1193:
1051:and the initial velocities
717:the scope of other articles
585:Results for prominent cases
432:Tsiolkovsky rocket equation
68:. This model is typical of
10:
4511:
4283:Euler's three-body problem
4091:, i.e., it is of the form
4069:
692:
686:
570:(and, more generally, the
401:Engineering and efficiency
220:Bi-elliptic transfer orbit
18:
2885:is the relative position
2763:Two-body motion is planar
2420:Solving the equation for
2106:where we have again used
25:Two-body problem (career)
4356:10.1088/2399-6528/ab9c30
3210:are always in the plane
3190:angular momentum vector
1876:conservation of momentum
1801:shows that the velocity
1764:The resulting equation:
638:Sturm-Liouville equation
630:electrostatic attraction
427:Propellant mass fraction
326:Gravitational influences
4370:David Betounes (2001).
3214:to the constant vector
1878:). Hence, the position
1438:be the position of the
1356:) motion. By contrast,
298:Specific orbital energy
4374:Differential Equations
4278:Equation of the center
4249:
4147:
4057:
3722:
3666:
3594:
3514:
3243:then the system has a
3140:
3033:
2863:
2745:
2635:
2491:
2458:
2412:
2281:
2261:
2164:translational symmetry
2100:
1840:
1795:
1756:
1604:
1432:
1295:
1181:
956:
949:
900:
848:
215:Hohmann transfer orbit
56:Two bodies of similar
4250:
4180:is the corresponding
4148:
4058:
3723:
3667:
3595:
3515:
3141:
3034:
2864:
2746:
2636:
2492:
2464:. For the case where
2459:
2413:
2282:
2262:
2101:
1841:
1796:
1757:
1605:
1433:
1296:
1182:
950:
901:
849:
764:
735:splitting the content
729:and help introduce a
554:central-force problem
411:Preflight engineering
143:Argument of periapsis
4188:
4095:
3754:
3675:
3622:
3618:can be expressed as
3530:
3265:
3075:
3041:vector cross product
2925:
2796:
2774:Proof: Defining the
2769:center of mass frame
2644:
2534:
2468:
2435:
2296:
2280:{\displaystyle \mu }
2271:
2170:
1901:
1805:
1768:
1633:
1450:
1420:
1209:
1095:
910:
858:
772:
467:Propulsive maneuvers
4442:Classical Mechanics
4347:2020JPhCo...4f1001L
3970:
3846:
3375:
3326:
2133:displacement vector
1610:where we have used
1342:equations (1) and (
1084:Newton's second law
737:into a new article.
598:Under the force of
520:classical mechanics
444:Efficiency measures
347:Sphere of influence
316:Celestial mechanics
98:Part of a series on
79:Pluto–Charon system
4306:Three-body problem
4245:
4143:
4053:
4051:
3945:
3821:
3718:
3662:
3590:
3510:
3350:
3301:
3261:can be written as
3136:
3029:
2859:
2741:
2631:
2499:inverse-square law
2487:
2454:
2408:
2277:
2257:
2108:Newton's third law
2096:
1874:is also constant (
1836:
1791:
1752:
1612:Newton's third law
1600:
1428:
1291:
1177:
957:
945:
896:
844:
766:Jacobi coordinates
727:discuss this issue
626:inverse-square law
622:scalar force field
568:three-body problem
263:Dynamical friction
60:orbiting a common
4495:Dynamical systems
4241:
4236:
4205:
4140:
4016:
3989:
3957:
3933:
3917:
3865:
3833:
3809:
3793:
3711:
3655:
3564:
3547:
3484:
3467:
3447:
3404:
3362:
3338:
3313:
3289:
3275:
3204:and its velocity
3132:
3112:
3091:
3085:
3025:
3020:
2992:
2972:
2956:
2854:
2725:
2615:
2403:
2349:
2346:
2326:
2187:
2077:
2057:
2027:
1998:
1957:
1933:
1915:
1834:
1782:
1747:
1712:
1678:
1647:
1561:
1509:
1475:
1315:
1314:
1282:
1201:
1200:
1168:
944:
830:
798:
752:
751:
676:quantum mechanics
