1321:
976:
5108:
1316:{\displaystyle {\frac {d}{ds}}{\Big |}_{s=0}{\mathcal {S}}(\gamma _{s})={\Big |}_{a}^{b}{\frac {\partial L}{\partial \xi ^{i}}}X^{i}-\int _{a}^{b}{\Big (}{\frac {\partial ^{2}L}{\partial \xi ^{j}\partial \xi ^{i}}}{\ddot {\gamma }}^{j}+{\frac {\partial ^{2}L}{\partial x^{j}\partial \xi ^{i}}}{\dot {\gamma }}^{j}-{\frac {\partial L}{\partial x^{i}}}{\Big )}X^{i}dt,}
2759:
2540:
2942:
3822:
3156:
702:
2551:
5086:
3605:
4556:
4424:
4083:
2366:
1976:
2778:
3624:
2953:
960:
537:
1752:
820:
1548:
4861:
2754:{\displaystyle dE={\Big (}{\frac {\partial ^{2}L}{\partial x^{i}\partial \xi ^{j}}}\xi ^{j}-{\frac {\partial L}{\partial x^{i}}}{\Big )}dx^{i}+\xi ^{j}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}d\xi ^{i}}
1459:
4290:
3432:
4922:
2257:
1864:
3911:
4937:
3443:
3288:
2338:
4765:
4430:
4298:
2182:
2135:
4691:
1636:
2052:
3831:
for this mechanical system is a full spray. The constant speed geodesics of the underlying
Finsler (or Riemannian) manifold are described by this spray for the following reasons:
3967:
2535:{\displaystyle \iota _{H}d\alpha =Y^{i}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}dx^{j}-X^{i}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}d\xi ^{j}}
416:
2088:
2016:
1588:
4130:
3958:
1869:
2937:{\displaystyle G^{k}(x,\xi )={\frac {g^{ki}}{2}}{\Big (}{\frac {\partial ^{2}L}{\partial \xi ^{i}\partial x^{j}}}\xi ^{j}-{\frac {\partial L}{\partial x^{i}}}{\Big )}.}
3817:{\displaystyle g_{ij}(\lambda \xi )=g_{ij}(\xi ),\quad \alpha _{i}(x,\lambda \xi )=\lambda \alpha _{i}(x,\xi ),\quad G^{i}(x,\lambda \xi )=\lambda ^{2}G^{i}(x,\xi ),}
469:
3151:{\displaystyle {\frac {d}{ds}}{\Big |}_{s=0}{\mathcal {S}}(\gamma _{s})={\Big |}_{a}^{b}\alpha _{i}X^{i}-\int _{a}^{b}g_{ik}({\ddot {\gamma }}^{k}+2G^{k})X^{i}dt,}
439:
4204:
4184:
697:{\displaystyle H_{\xi }=\xi ^{i}{\frac {\partial }{\partial x^{i}}}{\Big |}_{(x,\xi )}-2G^{i}(x,\xi ){\frac {\partial }{\partial \xi ^{i}}}{\Big |}_{(x,\xi )}.}
529:
509:
489:
863:
1641:
728:
1464:
4776:
4132:
is stationary to the action integral if and only if it is of constant speed and stationary to the length functional. The
Hamiltonian vector field
1363:
96:
are precisely the tangent curves of locally length minimizing curves. Semisprays arise naturally as the extremal curves of action integrals in
4209:
3362:
4872:
2194:
1764:
3205:. Using the framework of Lagrangian mechanics one can describe these curves with spray structures. Define a Lagrangian function on
5081:{\displaystyle \epsilon |_{\xi }=\int \limits _{-\infty }^{0}e^{-s}(\Phi _{V}^{-s})_{*}(\tau \Theta V)|_{\Phi _{V}^{s}(\xi )}ds.}
3600:{\displaystyle \alpha _{i}=g_{ij}\xi ^{i},\quad F^{2}=g_{ij}\xi ^{i}\xi ^{j},\quad E=\alpha _{i}\xi ^{i}-L={\tfrac {1}{2}}F^{2}.}
3848:
5231:
3215:
3842:
is positive definite for
Finsler spaces, every short enough stationary curve for the length functional is length minimizing.
4551:{\displaystyle v:T(TM\setminus 0)\to T(TM\setminus 0)\quad ;\quad v={\tfrac {1}{2}}{\big (}I+{\mathcal {L}}_{H}J{\big )}.}
4419:{\displaystyle h:T(TM\setminus 0)\to T(TM\setminus 0)\quad ;\quad h={\tfrac {1}{2}}{\big (}I-{\mathcal {L}}_{H}J{\big )},}
2296:
4730:
3330:)ξξ. Now introduce the concepts from the section above. In the Riemannian case it turns out that the fundamental tensor
5166:
5148:
5158:
2140:
2093:
5236:
4704:\0 and the adjoint structure Θ of the induced connection the horizontal part of the semi-spray can be written as
4575:
1596:
1356:). This first variation formula can be recast in a more informative form by introducing the following concepts:
2276:
4078:{\displaystyle {\mathcal {S}}(\gamma )={\frac {(b-a)\lambda ^{2}}{2}}={\frac {\ell (\gamma )^{2}}{2(b-a)}}.}
2028:
133:
159:
853:. The dynamical law is obtained from the Hamiltonian principle, which states that the time evolution γ:→
254:
212:
163:
20:
3173:. Hence the dynamics of mechanical systems are described by semisprays arising from action integrals.
