4072:
773:
121:
3764:
3529:
2115:
567:", 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).
716:
3275:
1911:
4067:{\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}}}
56:
47:
3048:
643:
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
605:
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
1619:
1771:
2276:
863:
2878:
3524:{\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} )}
2427:
2935:
2110:{\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}}
1310:
1196:
1460:
3155:
2760:
2650:
613:, 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
3609:
915:
1643:
4264:
689:
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.
4162:
3737:
3681:
3769:
2180:
617:. 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.
2806:
2306:
1855:
1219:
1105:
2506:
2473:
570:
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
1810:
782:
3085:
2654:
2544:
765:
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
964:
1447:
659:
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
3540:
2296:
4198:
3043:{\displaystyle \mathbf {N} ={\frac {d\mathbf {L} }{dt}}={\dot {\mathbf {r} }}\times \mu {\dot {\mathbf {r} }}+\mathbf {r} \times \mu {\ddot {\mathbf {r} }}\ ,}
4105:
3685:
1614:{\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}
3632:
1391:
between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories
631:
In principle, the same solutions apply to macroscopic problems involving objects interacting not only through gravity, but through any other attractive
868:
516:
537:
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
31:
1908:
Dividing both force equations by the respective masses, subtracting the second equation from the first, and rearranging gives the equation
560:. A two-point-particle model of such a system nearly always describes its behavior well enough to provide useful insights and predictions.
288:
541:
that interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored.
357:
4465:
4437:
4394:
1766:{\displaystyle {\ddot {\mathbf {R} }}\equiv {\frac {m_{1}{\ddot {\mathbf {x} }}_{1}+m_{2}{\ddot {\mathbf {x} }}_{2}}{m_{1}+m_{2}}}.}
88:
Two bodies with a "slight" difference in mass orbiting a common barycenter. Their sizes and this type of orbit are similar to the
3200:
4452:
4082:
731:
699:
621:
564:
1778:
178:
4293:
2271:{\displaystyle \mu {\ddot {\mathbf {r} }}=\mathbf {F} _{12}(\mathbf {x} _{1},\mathbf {x} _{2})=\mathbf {F} (\mathbf {r} )}
1349:
Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently.
2442:
is the key to the two-body problem. The solution depends on the specific force between the bodies, which is defined by
2177:, and the laws of physics would have to change from place to place. The subtracted equation can therefore be written:
2149:
The force between the two objects, which originates in the two objects, should only be a function of their separation
509:
442:
648:
4505:
3182:
1815:
1622:
1094:
741:
858:{\displaystyle {\boldsymbol {R}}={\frac {m_{1}}{M}}{\boldsymbol {x}}_{1}+{\frac {m_{2}}{M}}{\boldsymbol {x}}_{2}}
35:
2873:{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {r} \times \mu {\frac {d\mathbf {r} }{dt}},}
2478:
2445:
1886:
437:
352:
769:. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved.
308:
4288:
2422:{\displaystyle \mu ={\frac {1}{{\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}}}={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}.}
2174:
502:
225:
2143:
737:
410:
245:
153:
2118:
4353:
3051:
2779:
1346:
position vectors denote their second derivative with respect to time, or their acceleration vectors.
920:
283:
240:
230:
158:
30:
This article is about the two-body problem in classical mechanics. For the relativistic version, see
3185:) that the force between two particles acts along the line between their positions, it follows that
1305:{\displaystyle \mathbf {F} _{21}(\mathbf {x} _{1},\mathbf {x} _{2})=m_{2}{\ddot {\mathbf {x} }}_{2}}
1191:{\displaystyle \mathbf {F} _{12}(\mathbf {x} _{1},\mathbf {x} _{2})=m_{1}{\ddot {\mathbf {x} }}_{1}}
663:
be predicted under the classical assumptions underlying this article or using the mathematics here.
574:, allowing it to be solved completely, and giving a solution simple enough to be used effectively.
530:
325:
163:
1430:
4316:
3251:
2509:
776:
636:
578:
544:
The most prominent example of the classical two-body problem is the gravitational case (see also
398:
273:
1900:
of the center of mass can be determined at all times from the initial positions and velocities.
745:
725:
4481:
3150:{\displaystyle \mathbf {N} \ =\ {\frac {d\mathbf {L} }{dt}}=\mathbf {r} \times \mathbf {F} \ ,}
2755:{\displaystyle \mathbf {x} _{2}(t)=\mathbf {R} (t)-{\frac {m_{1}}{m_{1}+m_{2}}}\mathbf {r} (t)}
2645:{\displaystyle \mathbf {x} _{1}(t)=\mathbf {R} (t)+{\frac {m_{2}}{m_{1}+m_{2}}}\mathbf {r} (t)}
4461:
4447:
4433:
4390:
3269:
772:
686:
313:
250:
129:
548:), arising in astronomy for predicting the orbits (or escapes from orbit) of objects such as
4361:
3255:
2795:
640:
632:
483:
432:
141:
4500:
4417:
2786:
2281:
647:
The dynamical system of a two-body system under the influence of torque turns out to be a
488:
393:
303:
278:
4357:
4457:
4429:
4383:
4321:
4308:
4303:
3534:
2912:(with these written taking the center of mass as the origin, and thus both parallel to
2513:
1450:
1360:
1342:
is the force on mass 2 due to its interactions with mass 1. The two dots on top of the
705:
682:
670:
625:
592: ≥ 3) cannot be solved in terms of first integrals, except in special cases.
582:
545:
538:
460:
376:
370:
293:
218:
212:
207:
76:
4494:
4099:
3222:
614:
465:
298:
255:
1371:
equation (2) from equation (1) results in an equation that describes how the vector
120:
4485:
4298:
4283:
2887:
2778:
The motion of two bodies with respect to each other always lies in a plane (in the
2300:
183:
173:
168:
80:
700:
Classical central-force problem § Relation to the classical two-body problem
4192:
685:"). However, electrons don't actually orbit nuclei in any meaningful sense, and
347:
4366:
4341:
17:
4422:
4413:
1857:
of the center of mass is constant, from which follows that the total momentum
1454:
1364:
678:
674:
427:
383:
342:
72:
3604:{\displaystyle E={\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}+U(\mathbf {r} )}
910:{\displaystyle {\boldsymbol {r}}={\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{2}}
766:
744:
to the section by replacing the section with a link and a summary or by
549:
732:
Classical central-force problem#Relation to the classical two-body problem
4259:{\displaystyle \mu {\ddot {\mathbf {r} }}={F}(r){\hat {\mathbf {r} }}\ ,}
666:
3181:
Introducing the assumption (true of most physical forces, as they obey
610:
2925:
553:
92:(in which the barycenter is external to both bodies), as well as the
4157:{\displaystyle \mathbf {F} (\mathbf {r} )=F(r){\hat {\mathbf {r} }}}
2803:
of the system, with respect to the center of mass, by the equations
1457:) of the system. Addition of the force equations (1) and (2) yields
571:
3732:{\displaystyle \mathbf {x} _{2}=-{\frac {\mu }{m_{2}}}\mathbf {r} }
3676:{\displaystyle \mathbf {x} _{1}={\frac {\mu }{m_{1}}}\mathbf {r} }
557:
148:
93:
89:
55:
46:
2541:
have been determined, the original trajectories may be obtained
1335:
is the force on mass 1 due to its interactions with mass 2, and
100:
system (in which the barycenter is internal to the larger body).