566:By contrast, the
516:
515:
366:Lagrangian points
303:Vis-viva equation
273:Kepler's equation
120:Orbital mechanics
4502:
4471:Two-body problem
4460:
4432:
4416:
4390:
4389:
4377:
4367:
4361:
4360:
4358:
4326:
4254:
4252:
4251:
4246:
4239:
4238:
4237:
4232:
4227:
4215:
4207:
4206:
4201:
4196:
4179:
4165:
4163:
4152:
4150:
4149:
4144:
4142:
4141:
4136:
4131:
4110:
4102:
4086:
4062:
4060:
4059:
4054:
4052:
4048:
4047:
4035:
4034:
4018:
4017:
4014:
4001:
3990:
3988:
3987:
3975:
3969:
3964:
3959:
3958:
3953:
3948:
3944:
3943:
3934:
3926:
3918:
3916:
3915:
3903:
3894:
3893:
3877:
3866:
3864:
3863:
3851:
3845:
3840:
3835:
3834:
3829:
3824:
3820:
3819:
3810:
3802:
3794:
3792:
3791:
3779:
3770:
3769:
3749:
3740:
3727:
3725:
3724:
3719:
3717:
3712:
3710:
3709:
3697:
3689:
3688:
3683:
3671:
3669:
3668:
3663:
3661:
3656:
3654:
3653:
3641:
3636:
3635:
3630:
3617:
3608:
3600:The coordinates
3599:
3597:
3596:
3591:
3586:
3572:
3571:
3566:
3565:
3560:
3555:
3548:
3540:
3519:
3517:
3516:
3511:
3506:
3492:
3491:
3486:
3485:
3480:
3475:
3468:
3460:
3455:
3454:
3449:
3448:
3443:
3438:
3431:
3430:
3418:
3417:
3405:
3397:
3389:
3374:
3369:
3364:
3363:
3358:
3353:
3349:
3348:
3339:
3331:
3325:
3320:
3315:
3314:
3309:
3304:
3300:
3299:
3290:
3282:
3277:
3276:
3273:
3256:
3245:potential energy
3238:
3219:
3209:
3203:
3195:
3187:
3166:
3145:
3143:
3142:
3137:
3130:
3129:
3121:
3113:
3111:
3103:
3102:
3093:
3089:
3083:
3082:
3068:
3062:
3057:for any vectors
3056:
3038:
3036:
3035:
3030:
3023:
3022:
3021:
3016:
3011:
3002:
2994:
2993:
2988:
2983:
2974:
2973:
2968:
2963:
2957:
2955:
2947:
2946:
2937:
2932:
2921:
2912:
2906:
2900:
2884:
2874:
2868:
2866:
2865:
2860:
2855:
2853:
2845:
2844:
2835:
2827:
2819:
2811:
2803:
2791:
2785:angular momentum
2782:
2750:
2748:
2747:
2742:
2731:
2726:
2724:
2723:
2722:
2710:
2709:
2699:
2698:
2689:
2675:
2658:
2657:
2652:
2640:
2638:
2637:
2632:
2621:
2616:
2614:
2613:
2612:
2600:
2599:
2589:
2588:
2579:
2565:
2548:
2547:
2542:
2529:
2518:
2496:
2494:
2493:
2488:
2483:
2475:
2463:
2461:
2460:
2455:
2450:
2442:
2430:
2417:
2415:
2414:
2409:
2404:
2402:
2401:
2400:
2388:
2387:
2377:
2376:
2375:
2366:
2365:
2355:
2350:
2348:
2347:
2345:
2344:
2332:
2327:
2325:
2324:
2312:
2306:
2286:
2284:
2283:
2278:
2266:
2264:
2263:
2258:
2253:
2245:
2234:
2233:
2228:
2219:
2218:
2213:
2204:
2203:
2198:
2189:
2188:
2183:
2178:
2161:
2152:
2143:
2130:
2124:
2105:
2103:
2102:
2097:
2095:
2094:
2089:
2083:
2079:
2078:
2076:
2075:
2063:
2058:
2056:
2055:
2043:
2033:
2029:
2028:
2026:
2025:
2016:
2015:
2010:
2004:
1999:
1997:
1996:
1987:
1986:
1981:
1975:
1965:
1964:
1959:
1958:
1953:
1948:
1941:
1940:
1935:
1934:
1929:
1924:
1917:
1916:
1911:
1906:
1888:
1873:
1845:
1843:
1842:
1837:
1835:
1833:
1825:
1817:
1812:
1800:
1798:
1797:
1792:
1784:
1783:
1778:
1773:
1761:
1759:
1758:
1753:
1748:
1746:
1745:
1744:
1732:
1731:
1721:
1720:
1719:
1714:
1713:
1708:
1703:
1699:
1698:
1686:
1685:
1680:
1679:
1674:
1669:
1665:
1664:
1654:
1649:
1648:
1643:
1638:
1628:
1609:
1607:
1606:
1601:
1593:
1592:
1587:
1578:
1577:
1572:
1563:
1562:
1557:
1552:
1546:
1545:
1533:
1532:
1517:
1516:
1511:
1510:
1505:
1500:
1496:
1495:
1483:
1482:
1477:
1476:
1471:
1466:
1462:
1461:
1437:
1435:
1434:
1429:
1427:
1407:
1393:
1379:
1309:
1300:
1298:
1297:
1292:
1290:
1289:
1284:
1283:
1278:
1273:
1269:
1268:
1253:
1252:
1247:
1238:
1237:
1232:
1223:
1222:
1217:
1203:
1195:
1186:
1184:
1183:
1178:
1176:
1175:
1170:
1169:
1164:
1159:
1155:
1154:
1139:
1138:
1133:
1124:
1123:
1118:
1109:
1108:
1103:
1089:
1078:
1064:
1050:
1036:
1018:
1004:
976:
967:
954:
952:
951:
946:
942:
941:
940:
928:
927:
905:
903:
902:
897:
895:
894:
889:
880:
879:
874:
865:
853:
851:
850:
845:
843:
842:
837:
831:
826:
825:
816:
811:
810:
805:
799:
794:
793:
784:
779:
747:
744:
738:
707:
706:
699:
664:early conjecture
524:two-body problem
508:
501:
494:
473:Orbital maneuver
422:Payload fraction
402:
383:Lissajous orbits
317:
288:Orbital velocity
235:Hyperbolic orbit
131:Orbital elements
121:
112:
95:
94:
47:
38:
4510:
4509:
4505:
4504:
4503:
4501:
4500:
4499:
4480:
4479:
4467:
4457:
4429:
4398:
4393:
4386:
4368:
4364:
4327:
4323:
4319:
4269:
4228:
4226:
4225:
4211:
4197:
4195:
4194:
4189:
4186:
4185:
4184:. We now have:
4167:
4159:
4154:
4132:
4130:
4129:
4106:
4098:
4096:
4093:
4092:
4077:
4074:
4068:
4050:
4049:
4043:
4039:
4030:
4026:
4019:
4013:
4009:
4006:
4005:
3997:
3983:
3979:
3974:
3965:
3960:
3949:
3947:
3946:
3939:
3935:
3925:
3911:
3907:
3902:
3895:
3889:
3885:
3882:
3881:
3873:
3859:
3855:
3850:
3841:
3836:
3825:
3823:
3822:
3815:
3811:
3801:
3787:
3783:
3778:
3771:
3765:
3761:
3757:
3755:
3752:
3751:
3748:
3742:
3739:
3733:
3713:
3705:
3701:
3696:
3684:
3679:
3678:
3676:
3673:
3672:
3657:
3649:
3645:
3640:
3631:
3626:
3625:
3623:
3620:
3619:
3616:
3610:
3607:
3601:
3582:
3567:
3556:
3554:
3553:
3552:
3539:
3531:
3528:
3527:
3502:
3487:
3476:
3474:
3473:
3472:
3459:
3450:
3439:
3437:
3436:
3435:
3426:
3422:
3413:
3409:
3396:
3385:
3370:
3365:
3354:
3352:
3351:
3344:
3340:
3330:
3321:
3316:
3305:
3303:
3302:
3295:
3291:
3281:
3272:
3268:
3266:
3263:
3262:
3257:, so the total
3247:
3229:
3226:
3215:
3205:
3199:
3191:
3175:
3147:
3125:
3117:
3104:
3098:
3094:
3092:
3078:
3076:
3073:
3072:
3064:
3058:
3044:
3012:
3010:
3009:
2998:
2984:
2982:
2981:
2964:
2962:
2961:
2948:
2942:
2938:
2936:
2928:
2926:
2923:
2922:
2917:
2913:equals the net
2908:
2902:
2899:
2892:
2886:
2880:
2872:
2846:
2840:
2836:
2834:
2823:
2815:
2807:
2799:
2797:
2794:
2793:
2787:
2778:
2776:linear momentum
2765:
2727:
2718:
2714:
2705:
2701:
2700:
2694:
2690:
2688:
2671:
2653:
2648:
2647:
2645:
2642:
2641:
2617:
2608:
2604:
2595:
2591:
2590:
2584:
2580:
2578:
2561:
2543:
2538:
2537:
2535:
2532:
2531:
2520:
2509:
2479:
2471:
2469:
2466:
2465:
2446:
2438:
2436:
2433:
2432:
2421:
2396:
2392:
2383:
2379:
2378:
2371:
2367:
2361:
2357:
2356:
2354:
2340:
2336:
2331:
2320:
2316:
2311:
2310:
2305:
2297:
2294:
2293:
2272:
2269:
2268:
2249:
2241:
2229:
2224:
2223:
2214:
2209:
2208:
2199:
2194:
2193:
2179:
2177:
2176:
2171:
2168:
2167:
2160:
2154:
2151:
2145:
2139:
2126:
2123:
2116:
2110:
2090:
2085:
2084:
2071:
2067:
2062:
2051:
2047:
2042:
2041:
2037:
2021:
2017:
2011:
2006:
2005:
2003:
1992:
1988:
1982:
1977:
1976:
1974:
1973:
1969:
1960:
1949:
1947:
1946:
1945:
1936:
1925:
1923:
1922:
1921:
1907:
1905:
1904:
1902:
1899:
1898:
1895:
1879:
1872:
1866:
1859:
1853:
1847:
1826:
1818:
1816:
1808:
1806:
1803:
1802:
1774:
1772:
1771:
1769:
1766:
1765:
1740:
1736:
1727:
1723:
1722:
1715:
1704:
1702:
1701:
1700:
1694:
1690:
1681:
1670:
1668:
1667:
1666:
1660:
1656:
1655:
1653:
1639:
1637:
1636:
1634:
1631:
1630:
1627:
1620:
1614:
1588:
1583:
1582:
1573:
1568:
1567:
1553:
1551:
1550:
1541:
1537:
1528:
1524:
1512:
1501:
1499:
1498:
1497:
1491:
1487:
1478:
1467:
1465:
1464:
1463:
1457:
1453:
1451:
1448:
1447:
1423:
1421:
1418:
1417:
1414:
1401:
1395:
1387:
1381:
1378:
1371:
1361:
1330:
1323:
1285:
1274:
1272:
1271:
1270:
1264:
1260:
1248:
1243:
1242:
1233:
1228:
1227:
1218:
1213:
1212:
1210:
1207:
1206:
1171:
1160:
1158:
1157:
1156:
1150:
1146:
1134:
1129:
1128:
1119:
1114:
1113:
1104:
1099:
1098:
1096:
1093:
1092:
1072:
1066:
1058:
1052:
1044:
1038:
1030:
1024:
1012:
1006:
998:
992:
990:
983:
975:
969:
966:
960:
936:
932:
923:
919:
911:
908:
907:
890:
885:
884:
875:
870:
869:
861:
859:
856:
855:
838:
833:
832:
821:
817:
815:
806:
801:
800:
789:
785:
783:
775:
773:
770:
769:
748:
742:
739:
724:
719:, specifically
708:
704:
697:
691:
685:
662:, following an
646:
592:
587:
528:point particles
512:
483:
482:
478:Orbit insertion
468:
460:
459:
445:
437:
436:
412:
404:
400:
393:
392:
388:Lyapunov orbits
379:
378:
362:
352:
351:
327:
319:
315:
308:
307:
293:Surface gravity
268:Escape velocity
258:
250:
249:
230:Parabolic orbit
226:
225:
192:
190:
187:two-body orbits
178:
177:
168:Semi-major axis
133:
123:
119:
93:
92:
91:
90:
73:
66:elliptic orbits
50:
49:
48:
40:
39:
28:
17:
12:
11:
5:
4508:
4498:
4497:
4492:
4478:
4477:
4466:
4465:External links
4463:
4462:
4461:
4455:
4447:Addison-Wesley
4433:
4427:
4419:Pergamon Press
4397:
4394:
4392:
4391:
4384:
4362:
4320:
4318:
4315:
4314:
4313:
4311:Virial theorem
4308:
4303:
4295:
4293:Kepler problem
4290:
4285:
4280:
4275:
4268:
4265:
4244:
4235:
4231:
4224:
4221:
4218:
4214:
4210:
4204:
4200:
4193:
4139:
4135:
4128:
4125:
4122:
4119:
4116:
4113:
4109:
4105:
4101:
4070:Main article:
4067:
4066:Central forces
4064:
4046:
4042:
4038:
4033:
4029:
4025:
4022:
4020:
4012:
4008:
4007:
4004:
4000:
3996:
3993:
3986:
3982:
3978:
3973:
3968:
3963:
3956:
3952:
3942:
3938:
3932:
3929:
3924:
3921:
3914:
3910:
3906:
3901:
3898:
3896:
3892:
3888:
3884:
3883:
3880:
3876:
3872:
3869:
3862:
3858:
3854:
3849:
3844:
3839:
3832:
3828:
3818:
3814:
3808:
3805:
3800:
3797:
3790:
3786:
3782:
3777:
3774:
3772:
3768:
3764:
3760:
3759:
3746:
3737:
3716:
3708:
3704:
3700:
3695:
3692:
3687:
3682:
3660:
3652:
3648:
3644:
3639:
3634:
3629:
3614:
3605:
3589:
3585:
3581:
3578:
3575:
3570:
3563:
3559:
3551:
3546:
3543:
3538:
3535:
3524:kinetic energy
3509:
3505:
3501:
3498:
3495:
3490:
3483:
3479:
3471:
3466:
3463:
3458:
3453:
3446:
3442:
3434:
3429:
3425:
3421:
3416:
3412:
3408:
3403:
3400:
3395:
3392:
3388:
3384:
3381:
3378:
3373:
3368:
3361:
3357:
3347:
3343:
3337:
3334:
3329:
3324:
3319:
3312:
3308:
3298:
3294:
3288:
3285:
3280:
3271:
3225:
3222:
3135:
3128:
3124:
3120:
3116:
3110:
3107:
3101:
3097:
3088:
3081:
3028:
3019:
3015:
3008:
3005:
3001:
2997:
2991:
2987:
2980:
2977:
2971:
2967:
2960:
2954:
2951:
2945:
2941:
2935:
2931:
2897:
2890:
2858:
2852:
2849:
2843:
2839:
2833:
2830:
2826:
2822:
2818:
2814:
2810:
2806:
2802:
2764:
2761:
2740:
2737:
2734:
2730:
2721:
2717:
2713:
2708:
2704:
2697:
2693:
2687:
2684:
2681:
2678:
2674:
2670:
2667:
2664:
2661:
2656:
2651:
2630:
2627:
2624:
2620:
2611:
2607:
2603:
2598:
2594:
2587:
2583:
2577:
2574:
2571:
2568:
2564:
2560:
2557:
2554:
2551:
2546:
2541:
2503:Kepler problem