375:
97:
42:
2057:
1985:
1971:{\displaystyle g_{ij}(x,\xi )={\tfrac {\partial ^{2}L}{\partial \xi ^{i}\partial \xi ^{j}}}(x,\xi )}
1557:
4091:
3919:
3194:
831:
81:
3165:
is stationary for the action integral with fixed end points if and only if its tangent curve γ':→
4565:\0 always has a vanishing torsion tensor, which is defined as the Frölicher-Nijenhuis bracket
120:, and homogeneous torsion-free connections are in one-to-one correspondence with full sprays.
3610:
In terms of classical mechanics, the last equation states that all the energy in the system (
3306:
3198:
444:
85:
5091:
From this relation one also sees that the induced connection is homogeneous if and only if
2272:
955:{\displaystyle {\mathcal {S}}(\gamma ):=\int _{a}^{b}L(\gamma (t),{\dot {\gamma }}(t))dt}
116:. If the original connection is torsion-free it coincides with the connection induced by
421:
4189:
4169:
3182:
846:
514:
494:
474:
93:
89:
35:
5107:
1747:{\displaystyle \alpha _{\xi }=\alpha _{i}(x,\xi )dx^{i}|_{(x,\xi )}\in T_{\xi }^{*}TM}
815:{\displaystyle G^{i}(x,\lambda \xi )=\lambda ^{2}G^{i}(x,\xi ),\quad \lambda >0.\,}
49:. Usually a spray is required to be homogeneous in the sense that its integral curves
5225:
3188:
1543:{\displaystyle \alpha _{i}(x,\xi )={\tfrac {\partial L}{\partial \xi ^{i}}}(x,\xi )}
28:
4866:
of the induced non-linear connection through the ordinary differential equation
3960:
of constant speed the action integral and the length functional are related by
3618:) is in the kinetic form. Furthermore, one obtains the homogeneity properties
4856:{\displaystyle \tau ={\mathcal {L}}_{V}v={\tfrac {1}{2}}{\mathcal {L}}_{-H}J}
4292:
on the slit tangent bundle through its horizontal and vertical projections
3202:
45:
second order system of ordinary differential equations on the base manifold
1454:{\displaystyle \alpha _{\xi }=\alpha _{i}(x,\xi )dx^{i}|_{x}\in T_{x}^{*}M}
5171:
100:. Generalizing all these examples, any (possibly nonlinear) connection on
4285:{\displaystyle T(TM\setminus 0)=H(TM\setminus 0)\oplus V(TM\setminus 0)}
4927:
Therefore, the first spray invariant ε (and hence the whole semi-spray
3427:{\displaystyle F(x,\lambda \xi )=\lambda F(x,\xi ),\quad \lambda >0}
69:(ξ) in positive re-parameterizations. If this requirement is dropped,
5156:
Bucătaru, Constantinescu and Dahl relate this concept to that of the
4917:{\displaystyle {\mathcal {L}}_{V}\epsilon +\epsilon =\tau \Theta V.}
3845:
Every stationary curve for the action integral is of constant speed
5153:
2252:{\displaystyle \displaystyle E(\xi )=\alpha _{\xi }(\xi )-L(\xi )}
1859:{\displaystyle g_{\xi }=g_{ij}(x,\xi )(dx^{i}\otimes dx^{j})|_{x}}
857:
of the state of the system is stationary for the action integral
5146:
by Bucătaru and Miron. Of particular note is their concept of a
5138:
A good source for Jacobi fields of semisprays is
Section 4.4,
5102:
1334:
is the variation vector field associated with the variation γ
3827:
of which the last one says that the
Hamiltonian vector field
177:, if any of the three following equivalent conditions holds:
4879:
4821:
4789:
4524:
4392:
3973:
2994:
1017:
869:
4569:=. In more elementary terms the torsion can be defined as
3906:{\displaystyle F(\gamma (t),{\dot {\gamma }}(t))=\lambda }
830:
A physical system is modeled in
Lagrangian mechanics by a
3913:, since the energy is automatically a constant of motion.