97:
68:
709:
34:. For the career management problem of working couples, see
1002:
be their masses. The goal is to determine the trajectories
3761:
that separately contain the kinetic energy of each body:
620:
For the derivation of the solutions to the problem, see
606:
influences will be small enough to be ignored safely).
2762:
as may be verified by substituting the definitions of
4201:
4108:
3767:
3688:
3635:
3543:
3278:
3088:
2938:
2809:
2657:
2547:
2481:
2448:
2309:
2284:
2183:
1914:
1818:
1781:
1646:
1463:
1433:
1222:
1108:
923:
871:
785:
669:
in an atom are sometimes described as "orbiting" its
4421:
4382:
4274:) is negative in the case of an attractive force.
4258:
4156:
4066:
3731:
3675:
3603:
3523:
3149:
3042:
2872:
2770:into the right-hand sides of these two equations.
2754:
2644:
2500:
2467:
2421:
2290:
2270:
2109:
1849:
1804:
1765:
1613:
1441:
1304:
1190:
958:
909:
857:
1904:Displacement vector motion (2nd one-body problem)
4342:"The Sturm-Liouville problem of two-body system"
3209:(conserved). Therefore, the displacement vector
655:Inapplicability to atoms and subatomic particles
988:be the vector positions of the two bodies, and
694:Reduction to two independent, one-body problems
2918:) the rate of change of the angular momentum
1850:{\displaystyle \mathbf {v} ={\frac {dR}{dt}}}
779:for two-body problem; Jacobi coordinates are
644:are isolated enough from their surroundings.
601:Gravitation and other inverse-square examples
510:
8:
1423:Center of mass motion (1st one-body problem)
3537:is the lowest and the total energy becomes
2501:{\displaystyle \mathbf {F} (\mathbf {r} )}
2468:{\displaystyle \mathbf {F} (\mathbf {r} )}
517:
503:
105:
4365:
4239:
4237:
4236:
4222:
4208:
4206:
4205:
4200:
4143:
4141:
4140:
4117:
4109:
4107:
4054:
4041:
4024:
4008:
3994:
3985:
3976:
3971:
3960:
3958:
3957:
3950:
3936:
3922:
3913:
3900:
3884:
3870:
3861:
3852:
3847:
3836:
3834:
3833:
3826:
3812:
3798:
3789:
3776:
3768:
3766:
3724:
3716:
3707:
3695:
3690:
3687:
3668:
3660:
3651:
3642:
3637:
3634:
3593:
3578:
3567:
3565:
3564:
3550:
3542:
3513:
3498:
3487:
3485:
3484:
3470:
3461:
3450:
3448:
3447:
3437:
3424:
3407:
3396:
3381:
3376:
3365:
3363:
3362:
3355:
3341:
3332:
3327:
3316:
3314:
3313:
3306:
3292:
3283:
3277:
3136:
3128:
3109:
3103:
3089:
3087:
3023:
3021:
3020:
3009:
2995:
2993:
2992:
2975:
2973:
2972:
2953:
2947:
2939:
2937:
2851:
2845:
2834:
2826:
2818:
2810:
2808:
2738:
2729:
2716:
2705:
2699:
2682:
2664:
2659:
2656:
2628:
2619:
2606:
2595:
2589:
2572:
2554:
2549:
2546:
2490:
2482:
2480:
2457:
2449:
2447:
2407:
2394:
2382:
2372:
2365:
2351:
2342:
2331:
2322:
2316:
2308:
2283:
2260:
2252:
2240:
2235:
2225:
2220:
2210:
2205:
2190:
2188:
2187:
2182:
2146:from mass 2 to mass 1, as defined above.
2101:
2096:
2082:
2073:
2062:
2053:
2032:
2022:
2017:
2014:
2003:
1993:
1988:
1985:
1971:
1960:
1958:
1957:
1947:
1936:
1934:
1933:
1918:
1916:
1915:
1913:
1827:
1819:
1817:
1785:
1783:
1782:
1780:
1751:
1738:
1726:
1715:
1713:
1712:
1705:
1692:
1681:
1679:
1678:
1671:
1664:
1650:
1648:
1647:
1645:
1599:
1594:
1584:
1579:
1564:
1562:
1561:
1552:
1539:
1523:
1512:
1510:
1509:
1502:
1489:
1478:
1476:
1475:
1468:
1462:
1434:
1432:
1296:
1285:
1283:
1282:
1275:
1259:
1254:
1244:
1239:
1229:
1224:
1221:
1182:
1171:
1169:
1168:
1161:
1145:
1140:
1130:
1125:
1115:
1110:
1107:
947:
934:
922:
901:
896:
886:
881:
872:
870:
849:
844:
832:
826:
817:
812:
800:
794:
786:
784:
1805:{\displaystyle {\ddot {\mathbf {R} }}=0}
1359:) results in an equation describing the
771:
4332:
897:
882:
873:
845:
813:
787:
408:
323:
127:
113:
4087:For many physical problems, the force
572:reduced to a pair of one-body problems
32:Two-body problem in general relativity
4389:. Springer. p. 58; Figure 2.15.
27:Motion problem in classical mechanics
7:
2155:and not of their absolute positions
1213:
1099:
3183:Newton's strong third law of motion
25:
4486:Eric Weisstein's World of Physics
4346:Journal of Physics Communications
681:(this is the source of the term "
563:A simpler "one body" model, the "
289:Kepler's laws of planetary motion
4240:
4209:
4144:
4118:
4110:
4009:
3961:
3885:
3837:
3739:and in a similar way the energy
3725:
3691:
3669:
3638:
3594:
3568:
3533:In the center of mass frame the
3514:
3488:
3451:
3397:
3366:
3317:
3137:
3129:
3110:
3090:
3080:pointing in the same direction,
3024:
3010:
2996:
2976:
2954:
2940:
2852:
2835:
2827:
2819:
2811:
2739:
2683:
2660:
2629:
2573:
2550:
2491:
2483:
2458:
2450:
2261:
2253:
2236:
2221:
2206:
2191:
2173:; otherwise, there would not be
2097:
2018:
1989:
1961:
1937:
1919:
1820:
1786:
1716:
1682:
1651:
1595:
1580:
1565:
1513:
1479:
1435:
1286:
1255:
1240:
1225:
1172:
1141:
1126:
1111:
1093:When applied to the two masses,
714:
119:
54:
45:
4083:Classical central-force problem
959:{\displaystyle M=m_{1}+m_{2}\ }
622:Classical central-force problem
4244:
4233:
4227:
4148:
4137:
4131:
4122:
4114:
4013:
4005:
3889:
3881:
3598:
3590:
3518:
3510:
3443:
3417:
3401:
3393:
3050:and using the property of the
2749:
2743:
2693:
2687:
2676:
2670:
2639:
2633:
2583:
2577:
2566:
2560:
2495:
2487:
2462:
2454:
2265:
2257:
2246:
2216:
1558:
1532:
1265:
1235:
1151:
1121:
1034:, given the initial positions
75:external to both bodies, with
1:
3235:Energy of the two-body system
1442:{\displaystyle \mathbf {R} }
4456:(2nd. ed.). New York:
4428:(3rd. ed.). New York:
4340:Luo, Siwei (22 June 2020).