2486:
2482:
2478:
2474:
2453:
2449:
2445:
2441:
2407:
2399:
2395:
2391:
2386:
2382:
2374:
2370:
2364:
2360:
2353:
2343:
2339:
2335:
2330:
2323:
2319:
2315:
2309:
2304:
2301:
2276:
2256:
2252:
2248:
2244:
2240:
2237:
2232:
2227:
2222:
2217:
2212:
2207:
2202:
2197:
2192:
2186:
2182:
2175:
2158:
2149:
2121:
2114:
2093:
2088:
2082:
2074:
2070:
2066:
2061:
2054:
2050:
2046:
2040:
2036:
2032:
2024:
2020:
2014:
2009:
2002:
1995:
1991:
1985:
1980:
1972:
1968:
1963:
1956:
1952:
1944:
1939:
1932:
1928:
1920:
1914:
1910:
1894:
1891:
1870:
1864:
1857:
1851:
1832:
1829:
1824:
1821:
1815:
1811:
1790:
1787:
1781:
1777:
1751:
1743:
1739:
1735:
1730:
1726:
1718:
1711:
1707:
1697:
1693:
1689:
1684:
1677:
1673:
1663:
1659:
1652:
1646:
1642:
1625:
1618:
1599:
1596:
1591:
1586:
1581:
1576:
1571:
1566:
1560:
1556:
1549:
1544:
1540:
1536:
1531:
1527:
1523:
1520:
1515:
1508:
1504:
1494:
1490:
1486:
1481:
1474:
1470:
1460:
1456:
1440:center of mass
1426:
1413:
1410:
1399:
1385:
1376:
1369:
1350:center of mass
1328:
1321:
1313:
1312:
1303:
1301:
1288:
1281:
1277:
1267:
1263:
1259:
1256:
1251:
1246:
1241:
1236:
1231:
1226:
1221:
1216:
1199:
1198:
1189:
1187:
1174:
1167:
1163:
1153:
1149:
1145:
1142:
1137:
1132:
1127:
1122:
1117:
1112:
1107:
1102:
1070:
1056:
1042:
1028:
1019:for all times
1010:
996:
988:
981:
973:
964:
939:
935:
931:
926:
922:
918:
915:
893:
888:
883:
878:
873:
868:
864:
841:
836:
829:
824:
820:
814:
809:
804:
797:
792:
788:
782:
778:
750:
749:
711:
709:
702:
695:Kepler problem
684:
681:
645:
642:
615:Kepler problem
604:conic sections
591:
588:
586:
583:
535:Kepler problem
514:
513:
511:
510:
503:
496:
488:
485:
484:
481:
480:
475:
469:
466:
465:
462:
461:
458:
457:
452:
450:Gravity assist
446:
443:
442:
439:
438:
435:
434:
429:
424:
419:
413:
410:
409:
406:
405:
398:
395:
394:
391:
390:
385:
377:
376:
368:
364:
363:
358:
357:
354:
353:
350:
349:
344:
339:
334:
328:
325:
324:
321:
320:
313:
310:
309:
306:
305:
300:
295:
290:
285:
283:Orbital period
280:
275:
270:
265:
259:
256:
255:
252:
251:
248:
247:
245:Decaying orbit
242:
237:
232:
224:
223:
217:
210:
208:Transfer orbit
206:
205:
204:
202:Elliptic orbit
199:
197:Circular orbit
193:
184:
183:
180:
179:
176:
175:
170:
165:
160:
155:
150:
145:
140:
134:
129:
128:
125:
124:
117:
114:
113:
105:
104:
100:
99:
52:
51:
42:
41:
33:
32:
31:
30:
29:
15:
9:
6:
4:
3:
2:
4507:
4496:
4493:
4491:
4488:
4487:
4485:
4476:
4472:
4469:
4468:
4458:
4456:0-201-02918-9
4452:
4448:
4444:
4443:
4438:
4434:
4430:
4428:0-08-029141-4
4424:
4420:
4415:
4414:
4408:
4404:
4400:
4399:
4387:
4385:0-387-95140-7
4381:
4376:
4375:
4366:
4357:
4352:
4348:
4344:
4341:(6): 061001.
4340:
4336:
4332:
4325:
4321:
4312:
4309:
4307:
4304:
4302:
4301:-body problem
4300:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4270:
4264:
4262:
4258:
4242:
4219:
4212:
4208:
4202:
4191:
4183:
4178:
4174:
4170:
4162:
4157:
4123:
4117:
4114:
4090:
4089:central force
4084:
4080:
4073:
4063:
4044:
4040:
4036:
4031:
4027:
4023:
4021:
4010:
3991:
3984:
3980:
3976:
3971:
3966:
3961:
3954:
3940:
3936:
3930:
3927:
3922:
3919:
3912:
3908:
3904:
3899:
3897:
3890:
3886:
3867:
3860:
3856:
3852:
3847:
3842:
3837:
3830:
3816:
3812:
3806:
3803:
3798:
3795:
3788:
3784:
3780:
3775:
3773:
3766:
3762:
3745:
3736:
3731:
3706:
3702:
3698:
3693:
3690:
3685:
3650:
3646:
3642:
3637:
3632:
3613:
3604:
3576:
3573:
3568:
3561:
3549:
3544:
3541:
3536:
3533:
3525:
3520:
3496:
3493:
3488:
3481:
3469:
3464:
3461:
3456:
3451:
3444:
3427:
3423:
3419:
3414:
3410:
3401:
3398:
3393:
3379:
3376:
3371:
3366:
3359:
3345:
3341:
3335:
3332:
3327:
3322:
3317:
3310:
3296:
3292:
3286:
3283:
3278:
3269:
3260:
3254:
3250:
3246:
3242:
3236:
3232:
3228:If the force
3221:
3218:
3213:
3212:perpendicular
3208:
3202:
3197:
3194:
3186:
3182:
3178:
3173:
3168:
3165:
3161:
3158:
3154:
3150:
3133:
3122:
3114:
3108:
3105:
3095:
3086:
3070:
3067:
3061:
3055:
3051:
3047:
3042:
3026:
3017:
3006:
3003:
2995:
2989:
2978:
2975:
2969:
2958:
2952:
2949:
2939:
2933:
2920:
2916:
2911:
2905:
2896:
2889:
2883:
2878:
2869:
2856:
2850:
2847:
2837:
2831:
2828:
2820:
2812:
2804:
2790:
2786:
2781:
2777:
2772:
2770:
2760:
2758:
2754:
2735:
2719:
2715:
2711:
2706:
2702:
2695:
2691:
2685:
2679:
2668:
2662:
2654:
2625:
2609:
2605:
2601:
2596:
2592:
2585:
2581:
2575:
2569:
2558:
2552:
2544:
2527:
2523:
2516:
2512:
2506:
2504:
2500:
2428:
2424:
2418:
2405:
2397:
2393:
2389:
2384:
2380:
2372:
2368:
2362:
2358:
2351:
2341:
2337:
2333:
2328:
2321:
2317:
2313:
2307:
2302:
2299:
2292:
2291:
2274:
2238:
2230:
2220:
2215:
2200:
2190:
2184:
2173:
2165:
2157:
2148:
2142:
2136:
2134:
2129:
2120:
2113:
2109:
2091:
2080:
2072:
2068:
2064:
2059:
2052:
2048:
2044:
2038:
2034:
2030:
2022:
2018:
2012:
2000:
1993:
1989:
1983:
1970:
1966:
1961:
1954:
1942:
1937:
1930:
1918:
1912:
1890:
1886:
1882:
1877:
1869:
1863:
1856:
1850:
1830:
1827:
1822:
1819:
1813:
1788:
1785:
1779:
1762:
1749:
1741:
1737:
1733:
1728:
1724:
1716:
1709:
1695:
1691:
1687:
1682:
1675:
1661:
1657:
1650:
1644:
1624:
1617:
1613:
1597:
1594:
1589:
1579:
1574:
1564:
1558:
1542:
1538:
1534:
1529:
1525:
1518:
1513:
1506:
1492:
1488:
1484:
1479:
1472:
1458:
1454:
1445:
1441:
1409:
1405:
1398:
1391:
1384:
1375:
1368:
1364:
1359:
1355:
1351:
1347:
1346:
1341:
1336:
1334:
1327:
1320:
1311:
1304:
1302:
1286:
1279:
1265:
1261:
1257:
1249:
1239:
1234:
1219:
1205:
1204:
1197:
1190:
1188:
1172:
1165:
1151:
1147:
1143:
1135:
1125:
1120:
1105:
1091:
1090:
1087:
1085:
1080:
1076:
1069:
1062:
1055:
1048:
1041:
1034:
1027:
1022:
1016:
1009:
1002:
995:
987:
980:
972:
963:
937:
933:
929:
924:
920:
916:
913:
891:
881:
876:
866:
839:
827:
822:
818:
812:
807:
795:
790:
786:
780:
767:
763:
759:
757:
746:
736:
732:
731:summary style
728:
722:
718:
716:
712:This section
710:
701:
700:
696:
690:
680:
677:
673:
669:
665:
661:
657:
653:
651:
641:
639:
634:
631:
627:
623:
618:
616:
612:
607:
605:
601:
596:
582:
580:
576:
575:-body problem
574:
569:
564:
562:
557:
555:
550:
548:
544:
540:
536:
531:
529:
525:
521:
509:
504:
502:
497:
495:
490:
489:
487:
486:
479:
476:
474:
471:
470:
464:
463:
456:
455:Oberth effect
453:
451:
448:
447:
441:
440:
433:
430:
428:
425:
423:
420:
418:
415:
414:
408:
407:
403:
397:
396:
389:
386:
384:
381:
380:
374:
370:
369:
367:
361:
360:N-body orbits
356:
355:
348:
345:
343:
342:Perturbations
340:
338:
335:
333:
330:
329:
323:
322:
318:
312:
311:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
274:
271:
269:
266:
264:
261:
260:
254:
253:
246:
243:
241:
238:
236:
233:
231:
228:
227:
221:
218:
216:
212:
211:
209:
203:
200:
198:
195:
194:
188:
182:
181:
174:
171:
169:
166:
164:
163:Orbital nodes
161:
159:
156:
154:
151:
149:
146:
144:
141:
139:
136:
135:
132:
127:
126:
122:
116:
115:
111:
107:
106:
103:Astrodynamics
102:
101:
97:
96:
88:
84:
80:
76:
71:
67:
63:
59:
55:
46:
37:
26:
22:
4440:
4412:
4396:Bibliography
4373:
4365:
4338:
4334:
4324:
4298:
4288:Kepler orbit
4273:Energy drift
4260:
4256:
4176:
4172:
4168:
4160:
4155:
4082:
4078:
4075:
3743:
3734:
3729:
3611:
3602:
3521:
3252:
3248:
3241:conservative
3234:
3230:
3227:
3216:
3206:
3200:
3192:
3184:
3180:
3176:
3169:
3163:
3159:
3156:
3152:
3148:
3071:
3065:
3059:
3053:
3049:
3045:
2918:
2909:
2903:
2894:
2887:
2881:
2877:reduced mass
2870:
2788:
2779:
2773:
2766:
2756:
2752:
2525:
2521:
2514:
2510:
2507:
2426:
2422:
2419:
2290:reduced mass
2288:
2155:
2146:
2140:
2137:
2127:
2118:
2111:
1896:
1884:
1880:
1867:
1861:
1854:
1848:
1763:
1622:
1615:
1415:
1403:
1396:
1389:
1382:
1373:
1366:
1362:
1357:
1343:
1339:
1337:
1332:
1325:
1318:
1316:
1305:
1191:
1086:states that
1081:
1074:
1067:
1060:
1053:
1046:
1039:
1032:
1025:
1020:
1014:
1007:
1000:
993:
985:
978:
970:
961:
958:
753:
740:
713:
654:
649:
647:
635:
619:
608:
597:
593:
578:
572:
565:
558:
551:
532:
523:
517:
240:Radial orbit
191:eccentricity
186:
173:True anomaly
158:Mean anomaly
148:Eccentricity
74:
70:binary stars
53:
4437:Goldstein H
4407:Lifshitz EM
4182:unit vector
3196:is constant
2497:follows an
1358:subtracting
624:obeying an
373:Halo orbits
337:Hill sphere
153:Inclination
4484:Categories
4317:References
2501:, see the
2125:and where
1629:and where
1444:barycenter
1354:barycenter
1306:(Equation
1192:(Equation
715:duplicates
693:See also:
687:See also:
668:Niels Bohr
539:satellites
417:Mass ratio
332:Barycenter
62:barycenter
4413:Mechanics
4403:Landau LD
4234:^
4203:¨
4192:μ
4138:^
3977:μ
3955:˙
3905:μ
3853:μ
3831:˙
3781:μ
3699:μ
3694:−
3643:μ
3562:˙
3550:μ
3482:˙
3470:μ
3445:˙
3360:˙
3311:˙
3123:×
3018:¨
3007:μ
3004:×
2990:˙
2979:μ
2976:×
2970:˙
2832:μ
2829:×
2813:×
2686:−
2300:μ
2275:μ
2185:¨
2174:μ
2001:−
1955:¨
1943:−
1931:¨
1913:¨
1780:¨
1710:¨
1676:¨
1651:≡
1645:¨
1559:¨
1507:¨
1473:¨
1280:¨
1166:¨
882:−
756:potential
743:June 2019
656:Electrons
257:Equations
185:Types of
4439:(1980).
4409:(1976).
4267:See also
4158:= |
3188:and the
2783:and the
4343:Bibcode
3155:
2875:is the
2287:is the
2131:is the
725:Please
672:orbital
660:nucleus
628:, with
600:gravity
543:planets
4490:Orbits
4453:
4425:
4382:
4255:where
4240:
4164:|
4153:where
3259:energy
3131:
3090:
3084:
3024:
2915:torque
2871:where
2267:where
1340:Adding
1317:where
943:
650:cannot
545:, and
522:, the
75:Right:
4087:is a
3146:with
3043:that
2508:Once
906:with
547:stars
138:Apsis
83:Earth
54:Left:
4451:ISBN
4423:ISBN
4380:ISBN
4166:and
3741:and
3609:and
3063:and
2879:and
2755:and
2519:and
2153:and
1416:Let
1394:and
1077:= 0)
1065:and
1063:= 0)
1049:= 0)
1037:and
1035:= 0)
1005:and
984:and
968:and
959:Let
854:and
577:for
87:Moon
58:mass
4473:at
4351:doi
4015:tot
3274:tot
3239:is
3183:=
2771:).
2117:= −
1621:= −
666:of
613:or
518:In
189:by
4486::
4449:.
4421:.
4405:;
4349:.
4337:.
4333:.
4171:=
4169:r̂
3220:.
3179:×
3167:.
3164:dt
3151:=
3052:=
3048:×
2893:−
2505:.
2201:12
2122:21
2115:12
2092:12
2013:21
1984:12
1860:+
1626:21
1619:12
1590:21
1575:12
1408:.
1372:−
1365:=
1329:21
1322:12
1220:21
1106:12
1079:.
640:.
617:.
541:,
4459:.
4431:.
4388:.
4359:.