3437:
of the
Finsler-function implies the following formulae:
5119:
3283:{\displaystyle L(x,\xi )={\tfrac {1}{2}}F^{2}(x,\xi ),}
491:
using the coordinate basis on each tangent space. Then
4807:
4497:
4365:
3573:
3241:
3169:
2947:
Now the first variational formula can be rewritten as
1905:
1497:
4940:
4931:) can be recovered from the non-linear connection by
4875:
4779:
4734:
4733:
4579:
4578:
4433:
4301:
4212:
4192:
4172:
4094:
3970:
3922:
3851:
3627:
3446:
3365:
3218:
2956:
2781:
2554:
2369:
2300:
2299:
2198:
2197:
2144:
2143:
2097:
2096:
2060:
2032:
2031:
1988:
1872:
1767:
1644:
1599:
1560:
1467:
1366:
979:
866:
731:
540:
517:
497:
477:
447:
424:
378:
4770:
The first spray invariant is related to the tension
2352:) be the components of the Hamiltonian vector field
2333:{\displaystyle \displaystyle dE=-\iota _{H}d\alpha }
970:
the first variation of the action integral reads as
285:
if any of the following equivalent conditions hold:
4760:{\displaystyle \displaystyle H=\Theta V+\epsilon .}
5080:
4916:
4855:
4759:
4685:
4550:
4418:
4284:
4198:
4178:
4124:
4077:
3952:
3905:
3816:
3599:
3426:
3282:
3150:
2936:
2753:
2534:
2332:
2251:
2176:
2129:
2082:
2046:
2010:
1970:
1858:
1746:
1630:
1582:
1542:
1453:
1315:
954:
814:
696:
523:
503:
483:
463:
433:
410:
3356:). In the general case the homogeneity condition
3021:
2975:
2926:
2832:
2660:
2566:
1289:
1117:
1044:
998:
668:
589:
112:induces a torsion-free nonlinear connection on
4540:
4510:
4408:
4378:
2263:If the Legendre condition is satisfied, then
531:if it has a local representation of the form
8:
2764:so we see that the Hamiltonian vector field
2177:{\displaystyle \displaystyle g^{ij}(x,\xi )}
2130:{\displaystyle \displaystyle g_{ij}(x,\xi )}
318:is the push-forward of the multiplication λ:
4686:{\displaystyle \displaystyle T(X,Y)=J-v-v.}
2768:is a semi-spray on the configuration space
2286:corresponding to the Hamiltonian function
1631:{\displaystyle \alpha \in \Omega ^{1}(TM)}
150:) its tangent bundle. Then a vector field
5052:
5047:
5042:
5037:
5015:
5002:
4997:
4981:
4971:
4963:
4950:
4945:
4939:
4884:
4878:
4877:
4874:
4826:
4820:
4819:
4806:
4794:
4788:
4787:
4778:
4732:
4577:
4539:
4538:
4529:
4523:
4522:
4509:
4508:
4496:
4432:
4407:
4406:
4397:
4391:
4390:
4377:
4376:
4364:
4300:
4211:
4191:
4171:
4162:Correspondence with nonlinear connections
4093:
4043:
4027:
4012:
3990:
3972:
3971:
3969:
3921:
3874:
3873:
3850:
3790:
3780:
3749:
3720:
3686:
3660:
3632:
3626:
3588:
3572:
3557:
3547:
3527:
3517:
3504:
3491:
3477:
3464:
3451:
3445:
3364:
3256:
3240:
3217:
3133:
3120:
3104:
3093:
3092:
3079:
3069:
3064:
3051:
3041:
3031:
3026:
3020:
3019:
3006:
2993:
2992:
2980:
2974:
2973:
2957:
2955:
2925:
2924:
2915:
2897:
2888:
2875:
2862:
2844:
2837:
2831:
2830:
2816:
2810:
2786:
2780:
2745:
2729:
2716:
2698:
2691:
2685:
2672:
2659:
2658:
2649:
2631:
2622:
2609:
2596:
2578:
2571:
2565:
2564:
2553:
2526:
2510:
2497:
2479:
2472:
2466:
2453:
2437:
2424:
2406:
2399:
2393:
2374:
2368:
2317:
2298:
2218:
2196:
2149:
2142:
2102:
2095:
2071:
2059:
2037:
2030:
1999:
1987:
1943:
1930:
1912:
1904:
1877:
1871:
1850:
1845:
1835:
1819:
1785:
1772:
1766:
1732:
1727:
1702:
1697:
1690:
1662:
1649:
1643:
1610:
1598:
1571:
1559:
1515:
1496:
1472:
1466:
1442:
1437:
1424:
1419:
1412:
1384:
1371:
1365:
1298:
1288:
1287:
1278:
1260:
1251:
1240:
1239:
1229:
1216:
1198:
1191:
1182:
1171:
1170:
1160:
1147:
1129:
1122:
1116:
1115:
1109:
1104:
1091:
1078:
1060:
1054:
1049:
1043:
1042:
1029:
1016:
1015:
1003:
997:
996:
980:
978:
923:
922:
895:
890:
868:
867:
865:
811:
777:
767:
736:
730:
673:
667:
666:
656:
643:
622:
594:
588:
587:
577:
564:
558:
545:
539:
516:
496:
476:
455:
446:
423:
399:
386:
377:
3193:The locally length minimizing curves of
707:on each associated coordinate system on
5183:
5099:Jacobi fields of sprays and semi-sprays
4696:Introducing the canonical vector field
4476:
4452:
4344:
4320:
4273:
4249:
4225:
2047:{\displaystyle \displaystyle g_{\xi }}
715:is a (full) spray, if and only if the
441:associated with the local coordinates
5214:Fundamentals of Differential Geometry
7:
5172:What is Kosambi-Cartan-Chern theory?