3743:is related to the energies
1355:
1062:and the initial velocities
728:the scope of other articles
596:Results for prominent cases
443:Tsiolkovsky rocket equation
79:. This model is typical of
4522:
4294:Euler's three-body problem
4102:, i.e., it is of the form
4080:
703:
697:
581:(and, more generally, the
412:Engineering and efficiency
231:Bi-elliptic transfer orbit
29:
2896:is the relative position
2774:Two-body motion is planar
2431:Solving the equation for
2117:where we have again used
36:Two-body problem (career)
4367:10.1088/2399-6528/ab9c30
3221:are always in the plane
3201:angular momentum vector
1887:conservation of momentum
1812:shows that the velocity
1775:The resulting equation:
649:Sturm-Liouville equation
641:electrostatic attraction
438:Propellant mass fraction
337:Gravitational influences
4381:David Betounes (2001).
3225:to the constant vector
1889:). Hence, the position
1449:be the position of the
1367:) motion. By contrast,
309:Specific orbital energy
4385:Differential Equations
4289:Equation of the center
4260:
4158:
4068:
3733:
3677:
3605:
3525:
3254:then the system has a
3151:
3044:
2874:
2756:
2646:
2502:
2469:
2423:
2292:
2272:
2175:translational symmetry
2111:
1851:
1806:
1767:
1615:
1443:
1306:
1192:
967:
960:
911:
859:
226:Hohmann transfer orbit
67:Two bodies of similar
4261:
4191:is the corresponding
4159:
4069:
3734:
3678:
3606:
3526:
3152:
3045:
2875:
2757:
2647:
2503:
2475:. For the case where
2470:
2424:
2293:
2273:
2112:
1852:
1807:
1768:
1616:
1444:
1307:
1193:
961:
912:
860:
775:
746:splitting the content
740:and help introduce a
565:central-force problem
422:Preflight engineering
154:Argument of periapsis
4199:
4106:
3765:
3686:
3633:
3629:can be expressed as
3541:
3276:
3086:
3052:vector cross product
2936:
2807:
2785:Proof: Defining the
2780:center of mass frame
2655:
2545:
2479:
2446:
2307:
2291:{\displaystyle \mu }
2282:
2181:
1912:
1816:
1779:
1644:
1461:
1431:
1220:
1106:
921:
869:
783:
478:Propulsive maneuvers
4453:Classical Mechanics
4358:2020JPhCo...4f1001L
3981:
3857:
3386:
3337:
2144:displacement vector
1621:where we have used
1353:equations (1) and (
1095:Newton's second law
748:into a new article.
609:Under the force of
531:classical mechanics
455:Efficiency measures
358:Sphere of influence
327:Celestial mechanics
109:Part of a series on
90:Pluto–Charon system
4317:Three-body problem
4256:
4154:
4064:
4062:
3956:
3832:
3729:
3673:
3601:
3521:
3361:
3312:
3272:can be written as
3147:
3040:
2870:
2752:
2642:
2510:inverse-square law
2498:
2465:
2419:
2288:
2268:
2119:Newton's third law
2107:
1885:is also constant (
1847:
1802:
1763:
1623:Newton's third law
1611:
1439:
1302:
1188:
968:
956:
907:
855:
777:Jacobi coordinates
738:discuss this issue
637:inverse-square law
633:scalar force field
579:three-body problem
274:Dynamical friction
71:orbiting a common
4506:Dynamical systems
4252:
4247:
4216:
4151:
4027:
4000:
3968:
3944:
3928:
3876:
3844:
3820:
3804:
3722:
3666:
3575:
3558:
3495:
3478:
3458:
3415:
3373:
3349:
3324:
3300:
3286:
3215:and its velocity
3143:
3123:
3102:
3096:
3036:
3031:
3003:
2983:
2967:
2865:
2736:
2626:
2414:
2360:
2357:
2337:
2198:
2088:
2068:
2038:
2009:
1968:
1944:
1926:
1845:
1793:
1758:
1723:
1689:
1658:
1572:
1520:
1486:
1326:
1325:
1293:
1212:
1211:
1179:
955:
841:
809:
763:
762:
687:quantum mechanics
577:By contrast, the
527:
526:
377:Lagrangian points
314:Vis-viva equation
284:Kepler's equation
131:Orbital mechanics
16:(Redirected from
4513:
4482:Two-body problem
4471:
4443:
4427:
4401:
4400:
4388:
4378:
4372:
4371:
4369:
4337:
4265:
4263:
4262:
4257:
4250:
4249:
4248:
4243:
4238:
4226:
4218:
4217:
4212:
4207:
4190:
4176:
4174:
4163:
4161:
4160:
4155:
4153:
4152:
4147:
4142:
4121:
4113:
4097:
4073:
4071:
4070:
4065:
4063:
4059:
4058:
4046:
4045:
4029:
4028:
4025:
4012:
4001:
3999:
3998:
3986:
3980:
3975:
3970:
3969:
3964:
3959:
3955:
3954:
3945:
3937:
3929:
3927:
3926:
3914:
3905:
3904:
3888:
3877:
3875:
3874:
3862:
3856:
3851:
3846:
3845:
3840:
3835:
3831:
3830:
3821:
3813:
3805:
3803:
3802:
3790:
3781:
3780:
3760:
3751:
3738:
3736:
3735:
3730:
3728:
3723:
3721:
3720:
3708:
3700:
3699:
3694:
3682:
3680:
3679:
3674:
3672:
3667:
3665:
3664:
3652:
3647:
3646:
3641:
3628:
3619:
3611:The coordinates
3610:
3608:
3607:
3602:
3597:
3583:
3582:
3577:
3576:
3571:
3566:
3559:
3551:
3530:
3528:
3527:
3522:
3517:
3503:
3502:
3497:
3496:
3491:
3486:
3479:
3471:
3466:
3465:
3460:
3459:
3454:
3449:
3442:
3441:
3429:
3428:
3416:
3408:
3400:
3385:
3380:
3375:
3374:
3369:
3364:
3360:
3359:
3350:
3342:
3336:
3331:
3326:
3325:
3320:
3315:
3311:
3310:
3301:
3293:
3288:
3287:
3284:
3267:
3256:potential energy
3249:
3230:
3220:
3214:
3206:
3198:
3177:
3156:
3154:
3153:
3148:
3141:
3140:
3132:
3124:
3122:
3114:
3113:
3104:
3100:
3094:
3093:
3079:
3073:
3068:for any vectors
3067:
3049:
3047:
3046:
3041:
3034:
3033:
3032:
3027:
3022:
3013:
3005:
3004:
2999:
2994:
2985:
2984:
2979:
2974:
2968:
2966:
2958:
2957:
2948:
2943:
2932:
2923:
2917:
2911:
2895:
2885:
2879:
2877:
2876:
2871:
2866:
2864:
2856:
2855:
2846:
2838:
2830:
2822:
2814:
2802:
2796:angular momentum
2793:
2761:
2759:
2758:
2753:
2742:
2737:
2735:
2734:
2733:
2721:
2720:
2710:
2709:
2700:
2686:
2669:
2668:
2663:
2651:
2649:
2648:
2643:
2632:
2627:
2625:
2624:
2623:
2611:
2610:
2600:
2599:
2590:
2576:
2559:
2558:
2553:
2540:
2529:
2507:
2505:
2504:
2499:
2494:
2486:
2474:
2472:
2471:
2466:
2461:
2453:
2441:
2428:
2426:
2425:
2420:
2415:
2413:
2412:
2411:
2399:
2398:
2388:
2387:
2386:
2377:
2376:
2366:
2361:
2359:
2358:
2356:
2355:
2343:
2338:
2336:
2335:
2323:
2317:
2297:
2295:
2294:
2289:
2277:
2275:
2274:
2269:
2264:
2256:
2245:
2244:
2239:
2230:
2229:
2224:
2215:
2214:
2209:
2200:
2199:
2194:
2189:
2172:
2163:
2154:
2141:
2135:
2116:
2114:
2113:
2108:
2106:
2105:
2100:
2094:
2090:
2089:
2087:
2086:
2074:
2069:
2067:
2066:
2054:
2044:
2040:
2039:
2037:
2036:
2027:
2026:
2021:
2015:
2010:
2008:
2007:
1998:
1997:
1992:
1986:
1976:
1975:
1970:
1969:
1964:
1959:
1952:
1951:
1946:
1945:
1940:
1935:
1928:
1927:
1922:
1917:
1899:
1884:
1856:
1854:
1853:
1848:
1846:
1844:
1836:
1828:
1823:
1811:
1809:
1808:
1803:
1795:
1794:
1789:
1784:
1772:
1770:
1769:
1764:
1759:
1757:
1756:
1755:
1743:
1742:
1732:
1731:
1730:
1725:
1724:
1719:
1714:
1710:
1709:
1697:
1696:
1691:
1690:
1685:
1680:
1676:
1675:
1665:
1660:
1659:
1654:
1649:
1639:
1620:
1618:
1617:
1612:
1604:
1603:
1598:
1589:
1588:
1583:
1574:
1573:
1568:
1563:
1557:
1556:
1544:
1543:
1528:
1527:
1522:
1521:
1516:
1511:
1507:
1506:
1494:
1493:
1488:
1487:
1482:
1477:
1473:
1472:
1448:
1446:
1445:
1440:
1438:
1418:
1404:
1390:
1320:
1311:
1309:
1308:
1303:
1301:
1300:
1295:
1294:
1289:
1284:
1280:
1279:
1264:
1263:
1258:
1249:
1248:
1243:
1234:
1233:
1228:
1214:
1206:
1197:
1195:
1194:
1189:
1187:
1186:
1181:
1180:
1175:
1170:
1166:
1165:
1150:
1149:
1144:
1135:
1134:
1129:
1120:
1119:
1114:
1100:
1089:
1075:
1061:
1047:
1029:
1015:
987:
978:
965:
963:
962:
957:
953:
952:
951:
939:
938:
916:
914:
913:
908:
906:
905:
900:
891:
890:
885:
876:
864:
862:
861:
856:
854:
853:
848:
842:
837:
836:
827:
822:
821:
816:
810:
805:
804:
795:
790:
758:
755:
749:
718:
717:
710:
675:early conjecture
535:two-body problem
519:
512:
505:
484:Orbital maneuver
433:Payload fraction
413:
394:Lissajous orbits
328:
299:Orbital velocity
246:Hyperbolic orbit
142:Orbital elements
132:
123:
106:
58:
49:
21:
4521:
4520:
4516:
4515:
4514:
4512:
4511:
4510:
4491:
4490:
4478:
4468:
4446:
4440:
4412:
4409:
4404:
4397:
4380:
4379:
4375:
4339:
4338:
4334:
4330:
4280:
4197:
4196:
4195:. We now have:
4178:
4170:
4165:
4104:
4103:
4088:
4085:
4079:
4061:
4060:
4050:
4037:
4030:
4020:
4017:
4016:
3990:
3946:
3918:
3906:
3896:
3893:
3892:
3866:
3822:
3794:
3782:
3772:
3763:
3762:
3759:
3753:
3750:
3744:
3712:
3689:
3684:
3683:
3656:
3636:
3631:
3630:
3627:
3621:
3618:
3612:
3563:
3539:
3538:
3483:
3446:
3433:
3420:
3351:
3302:
3279:
3274:
3273:
3268:, so the total
3258:
3240:
3237:
3226:
3216:
3210:
3202:
3186:
3158:
3115:
3105:
3084:
3083:
3075:
3069:
3055:
2959:
2949:
2934:
2933:
2928:
2924:equals the net
2919:
2913:
2910:
2903:
2897:
2891:
2883:
2857:
2847:
2805:
2804:
2798:
2789:
2787:linear momentum
2776:
2725:
2712:
2711:
2701:
2658:
2653:
2652:
2615:
2602:
2601:
2591:
2548:
2543:
2542:
2531:
2520:
2477:
2476:
2444:
2443:
2432:
2403:
2390:
2389:
2378:
2368:
2367:
2347:
2327:
2321:
2305:
2304:
2280:
2279:
2234:
2219:
2204:
2179:
2178:
2171:
2165:
2162:
2156:
2150:
2137:
2134:
2127:
2121:
2095:
2078:
2058:
2052:
2048:
2028:
2016:
1999:
1987:
1984:
1980:
1956:
1932:
1910:
1909:
1906:
1890:
1883:
1877:
1870:
1864:
1858:
1837:
1829:
1814:
1813:
1777:
1776:
1747:
1734:
1733:
1711:
1701:
1677:
1667:
1666:
1642:
1641:
1638:
1631:
1625:
1593:
1578:
1548:
1535:
1508:
1498:
1474:
1464:
1459:
1458:
1429:
1428:
1425:
1412:
1406:
1398:
1392:
1389:
1382:
1372:
1341:
1334:
1318:
1281:
1271:
1253:
1238:
1223:
1218:
1217:
1204:
1167:
1157:
1139:
1124:
1109:
1104:
1103:
1083:
1077:
1069:
1063:
1055:
1049:
1041:
1035:
1023:
1017:
1009:
1003:
1001:
994:
986:
980:
977:
971:
943:
930:
919:
918:
895:
880:
867:
866:
843:
828:
811:
796:
781:
780:
759:
753:
750:
735:
730:, specifically
719:
715:
708:
702:
696:
673:, following an
657:
603:
598:
539:point particles
523:
494:
493:
489:Orbit insertion
479:
471:
470:
456:
448:
447:
423:
415:
411:
404:
403:
399:Lyapunov orbits
390:
389:
373:
363:
362:
338:
330:
326:
319:
318:
304:Surface gravity
279:Escape velocity
269:
261:
260:
241:Parabolic orbit
237:
236:
203:
201:
198:two-body orbits
189:
188:
179:Semi-major axis
144:
134:
130:
104:
103:
102:
101:
84:
77:elliptic orbits
61:
60:
59:
51:
50:
39:
28:
23:
22:
18:Two-body motion
15:
12:
11:
5:
4519:
4517:
4509:
4508:
4503:
4493:
4492:
4489:
4488:
4477:
4476:External links
4474:
4473:
4472:
4466:
4458:Addison-Wesley
4444:
4438:
4430:Pergamon Press
4408:
4405:
4403:
4402:
4395:
4373:
4331:
4329:
4326:
4325:
4324:
4322:Virial theorem
4319:
4314:
4306:
4304:Kepler problem
4301:
4296:
4291:
4286:
4279:
4276:
4255:
4246:
4242:
4235:
4232:
4229:
4225:
4221:
4215:
4211:
4204:
4150:
4146:
4139:
4136:
4133:
4130:
4127:
4124:
4120:
4116:
4112:
4081:Main article:
4078:
4077:Central forces
4075:
4057:
4053:
4049:
4044:
4040:
4036:
4033:
4031:
4023:
4019:
4018:
4015:
4011:
4007:
4004:
3997:
3993:
3989:
3984:
3979:
3974:
3967:
3963:
3953:
3949:
3943:
3940:
3935:
3932:
3925:
3921:
3917:
3912:
3909:
3907:
3903:
3899:
3895:
3894:
3891:
3887:
3883:
3880:
3873:
3869:
3865:
3860:
3855:
3850:
3843:
3839:
3829:
3825:
3819:
3816:
3811:
3808:
3801:
3797:
3793:
3788:
3785:
3783:
3779:
3775:
3771:
3770:
3757:
3748:
3727:
3719:
3715:
3711:
3706:
3703:
3698:
3693:
3671:
3663:
3659:
3655:
3650:
3645:
3640:
3625:
3616:
3600:
3596:
3592:
3589:
3586:
3581:
3574:
3570:
3562:
3557:
3554:
3549:
3546:
3535:kinetic energy
3520:
3516:
3512:
3509:
3506:
3501:
3494:
3490:
3482:
3477:
3474:
3469:
3464:
3457:
3453:
3445:
3440:
3436:
3432:
3427:
3423:
3419:
3414:
3411:
3406:
3403:
3399:
3395:
3392:
3389:
3384:
3379:
3372:
3368:
3358:
3354:
3348:
3345:
3340:
3335:
3330:
3323:
3319:
3309:
3305:
3299:
3296:
3291:
3282:
3236:
3233:
3146:
3139:
3135:
3131:
3127:
3121:
3118:
3112:
3108:
3099:
3092:
3039:
3030:
3026:
3019:
3016:
3012:
3008:
3002:
2998:
2991:
2988:
2982:
2978:
2971:
2965:
2962:
2956:
2952:
2946:
2942:
2908:
2901:
2869:
2863:
2860:
2854:
2850:
2844:
2841:
2837:
2833:
2829:
2825:
2821:
2817:
2813:
2775:
2772:
2751:
2748:
2745:
2741:
2732:
2728:
2724:
2719:
2715:
2708:
2704:
2698:
2695:
2692:
2689:
2685:
2681:
2678:
2675:
2672:
2667:
2662:
2641:
2638:
2635:
2631:
2622:
2618:
2614:
2609:
2605:
2598:
2594:
2588:
2585:
2582:
2579:
2575:
2571:
2568:
2565:
2562:
2557:
2552:
2514:Kepler problem
2497:
2493:
2489:
2485:
2464:
2460:
2456:
2452:
2418:
2410:
2406:
2402:
2397:
2393:
2385:
2381:
2375:
2371:
2364:
2354:
2350:
2346:
2341:
2334:
2330:
2326:
2320:
2315:
2312:
2287:
2267:
2263:
2259:
2255:
2251:
2248:
2243:
2238:
2233:
2228:
2223:
2218:
2213:
2208:
2203:
2197:
2193:
2186:
2169:
2160:
2132:
2125:
2104:
2099:
2093:
2085:
2081:
2077:
2072:
2065:
2061:
2057:
2051:
2047:
2043:
2035:
2031:
2025:
2020:
2013:
2006:
2002:
1996:
1991:
1983:
1979:
1974:
1967:
1963:
1955:
1950:
1943:
1939:
1931:
1925:
1921:
1905:
1902:
1881:
1875:
1868:
1862:
1843:
1840:
1835:
1832:
1826:
1822:
1801:
1798:
1792:
1788:
1762:
1754:
1750:
1746:
1741:
1737:
1729:
1722:
1718:
1708:
1704:
1700:
1695:
1688:
1684:
1674:
1670:
1663:
1657:
1653:
1636:
1629:
1610:
1607:
1602:
1597:
1592:
1587:
1582:
1577:
1571:
1567:
1560:
1555:
1551:
1547:
1542:
1538:
1534:
1531:
1526:
1519:
1515:
1505:
1501:
1497:
1492:
1485:
1481:
1471:
1467:
1451:center of mass
1437:
1424:
1421:
1410:
1396:
1387:
1380:
1361:center of mass
1339:
1332:
1324:
1323:
1314:
1312:
1299:
1292:
1288:
1278:
1274:
1270:
1267:
1262:
1257:
1252:
1247:
1242:
1237:
1232:
1227:
1210:
1209:
1200:
1198:
1185:
1178:
1174:
1164:
1160:
1156:
1153:
1148:
1143:
1138:
1133:
1128:
1123:
1118:
1113:
1081:
1067:
1053:
1039:
1030:for all times
1021:
1007:
999:
992:
984:
975:
950:
946:
942:
937:
933:
929:
926:
904:
899:
894:
889:
884:
879:
875:
852:
847:
840:
835:
831:
825:
820:
815:
808:
803:
799:
793:
789:
761:
760:
722:
720:
713:
706:Kepler problem
695:
692:
656:
653:
626:Kepler problem
615:conic sections
602:
599:
597:
594:
546:Kepler problem
525:
524:
522:
521:
514:
507:
499:
496:
495:
492:
491:
486:
480:
477:
476:
473:
472:
469:
468:
463:
461:Gravity assist
457:
454:
453:
450:
449:
446:
445:
440:
435:
430:
424:
421:
420:
417:
416:
409:
406:
405:
402:
401:
396:
388:
387:
379:
375:
374:
369:
368:
365:
364:
361:
360:
355:
350:
345:
339:
336:
335:
332:
331:
324:
321:
320:
317:
316:
311:
306:
301:
296:
294:Orbital period
291:
286:
281:
276:
270:
267:
266:
263:
262:
259:
258:
256:Decaying orbit
253:
248:
243:
235:
234:
228:
221:
219:Transfer orbit
217:
216:
215:
213:Elliptic orbit
210:
208:Circular orbit
204:
195:
194:
191:
190:
187:
186:
181:
176:
171:
166:
161:
156:
151:
145:
140:
139:
136:
135:
128:
125:
124:
116:
115:
111:
110:
63:
62:
53:
52:
44:
43:
42:
41:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4518:
4507:
4504:
4502:
4499:
4498:
4496:
4487:
4483:
4480:
4479:
4475:
4469:
4467:0-201-02918-9
4463:
4459:
4455:
4454:
4449:
4445:
4441:
4439:0-08-029141-4
4435:
4431:
4426:
4425:
4419:
4415:
4411:
4410:
4406:
4398:
4396:0-387-95140-7
4392:
4387:
4386:
4377:
4374:
4368:
4363:
4359:
4355:
4352:(6): 061001.
4351:
4347:
4343:
4336:
4333:
4327:
4323:
4320:
4318:
4315:
4313:
4312:-body problem
4311:
4307:
4305:
4302:
4300:
4297:
4295:
4292:
4290:
4287:
4285:
4282:
4281:
4277:
4275:
4273:
4269:
4253:
4230:
4223:
4219:
4213:
4202:
4194:
4189:
4185:
4181:
4173:
4168:
4134:
4128:
4125:
4101:
4100:central force
4095:
4091:
4084:
4076:
4074:
4055:
4051:
4047:
4042:
4038:
4034:
4032:
4021:
4002:
3995:
3991:
3987:
3982:
3977:
3972:
3965:
3951:
3947:
3941:
3938:
3933:
3930:
3923:
3919:
3915:
3910:
3908:
3901:
3897:
3878:
3871:
3867:
3863:
3858:
3853:
3848:
3841:
3827:
3823:
3817:
3814:
3809:
3806:
3799:
3795:
3791:
3786:
3784:
3777:
3773:
3756:
3747:
3742:
3717:
3713:
3709:
3704:
3701:
3696:
3661:
3657:
3653:
3648:
3643:
3624:
3615:
3587:
3584:
3579:
3572:
3560:
3555:
3552:
3547:
3544:
3536:
3531:
3507:
3504:
3499:
3492:
3480:
3475:
3472:
3467:
3462:
3455:
3438:
3434:
3430:
3425:
3421:
3412:
3409:
3404:
3390:
3387:
3382:
3377:
3370:
3356:
3352:
3346:
3343:
3338:
3333:
3328:
3321:
3307:
3303:
3297:
3294:
3289:
3280:
3271:
3265:
3261:
3257:
3253:
3247:
3243:
3239:If the force
3234:
3232:
3229:
3224:
3223:perpendicular
3219:
3213:
3208:
3205:
3197:
3193:
3189:
3184:
3179:
3176:
3172:
3169:
3165:
3161:
3144:
3133:
3125:
3119:
3116:
3106:
3097:
3081:
3078:
3072:
3066:
3062:
3058:
3053:
3037:
3028:
3017:
3014:
3006:
3000:
2989:
2986:
2980:
2969:
2963:
2960:
2950:
2944:
2931:
2927:
2922:
2916:
2907:
2900:
2894:
2889:
2880:
2867:
2861:
2858:
2848:
2842:
2839:
2831:
2823:
2815:
2801:
2797:
2792:
2788:
2783:
2781:
2773:
2771:
2769:
2765:
2746:
2730:
2726:
2722:
2717:
2713:
2706:
2702:
2696:
2690:
2679:
2673:
2665:
2636:
2620:
2616:
2612:
2607:
2603:
2596:
2592:
2586:
2580:
2569:
2563:
2555:
2538:
2534:
2527:
2523:
2517:
2515:
2511:
2439:
2435:
2429:
2416:
2408:
2404:
2400:
2395:
2391:
2383:
2379:
2373:
2369:
2362:
2352:
2348:
2344:
2339:
2332:
2328:
2324:
2318:
2313:
2310:
2303:
2302:
2285:
2249:
2241:
2231:
2226:
2211:
2201:
2195:
2184:
2176:
2168:
2159:
2153:
2147:
2145:
2140:
2131:
2124:
2120:
2102:
2091:
2083:
2079:
2075:
2070:
2063:
2059:
2055:
2049:
2045:
2041:
2033:
2029:
2023:
2011:
2004:
2000:
1994:
1981:
1977:
1972:
1965:
1953:
1948:
1941:
1929:
1923:
1903:
1901:
1897:
1893:
1888:
1880:
1874:
1867:
1861:
1841:
1838:
1833:
1830:
1824:
1799:
1796:
1790:
1773:
1760:
1752:
1748:
1744:
1739:
1735:
1727:
1720:
1706:
1702:
1698:
1693:
1686:
1672:
1668:
1661:
1655:
1635:
1628:
1624:
1608:
1605:
1600:
1590:
1585:
1575:
1569:
1553:
1549:
1545:
1540:
1536:
1529:
1524:
1517:
1503:
1499:
1495:
1490:
1483:
1469:
1465:
1456:
1452:
1422:
1420:
1416:
1409:
1402:
1395:
1386:
1379:
1375:
1370:
1366:
1362:
1358:
1357:
1352:
1347:
1345:
1338:
1331:
1322:
1315:
1313:
1297:
1290:
1276:
1272:
1268:
1260:
1250:
1245:
1230:
1216:
1215:
1208:
1201:
1199:
1183:
1176:
1162:
1158:
1154:
1146:
1136:
1131:
1116:
1102:
1101:
1098:
1096:
1091:
1087:
1080:
1073:
1066:
1059:
1052:
1045:
1038:
1033:
1027:
1020:
1013:
1006:
998:
991:
983:
974:
948:
944:
940:
935:
931:
927:
924:
902:
892:
887:
877:
850:
838:
833:
829:
823:
818:
806:
801:
797:
791:
778:
774:
770:
768:
757:
747:
743:
742:summary style
739:
733:
729:
727:
723:This section
721:
712:
711:
707:
701:
693:
691:
688:
684:
680:
676:
672:
668:
664:
662:
654:
652:
650:
645:
642:
638:
634:
629:
627:
623:
618:
616:
612:
607:
600:
595:
593:
591:
587:
586:-body problem
585:
580:
575:
573:
568:
566:
561:
559:
555:
551:
547:
542:
540:
536:
532:
520:
515:
513:
508:
506:
501:
500:
498:
497:
490:
487:
485:
482:
481:
475:
474:
467:
466:Oberth effect
464:
462:
459:
458:
452:
451:
444:
441:
439:
436:
434:
431:
429:
426:
425:
419:
418:
414:
407:
400:
397:
395:
392:
391:
385:
381:
380:
378:
372:
371:N-body orbits
367:
366:
359:
356:
354:
353:Perturbations
351:
349:
346:
344:
341:
340:
334:
333:
329:
322:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
285:
282:
280:
277:
275:
272:
271:
265:
264:
257:
254:
252:
249:
247:
244:
242:
239:
238:
232:
229:
227:
223:
222:
220:
214:
211:
209:
206:
205:
199:
193:
192:
185:
182:
180:
177:
175:
174:Orbital nodes
172:
170:
167:
165:
162:
160:
157:
155:
152:
150:
147:
146:
143:
138:
137:
133:
126:
122:
118:
117:
114:Astrodynamics
112:
108:
107:
99:
95:
91:
87:
82:
78:
74:
70:
66:
57:
48:
37:
33:
19:
4451:
4423:
4407:Bibliography
4384:
4376:
4349:
4345:
4335:
4309:
4299:Kepler orbit
4284:Energy drift
4271:
4267:
4187:
4183:
4179:
4171:
4166:
4093:
4089:
4086:
3754:
3745:
3740:
3622:
3613:
3532:
3263:
3259:
3252:conservative
3245:
3241:
3238:
3227:
3217:
3211:
3203:
3195:
3191:
3187:
3180:
3174:
3170:
3167:
3163:
3159:
3082:
3076:
3070:
3064:
3060:
3056:
2929:
2920:
2914:
2905:
2898:
2892:
2888:reduced mass
2881:
2799:
2790:
2784:
2777:
2767:
2763:
2536:
2532:
2525:
2521:
2518:
2437:
2433:
2430:
2301:reduced mass
2299:
2166:
2157:
2151:
2148:
2138:
2129:
2122:
1907:
1895:
1891:
1878:
1872:
1865:
1859:
1774:
1633:
1626:
1426:
1414:
1407:
1400:
1393:
1384:
1377:
1373:
1368:
1354:
1350:
1348:
1343:
1336:
1329:
1327:
1316:
1202:
1097:states that
1092:
1085:
1078:
1071:
1064:
1057:
1050:
1043:
1036:
1031:
1025:
1018:
1011:
1004:
996:
989:
981:
972:
969:
764:
751:
724:
665:
660:
658:
646:
630:
619:
608:
604:
589:
583:
576:
569:
562:
543:
534:
528:
251:Radial orbit
202:eccentricity
197:
184:True anomaly
169:Mean anomaly
159:Eccentricity
85:
81:binary stars
64:
4448:Goldstein H
4418:Lifshitz EM
4193:unit vector
3207:is constant
2508:follows an
1369:subtracting
635:obeying an
384:Halo orbits
348:Hill sphere
164:Inclination
4495:Categories
4328:References
2512:, see the
2136:and where
1640:and where
1455:barycenter
1365:barycenter
1317:(Equation
1203:(Equation
726:duplicates
704:See also:
698:See also:
679:Niels Bohr
550:satellites
428:Mass ratio
343:Barycenter
73:barycenter
4424:Mechanics
4414:Landau LD
4245:^
4214:¨
4203:μ
4149:^
3988:μ
3966:˙
3916:μ
3864:μ
3842:˙
3792:μ
3710:μ
3705:−
3654:μ
3573:˙
3561:μ
3493:˙
3481:μ
3456:˙
3371:˙
3322:˙
3134:×
3029:¨
3018:μ
3015:×
3001:˙
2990:μ
2987:×
2981:˙
2843:μ
2840:×
2824:×
2697:−
2311:μ
2286:μ
2196:¨
2185:μ
2012:−
1966:¨
1954:−
1942:¨
1924:¨
1791:¨
1721:¨
1687:¨
1662:≡
1656:¨
1570:¨
1518:¨
1484:¨
1291:¨
1177:¨
893:−
767:potential
754:June 2019
667:Electrons
268:Equations
196:Types of
4450:(1980).
4420:(1976).
4278:See also
4169:= |
3199:and the
2794:and the
4354:Bibcode
3166:
2886:is the
2298:is the
2142:is the
736:Please
683:orbital
671:nucleus
639:, with
611:gravity
554:planets
4501:Orbits
4464:
4436:
4393:
4266:where
4251:
4175:|
4164:where
3270:energy
3142:
3101:
3095:
3035:
2926:torque
2882:where
2278:where
1351:Adding
1328:where
954:
661:cannot
556:, and
533:, the
86:Right:
4098:is a
3157:with
3054:that
2519:Once
917:with
558:stars
149:Apsis
94:Earth
65:Left:
4462:ISBN
4434:ISBN
4391:ISBN
4177:and
3752:and
3620:and
3074:and
2890:and
2766:and
2530:and
2164:and
1427:Let
1405:and
1088:= 0)
1076:and
1074:= 0)
1060:= 0)
1048:and
1046:= 0)
1016:and
995:and
979:and
970:Let
865:and
588:for
98:Moon
69:mass
4484:at
4362:doi
4026:tot
3285:tot
3250:is
3194:=
2782:).
2128:= −
1632:= −
677:of
624:or
529:In
200:by
4497::
4460:.
4432:.
4416:;
4360:.
4348:.
4344:.
4182:=
4180:r̂
3231:.
3190:×
3178:.
3175:dt
3162:=
3063:=
3059:×
2904:−
2516:.
2212:12
2133:21
2126:12
2103:12
2024:21
1995:12
1871:+
1637:21
1630:12
1601:21
1586:12
1419:.
1383:−
1376:=
1340:21
1333:12
1231:21
1117:12
1090:.
651:.
628:.
552:,
4470:.
4442:.
4399:.
4370:.