4353::
4345::
4339:4
4299:n
4261:r
4259:(
4257:F
4243:,
4230:r
4223:)
4220:r
4217:(
4213:F
4209:=
4199:r
4177:r
4175:/
4173:r
4161:r
4156:r
4134:r
4127:)
4124:r
4121:(
4118:F
4115:=
4112:)
4108:r
4104:(
4100:F
4085:)
4083:r
4081:(
4079:F
4045:2
4041:E
4037:+
4032:1
4028:E
4024:=
4011:E
4003:)
3999:r
3995:(
3992:U
3985:2
3981:m
3972:+
3967:2
3962:2
3951:x
3941:2
3937:m
3931:2
3928:1
3923:=
3920:E
3913:2
3909:m
3900:=
3891:2
3887:E
3879:)
3875:r
3871:(
3868:U
3861:1
3857:m
3848:+
3843:2
3838:1
3827:x
3817:1
3813:m
3807:2
3804:1
3799:=
3796:E
3789:1
3785:m
3776:=
3767:1
3763:E
3747:2
3744:E
3738:1
3735:E
3730:E
3715:r
3707:2
3703:m
3691:=
3686:2
3681:x
3659:r
3651:1
3647:m
3638:=
3633:1
3628:x
3615:2
3612:x
3606:1
3603:x
3588:)
3584:r
3580:(
3577:U
3574:+
3569:2
3558:r
3545:2
3542:1
3537:=
3534:E
3508:)
3504:r
3500:(
3497:U
3494:+
3489:2
3478:r
3465:2
3462:1
3457:+
3452:2
3441:R
3433:)
3428:2
3424:m
3420:+
3415:1
3411:m
3407:(
3402:2
3399:1
3394:=
3391:)
3387:r
3383:(
3380:U
3377:+
3372:2
3367:2
3356:x
3346:2
3342:m
3336:2
3333:1
3328:+
3323:2
3318:1
3307:x
3297:1
3293:m
3287:2
3284:1
3279:=
3270:E
3255:)
3253:r
3251:(
3249:U
3237:)
3235:r
3233:(
3231:F
3217:L
3207:v
3201:r
3193:L
3185:0
3181:F
3177:r
3162:/
3160:r
3157:d
3153:μ
3149:F
3134:,
3127:F
3119:r
3115:=
3109:t
3106:d
3100:L
3096:d
3087:=
3080:N
3066:w
3060:v
3054:0
3050:w
3046:v
3027:,
3014:r
3000:r
2996:+
2986:r
2966:r
2959:=
2953:t
2950:d
2944:L
2940:d
2934:=
2930:N
2919:N
2910:L
2904:r
2898:1
2895:r
2891:2
2888:r
2882:r
2873:μ
2857:,
2851:t
2848:d
2842:r
2838:d
2825:r
2821:=
2817:p
2809:r
2805:=
2801:L
2789:L
2780:p
2757:r
2753:R
2739:)
2736:t
2733:(
2729:r
2720:2
2716:m
2712:+
2707:1
2703:m
2696:1
2692:m
2683:)
2680:t
2677:(
2673:R
2669:=
2666:)
2663:t
2660:(
2655:2
2650:x
2629:)
2626:t
2623:(
2619:r
2610:2
2606:m
2602:+
2597:1
2593:m
2586:2
2582:m
2576:+
2573:)
2570:t
2567:(
2563:R
2559:=
2556:)
2553:t
2550:(
2545:1
2540:x
2528:)
2526:t
2524:(
2522:r
2517:)
2515:t
2513:(
2511:R
2485:)
2481:r
2477:(
2473:F
2452:)
2448:r
2444:(
2440:F
2429:)
2427:t
2425:(
2423:r
2406:.
2398:2
2394:m
2390:+
2385:1
2381:m
2373:2
2369:m
2363:1
2359:m
2352:=
2342:2
2338:m
2334:1
2329:+
2322:1
2318:m
2314:1
2308:1
2303:=
2255:)
2251:r
2247:(
2243:F
2239:=
2236:)
2231:2
2226:x
2221:,
2216:1
2211:x
2206:(
2196:F
2191:=
2181:r
2159:2
2156:x
2150:1
2147:x
2141:r
2128:r
2119:F
2112:F
2087:F
2081:)
2073:2
2069:m
2065:1
2060:+
2053:1
2049:m
2045:1
2039:(
2035:=
2031:)
2023:2
2019:m
2008:F
1994:1
1990:m
1979:F
1971:(
1967:=
1962:2
1951:x
1938:1
1927:x
1919:=
1909:r
1887:)
1885:t
1883:(
1881:R
1871:2
1868:v
1865:2
1862:m
1858:1
1855:v
1852:1
1849:m
1831:t
1828:d
1823:R
1820:d
1814:=
1810:v
1789:0
1786:=
1776:R
1750:.
1742:2
1738:m
1734:+
1729:1
1725:m
1717:2
1706:x
1696:2
1692:m
1688:+
1683:1
1672:x
1662:1
1658:m
1641:R
1623:F
1616:F
1598:0
1595:=
1585:F
1580:+
1570:F
1565:=
1555:R
1548:)
1543:2
1539:m
1535:+
1530:1
1526:m
1522:(
1519:=
1514:2
1503:x
1493:2
1489:m
1485:+
1480:1
1469:x
1459:1
1455:m
1442:(
1425:R
1406:)
1404:t
1402:(
1400:2
1397:x
1392:)
1390:t
1388:(
1386:1
1383:x
1377:2
1374:x
1370:1
1367:x
1363:r
1352:(
1345:2
1333:x
1326:F
1319:F
1310:)
1308:2
1287:2
1276:x
1266:2
1262:m
1258:=
1255:)
1250:2
1245:x
1240:,
1235:1
1230:x
1225:(
1215:F
1196:)
1194:1
1173:1
1162:x
1152:1
1148:m
1144:=
1141:)
1136:2
1131:x
1126:,
1121:1
1116:x
1111:(
1101:F
1075:t
1073:(
1071:2
1068:v
1061:t
1059:(
1057:1
1054:v
1047:t
1045:(
1043:2
1040:x
1033:t
1031:(
1029:1
1026:x
1021:t
1017:)
1015:t
1013:(
1011:2
1008:x
1003:)
1001:t
999:(
997:1
994:x
989:2
986:m
982:1
979:m
974:2
971:x
965:1
962:x
955:.
938:2
934:m
930:+
925:1
921:m
917:=
914:M
892:2
887:x
877:1
872:x
867:=
863:r
840:2
835:x
828:M
823:2
819:m
813:+
808:1
803:x
796:M
791:1
787:m
781:=
777:R
745:)
741:(
723:.
579:n
573:n
507:e
500:t
493:v
375:)
371:(
222:)
213:(
85:–
72:.
27:.
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