3343:,ξ) is simply the Riemannian metric
826:Semi-sprays in Lagrangian mechanics
5044:
5027:
4994:
4967:
4905:
4741:
3309:. In the Riemannian case one uses
2908:
2900:
2868:
2855:
2841:
2722:
2709:
2695:
2642:
2634:
2602:
2589:
2575:
2503:
2490:
2476:
2430:
2417:
2403:
2191:associated with the Lagrangian is
1936:
1923:
1909:
1607:
1508:
1500:
1271:
1263:
1222:
1209:
1195:
1153:
1140:
1126:
1071:
1063:
649:
645:
570:
566:
14:
5204:Lectures on Differential Geometry
5194:, Editura Academiei Române, 2007.
4716:of the semispray is known as the
2356:in the associated coordinates on
966:In the associated coordinates on
333:along the canonical vector field
223:is the canonical vector field on
5140:Jacobi equations of a semi-spray
5106:
4206:defines an Ehresmann-connection
411:{\displaystyle (x^{i},\xi ^{i})}
108:, and conversely, any semispray
5142:of the publicly available book
4489:
4485:
4357:
4353:
3744:
3681:
3536:
3486:
3414:
1758:associated with the Lagrangian.
801:
5159:Kosambi bi-derivative operator
5064:
5058:
5038:
5033:
5021:
5012:
4990:
4946:
4839:
4827:
4676:
4658:
4649:
4631:
4622:
4604:
4595:
4583:
4482:
4467:
4461:
4458:
4443:
4350:
4335:
4329:
4326:
4311:
4279:
4264:
4255:
4240:
4231:
4216:
4148:) and the corresponding flow Φ
4116:
4113:
4101:
4066:
4054:
4040:
4033:
4005:
3993:
3984:
3978:
3944:
3941:
3929:
3894:
3891:
3885:
3867:
3861:
3855:
3808:
3796:
3770:
3755:
3738:
3726:
3707:
3692:
3675:
3669:
3650:
3641:
3408:
3396:
3384:
3369:
3274:
3262:
3234:
3222:
3126:
3088:
3012:
2999:
2804:
2792:
2245:
2239:
2230:
2224:
2208:
2202:
2170:
2158:
2123:
2111:
2083:{\displaystyle \xi \in T_{x}M}
2011:{\displaystyle \xi \in T_{x}M}
1965:
1953:
1898:
1886:
1846:
1841:
1809:
1806:
1794:
1715:
1703:
1698:
1680:
1668:
1625:
1616:
1583:{\displaystyle \xi \in T_{x}M}
1537:
1525:
1490:
1478:
1420:
1402:
1390:
1035:
1022:
943:
940:
934:
916:
910:
904:
880:
874:
795:
783:
757:
742:
686:
674:
640:
628:
607:
595:
448:
405:
379:
16:Vector field on tangent bundle
1:
5169:'s methods, see the article,
4125:{\displaystyle \gamma :\to M}
3953:{\displaystyle \gamma :\to M}
2090:. Then the inverse matrix of
2021:The Lagrangian satisfies the
5149:dynamic covariant derivative
2772:with the spray coefficients
2275:, and there exists a unique
849:of some configuration space
418:be the local coordinates on
326:by a positive scalar λ>0.