4364::
4356::
4350:4
4310:n
4272:r
4270:(
4268:F
4254:,
4241:r
4234:)
4231:r
4228:(
4224:F
4220:=
4210:r
4188:r
4186:/
4184:r
4172:r
4167:r
4145:r
4138:)
4135:r
4132:(
4129:F
4126:=
4123:)
4119:r
4115:(
4111:F
4096:)
4094:r
4092:(
4090:F
4056:2
4052:E
4048:+
4043:1
4039:E
4035:=
4022:E
4014:)
4010:r
4006:(
4003:U
3996:2
3992:m
3983:+
3978:2
3973:2
3962:x
3952:2
3948:m
3942:2
3939:1
3934:=
3931:E
3924:2
3920:m
3911:=
3902:2
3898:E
3890:)
3886:r
3882:(
3879:U
3872:1
3868:m
3859:+
3854:2
3849:1
3838:x
3828:1
3824:m
3818:2
3815:1
3810:=
3807:E
3800:1
3796:m
3787:=
3778:1
3774:E
3758:2
3755:E
3749:1
3746:E
3741:E
3726:r
3718:2
3714:m
3702:=
3697:2
3692:x
3670:r
3662:1
3658:m
3649:=
3644:1
3639:x
3626:2
3623:x
3617:1
3614:x
3599:)
3595:r
3591:(
3588:U
3585:+
3580:2
3569:r
3556:2
3553:1
3548:=
3545:E
3519:)
3515:r
3511:(
3508:U
3505:+
3500:2
3489:r
3476:2
3473:1
3468:+
3463:2
3452:R
3444:)
3439:2
3435:m
3431:+
3426:1
3422:m
3418:(
3413:2
3410:1
3405:=
3402:)
3398:r
3394:(
3391:U
3388:+
3383:2
3378:2
3367:x
3357:2
3353:m
3347:2
3344:1
3339:+
3334:2
3329:1
3318:x
3308:1
3304:m
3298:2
3295:1
3290:=
3281:E
3266:)
3264:r
3262:(
3260:U
3248:)
3246:r
3244:(
3242:F
3228:L
3218:v
3212:r
3204:L
3196:0
3192:F
3188:r
3173:/
3171:r
3168:d
3164:μ
3160:F
3145:,
3138:F
3130:r
3126:=
3120:t
3117:d
3111:L
3107:d
3098:=
3091:N
3077:w
3071:v
3065:0
3061:w
3057:v
3038:,
3025:r
3011:r
3007:+
2997:r
2977:r
2970:=
2964:t
2961:d
2955:L
2951:d
2945:=
2941:N
2930:N
2921:L
2915:r
2909:1
2906:r
2902:2
2899:r
2893:r
2884:μ
2868:,
2862:t
2859:d
2853:r
2849:d
2836:r
2832:=
2828:p
2820:r
2816:=
2812:L
2800:L
2791:p
2768:r
2764:R
2750:)
2747:t
2744:(
2740:r
2731:2
2727:m
2723:+
2718:1
2714:m
2707:1
2703:m
2694:)
2691:t
2688:(
2684:R
2680:=
2677:)
2674:t
2671:(
2666:2
2661:x
2640:)
2637:t
2634:(
2630:r
2621:2
2617:m
2613:+
2608:1
2604:m
2597:2
2593:m
2587:+
2584:)
2581:t
2578:(
2574:R
2570:=
2567:)
2564:t
2561:(
2556:1
2551:x
2539:)
2537:t
2535:(
2533:r
2528:)
2526:t
2524:(
2522:R
2496:)
2492:r
2488:(
2484:F
2463:)
2459:r
2455:(
2451:F
2440:)
2438:t
2436:(
2434:r
2417:.
2409:2
2405:m
2401:+
2396:1
2392:m
2384:2
2380:m
2374:1
2370:m
2363:=
2353:2
2349:m
2345:1
2340:+
2333:1
2329:m
2325:1
2319:1
2314:=
2266:)
2262:r
2258:(
2254:F
2250:=
2247:)
2242:2
2237:x
2232:,
2227:1
2222:x
2217:(
2207:F
2202:=
2192:r
2170:2
2167:x
2161:1
2158:x
2152:r
2139:r
2130:F
2123:F
2098:F
2092:)
2084:2
2080:m
2076:1
2071:+
2064:1
2060:m
2056:1
2050:(
2046:=
2042:)
2034:2
2030:m
2019:F
2005:1
2001:m
1990:F
1982:(
1978:=
1973:2
1962:x
1949:1
1938:x
1930:=
1920:r
1898:)
1896:t
1894:(
1892:R
1882:2
1879:v
1876:2
1873:m
1869:1
1866:v
1863:1
1860:m
1842:t
1839:d
1834:R
1831:d
1825:=
1821:v
1800:0
1797:=
1787:R
1761:.
1753:2
1749:m
1745:+
1740:1
1736:m
1728:2
1717:x
1707:2
1703:m
1699:+
1694:1
1683:x
1673:1
1669:m
1652:R
1634:F
1627:F
1609:0
1606:=
1596:F
1591:+
1581:F
1576:=
1566:R
1559:)
1554:2
1550:m
1546:+
1541:1
1537:m
1533:(
1530:=
1525:2
1514:x
1504:2
1500:m
1496:+
1491:1
1480:x
1470:1
1466:m
1453:(
1436:R
1417:)
1415:t
1413:(
1411:2
1408:x
1403:)
1401:t
1399:(
1397:1
1394:x
1388:2
1385:x
1381:1
1378:x
1374:r
1363:(
1356:2
1344:x
1337:F
1330:F
1321:)
1319:2
1298:2
1287:x
1277:2
1273:m
1269:=
1266:)
1261:2
1256:x
1251:,
1246:1
1241:x
1236:(
1226:F
1207:)
1205:1
1184:1
1173:x
1163:1
1159:m
1155:=
1152:)
1147:2
1142:x
1137:,
1132:1
1127:x
1122:(
1112:F
1086:t
1084:(
1082:2
1079:v
1072:t
1070:(
1068:1
1065:v
1058:t
1056:(
1054:2
1051:x
1044:t
1042:(
1040:1
1037:x
1032:t
1028:)
1026:t
1024:(
1022:2
1019:x
1014:)
1012:t
1010:(
1008:1
1005:x
1000:2
997:m
993:1
990:m
985:2
982:x
976:1
973:x
966:.
949:2
945:m
941:+
936:1
932:m
928:=
925:M
903:2
898:x
888:1
883:x
878:=
874:r
851:2
846:x
839:M
834:2
830:m
824:+
819:1
814:x
807:M
802:1
798:m
792:=
788:R
756:)
752:(
734:.
590:n
584:n
518:e
511:t
504:v
386:)
382:(
233:)
224:(
96:–
83:.
38:.
20:)
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