5165:For a good introduction to
2054:is non-degenerate at every
1593:The corresponding one-form
5253:
5202:Sternberg, Shlomo (1964),
3186:
3180:
2025:if the fundamental tensor
80:Sprays arise naturally in
5192:Finsler-Lagrange Geometry
5144:Finsler-Lagrange Geometry
4140:of the Finsler manifold (
2277:Hamiltonian vector field
5190:I. Bucataru, R. Miron,
4724:itself decomposes into
134:differentiable manifold
5082:
4976:
4918:
4857:
4761:
4712:. The vertical part ε=
4687:
4552:
4420:
4286:
4200:
4180:
4126:
4079:
3954:
3907:
3818:
3601:
3428:
3284:
3152:
2938:
2755:
2536:
2334:
2253:
2178:
2131:
2084:
2048:
2012:
1972:
1860:
1748:
1632:
1584:
1544:
1455:
1317:
956:
816:
698:
525:
505:
485:
465:
464:{\displaystyle (x^{i}}
435:
412:
329:The Lie-derivative of
5232:Differential geometry
5083:
4959:
4919:
4858:
4762:
4718:first spray invariant
4688:
4553:
4421:
4287:
4201:
4186:on a smooth manifold
4181:
4127:
4080:
3955:
3908:
3819:
3602:
3429:
3285:
3187:Further information:
3153:
2939:
2756:
2537:
2335:
2254:
2179:
2132:
2085:
2049:
2013:
1982:of the Lagrangian at
1973:
1861:
1749:
1633:
1585:
1545:
1456:
1318:
957:
817:
699:
526:
506:
486:
466:
436:
413:
261:is seen as a mapping
164:double tangent bundle
21:differential geometry
5212:Lang, Serge (1999),
4938:
4873:
4777:
4731:
4720:, and the semispray
4576:
4431:
4299:
4210:
4190:
4170:
4092:
3968:
3920:
3849:
3625:
3444:
3363:
3216:
2954:
2779:
2552:
2367:
2297:
2195:
2141:
2094:
2058:
2029:
1986:
1870:
1765:
1642:
1597:
1558:
1465:
1364:
977:
864:
729:
538:
515:
495:
475:
445:
422:
376:
344:The integral curves
104:induces a semispray
98:Lagrangian mechanics
5057:
5010:
4561:This connection on
4088:Therefore, a curve
3074:
3036:
1737:
1447:
1114:
1059:
900:
832:Lagrangian function
511:is a semi-spray on
368:(ξ) for any λ>0.
5118:. You can help by
5078:
5043:
4993:
4914:
4853:
4816:
4757:
4756:
4683:
4682:
4548:
4506:
4416:
4374:
4282:
4196:
4176:
4154:(ξ) is called the
4122:
4075:
3950:
3903:
3814:
3597:
3582:
3424:
3280:
3250:
3148:
3060:
3018:
2934:
2751:
2532:
2330:
2329:
2249:
2248:
2174:
2173:
2127:
2126:
2080:
2044:
2043:
2023:Legendre condition
2008:
1980:fundamental tensor
1968:
1951:
1856:
1761:The bilinear form
1744:
1723:
1628:
1580:
1552:conjugate momentum
1540:
1523:
1451:
1433:
1313:
1100:
1041:
952:
886:
812:
717:spray coefficients
694:
521:
501:
481:
461:
434:{\displaystyle TM}
431:
408:
124:Formal definitions
5216:, Springer-Verlag
5136:
5135:
5095:is a full spray.
4815:
4505:
4373:
4199:{\displaystyle M}
4179:{\displaystyle H}
4070:
4022:
3882:
3581:
3249:
3199:Finsler manifolds
3101:
2970:
2922:
2882:
2828:
2736:
2656:
2616:
2517:
2444:
1950:
1522:
1285:
1248:
1236:
1179:
1167:
1085:
993:
931:
663:
584:
524:{\displaystyle M}
504:{\displaystyle H}
484:{\displaystyle M}
213:tangent structure
5244:
5237:Finsler geometry
5217:
5207:
5195:
5188:
5131:
5128:
5110:
5103:
5087:
5085:
5084:
5079:
5068:
5067:
5056:
5051:
5041:
5020:
5019:
5009:
5001:
4989:
4988:
4975:
4970:
4955:
4954:
4949:
4923:
4921:
4920:
4915:
4889:
4888:
4883:
4882:
4862:
4860:
4859:
4854:
4849:
4848:
4825:
4824:
4817:
4808:
4799:
4798:
4793:
4792:
4766:
4764:
4763:
4758:
4692:
4690:
4689:
4684:
4557:
4555:
4554:
4549:
4544:
4543:
4534:
4533:
4528:
4527:
4514:
4513:
4507:
4498:
4425:
4423:
4422:
4417:
4412:
4411:
4402:
4401:
4396:
4395:
4382:
4381:
4375:
4366:
4291:
4289:
4288:
4283:
4205:
4203:
4202:
4197:
4185:
4183:
4182:
4177:
4131:
4129:
4128:
4123:
4084:
4082:
4081:
4076:
4071:
4069:
4049:
4048:
4047:
4028:
4023:
4018:
4017:
4016:
3991:
3977:
3976:
3959:
3957:
3956:
3951:
3912:
3910:
3909:
3904:
3884:
3883:
3875:
3823:
3821:
3820:
3815:
3795:
3794:
3785:
3784:
3754:
3753:
3725:
3724:
3691:
3690:
3668:
3667:
3640:
3639:
3606:
3604:
3603:
3598:
3593:
3592:
3583:
3574:
3562:
3561:
3552:
3551:
3532:
3531:
3522:
3521:
3512:
3511:
3496:
3495:
3482:
3481:
3472:
3471:
3456:
3455:
3433:
3431:
3430:
3425:
3307:Finsler function
3289:
3287:
3286:
3281:
3261:
3260:
3251:
3242:
3157:
3155:
3154:
3149:
3138:
3137:
3125:
3124:
3109:
3108:
3103:
3102:
3094:
3087:
3086:
3073:
3068:
3056:
3055:
3046:
3045:
3035:
3030:
3025:
3024:
3011:
3010:
2998:
2997:
2991:
2990:
2979:
2978:
2971:
2969:
2958:
2943:
2941:
2940:
2935:
2930:
2929:
2923:
2921:
2920:
2919:
2906:
2898:
2893:
2892:
2883:
2881:
2880:
2879:
2867:
2866:
2853:
2849:
2848:
2838:
2836:
2835:
2829:
2824:
2823:
2811:
2791:
2790:
2760:
2758:
2757:
2752:
2750:
2749:
2737:
2735:
2734:
2733:
2721:
2720:
2707:
2703:
2702:
2692:
2690:
2689:
2677:
2676:
2664:
2663:
2657:
2655:
2654:
2653:
2640:
2632:
2627:
2626:
2617:
2615:
2614:
2613:
2601:
2600:
2587:
2583:
2582:
2572:
2570:
2569:
2541:
2539:
2538:
2533:
2531:
2530:
2518:
2516:
2515:
2514:
2502:
2501:
2488:
2484:
2483:
2473:
2471:
2470:
2458:
2457:
2445:
2443:
2442:
2441:
2429:
2428:
2415:
2411:
2410:
2400:
2398:
2397:
2379:
2378:
2339:
2337:
2336:
2331:
2322:
2321:
2258:
2256:
2255:
2250:
2223:
2222:
2183:
2181:
2180:
2175:
2157:
2156:
2136:
2134:
2133:
2128:
2110:
2109:
2089:
2087:
2086:
2081:
2076:
2075:
2053:
2051:
2050:
2045:
2042:
2041:
2017:
2015:
2014:
2009:
2004:
2003:
1977:
1975:
1974:
1969:
1952:
1949:
1948:
1947:
1935:
1934:
1921:
1917:
1916:
1906:
1885:
1884:
1865:
1863:
1862:
1857:
1855:
1854:
1849:
1840:
1839:
1824:
1823:
1793:
1792:
1777:
1776:
1753:
1751:
1750:
1745:
1736:
1731:
1719:
1718:
1701:
1695:
1694:
1667:
1666:
1654:
1653:
1637:
1635:
1634:
1629:
1615:
1614:
1589:
1587:
1586:
1581:
1576:
1575:
1549:
1547:
1546:
1541:
1524:
1521:
1520:
1519:
1506:
1498:
1477:
1476:
1460:
1458:
1457:
1452:
1446:
1441:
1429:
1428:
1423:
1417:
1416:
1389:
1388:
1376:
1375:
1322:
1320:
1319:
1314:
1303:
1302:
1293:
1292:
1286:
1284:
1283:
1282:
1269:
1261:
1256:
1255:
1250:
1249:
1241:
1237:
1235:
1234:
1233:
1221:
1220:
1207:
1203:
1202:
1192:
1187:
1186:
1181:
1180:
1172:
1168:
1166:
1165:
1164:
1152:
1151:
1138:
1134:
1133:
1123:
1121:
1120:
1113:
1108:
1096:
1095:
1086:
1084:
1083:
1082:
1069:
1061:
1058:
1053:
1048:
1047:
1034:
1033:
1021:
1020:
1014:
1013:
1002:
1001:
994:
992:
981:
961:
959:
958:
953:
933:
932:
924:
899:
894:
873:
872:
821:
819:
818:
813:
782:
781:
772:
771:
741:
740:
711:. The semispray
703:
701:
700:
695:
690:
689:
672:
671:
664:
662:
661:
660:
644:
627:
626:
611:
610:
593:
592:
585:
583:
582:
581:
565:
563:
562:
550:
549:
530:
528:
527:
522:
510:
508:
507:
502:
490:
488:
487:
482:
470:
468:
467:
462:
460:
459:
440:
438:
437:
432:
417:
415:
414:
409:
404:
403:
391:
390:
86:Finsler geometry
5252:
5251:
5247:
5246:
5245:
5243:
5242:
5241:
5222:
5221:
5211:
5206:, Prentice-Hall
5201:
5198:
5189:
5185:
5181:
5132:
5126:
5123:
5116:needs expansion
5101:
5036:
5011:
4977:
4944:
4936:
4935:
4876:
4871:
4870:
4818:
4786:
4775:
4774:
4729:
4728:
4574:
4573:
4521:
4429:
4428:
4389:
4297:
4296:
4208:
4207:
4188:
4187:
4168:
4167:
4164:
4153:
4090:
4089:
4050:
4039:
4029:
4008:
3992:
3966:
3965:
3918:
3917:
3847:
3846:
3841:
3786:
3776:
3745:
3716:
3682:
3656:
3628:
3623:
3622:
3584:
3553:
3543:
3523:
3513:
3500:
3487:
3473:
3460:
3447:
3442:
3441:
3361:
3360:
3351:
3338:
3325:
3252:
3214:
3213:
3191:
3185:
3179:
3177:Geodesic spray
3129:
3116:
3091:
3075:
3047:
3037:
3002:
2972:
2962:
2952:
2951:
2911:
2907:
2899:
2884:
2871:
2858:
2854:
2840:
2839:
2812:
2782:
2777:
2776:
2741:
2725:
2712:
2708:
2694:
2693:
2681:
2668:
2645:
2641:
2633:
2618:
2605:
2592:
2588:
2574:
2573:
2550:
2549:
2522:
2506:
2493:
2489:
2475:
2474:
2462:
2449:
2433:
2420:
2416:
2402:
2401:
2389:
2370:
2365:
2364:
2313:
2295:
2294:
2273:symplectic form
2214:
2193:
2192:
2145:
2139:
2138:
2098:
2092:
2091:
2067:
2056:
2055:
2033:
2027:
2026:
1995:
1984:
1983:
1939:
1926:
1922:
1908:
1907:
1873:
1868:
1867:
1844:
1831:
1815:
1781:
1768:
1763:
1762:
1696:
1686:
1658:
1645:
1640:
1639:
1606:
1595:
1594:
1567:
1556:
1555:
1511:
1507:
1499:
1468:
1463:
1462:
1418:
1408:
1380:
1367:
1362:
1361:
1351:
1339:
1294:
1274:
1270:
1262:
1238:
1225:
1212:
1208:
1194:
1193:
1169:
1156:
1143:
1139:
1125:
1124:
1087:
1074:
1070:
1062:
1025:
995:
985:
975:
974:
862:
861:
828:
773:
763:
732:
727:
726:
665:
652:
648:
618:
586:
573:
569:
554:
541:
536:
535:
513:
512:
493:
492:
473:
472:
451:
443:
442:
420:
419:
395:
382:
374:
373:
367:
363:
351:
309:
305:
298:
294:
196:
190:
186:
145:
126:
94:integral curves
90:geodesic sprays
68:
64:
61:obey the rule Φ
56:
41:that encodes a
17:
12:
11:
5:
5250:
5248:
5240:
5239:
5234:
5224:
5223:
5220:
5219:
5209:
5197:
5196:
5182:
5180:
5177:
5134:
5133:
5113:
5111:
5100:
5097:
5089:
5088:
5077:
5074:
5071:
5066:
5063:
5060:
5055:
5050:
5046:
5040:
5035:
5032:
5029:
5026:
5023:
5018:
5014:
5008:
5005:
5000:
4996:
4992:
4987:
4984:
4980:
4974:
4969:
4966:
4962:
4958:
4953:
4948:
4943:
4925:
4924:
4913:
4910:
4907:
4904:
4901:
4898:
4895:
4892:
4887:
4881:
4864:
4863:
4852:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4823:
4814:
4811:
4805:
4802:
4797:
4791:
4785:
4782:
4768:
4767:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4694:
4693:
4681:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4559:
4558:
4547:
4542:
4537:
4532:
4526:
4520:
4517:
4512:
4504:
4501:
4495:
4492:
4488:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4454:
4451:
4448:
4445:
4442:
4439:
4436:
4426:
4415:
4410:
4405:
4400:
4394:
4388:
4385:
4380:
4372:
4369:
4363:
4360:
4356:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4304:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4195:
4175:
4163:
4160:
4149:
4138:geodesic spray
4136:is called the
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4086:
4085:
4074:
4068:
4065:
4062:
4059:
4056:
4053:
4046:
4042:
4038:
4035:
4032:
4026:
4021:
4015:
4011:
4007:
4004:
4001:
3998:
3995:
3989:
3986:
3983:
3980:
3975:
3962:
3961:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3916:For any curve
3914:
3902:
3899:
3896:
3893:
3890:
3887:
3881:
3878:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3843:
3839:
3825:
3824:
3813:
3810:
3807:
3804:
3801:
3798:
3793:
3789:
3783:
3779:
3775:
3772:
3769:
3766:
3763:
3760:
3757:
3752:
3748:
3743:
3740:
3737:
3734:
3731:
3728:
3723:
3719:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3689:
3685:
3680:
3677:
3674:
3671:
3666:
3663:
3659:
3655:
3652:
3649:
3646:
3643:
3638:
3635:
3631:
3608:
3607:
3596:
3591:
3587:
3580:
3577:
3571:
3568:
3565:
3560:
3556:
3550:
3546:
3542:
3539:
3535:
3530:
3526:
3520:
3516:
3510:
3507:
3503:
3499:
3494:
3490:
3485:
3480:
3476:
3470:
3467:
3463:
3459:
3454:
3450:
3435:
3434:
3423:
3420:
3417:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3347:
3334:
3321:
3291:
3290:
3279:
3276:
3273:
3270:
3267:
3264:
3259:
3255:
3248:
3245:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3183:Geodesic spray
3181:Main article:
3178:
3175:
3159:
3158:
3147:
3144:
3141:
3136:
3132:
3128:
3123:
3119:
3115:
3112:
3107:
3100:
3097:
3090:
3085:
3082:
3078:
3072:
3067:
3063:
3059:
3054:
3050:
3044:
3040:
3034:
3029:
3023:
3017:
3014:
3009:
3005:
3001:
2996:
2989:
2986:
2983:
2977:
2968:
2965:
2961:
2945:
2944:
2933:
2928:
2918:
2914:
2910:
2905:
2902:
2896:
2891:
2887:
2878:
2874:
2870:
2865:
2861:
2857:
2852:
2847:
2843:
2834:
2827:
2822:
2819:
2815:
2809:
2806:
2803:
2800:
2797:
2794:
2789:
2785:
2762:
2761:
2748:
2744:
2740:
2732:
2728:
2724:
2719:
2715:
2711:
2706:
2701:
2697:
2688:
2684:
2680:
2675:
2671:
2667:
2662:
2652:
2648:
2644:
2639:
2636:
2630:
2625:
2621:
2612:
2608:
2604:
2599:
2595:
2591:
2586:
2581:
2577:
2568:
2563:
2560:
2557:
2543:
2542:
2529:
2525:
2521:
2513:
2509:
2505:
2500:
2496:
2492:
2487:
2482:
2478:
2469:
2465:
2461:
2456:
2452:
2448:
2440:
2436:
2432:
2427:
2423:
2419:
2414:
2409:
2405:
2396:
2392:
2388:
2385:
2382:
2377:
2373:
2342:
2341:
2328:
2325:
2320:
2316:
2312:
2309:
2306:
2303:
2261:
2260:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2221:
2217:
2213:
2210:
2207:
2204:
2201:
2185:
2172:
2169:
2166:
2163:
2160:
2155:
2152:
2148:
2137:is denoted by
2125:
2122:
2119:
2116:
2113:
2108:
2105:
2101:
2079:
2074:
2070:
2066:
2063:
2040:
2036:
2019:
2007:
2002:
1998:
1994:
1991:
1967:
1964:
1961:
1958:
1955:
1946:
1942:
1938:
1933:
1929:
1925:
1920:
1915:
1911:
1903:
1900:
1897:
1894:
1891:
1888:
1883:
1880:
1876:
1853:
1848:
1843:
1838:
1834:
1830:
1827:
1822:
1818:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1791:
1788:
1784:
1780:
1775:
1771:
1759:
1743:
1740:
1735:
1730:
1726:
1722:
1717:
1714:
1711:
1708:
1705:
1700:
1693:
1689:
1685:
1682:
1679:
1676:
1673:
1670:
1665:
1661:
1657:
1652:
1648:
1627:
1624:
1621:
1618:
1613:
1609:
1605:
1602:
1591:
1579:
1574:
1570:
1566:
1563:
1539:
1536:
1533:
1530:
1527:
1518:
1514:
1510:
1505:
1502:
1495:
1492:
1489:
1486:
1483:
1480:
1475:
1471:
1450:
1445:
1440:
1436:
1432:
1427:
1422:
1415:
1411:
1407:
1404:
1401:
1398:
1395:
1392:
1387:
1383:
1379:
1374:
1370:
1349:
1335:
1324:
1323:
1312:
1309:
1306:
1301:
1297:
1291:
1281:
1277:
1273:
1268:
1265:
1259:
1254:
1247:
1244:
1232:
1228:
1224:
1219:
1215:
1211:
1206:
1201:
1197:
1190:
1185:
1178:
1175:
1163:
1159:
1155:
1150:
1146:
1142:
1137:
1132:
1128:
1119:
1112:
1107:
1103:
1099:
1094:
1090:
1081:
1077:
1073:
1068:
1065:
1057:
1052:
1046:
1040:
1037:
1032:
1028:
1024:
1019:
1012:
1009:
1006:
1000:
991:
988:
984:
964:
963:
951:
948:
945:
942:
939:
936:
930:
927:
921:
918:
915:
912:
909:
906:
903:
898:
893:
889:
885:
882:
879:
876:
871:
847:tangent bundle
827:
824:
823:
822:
810:
807:
804:
800:
797:
794:
791:
788:
785:
780:
776:
770:
766:
762:
759:
756:
753:
750:
747:
744:
739:
735:
705:
704:
693:
688:
685:
682:
679:
676:
670:
659:
655:
651:
647:
642:
639:
636:
633:
630:
625:
621:
617:
614:
609:
606:
603:
600:
597:
591:
580:
576:
572:
568:
561:
557:
553:
548:
544:
520:
500:
480:
458:
454:
450:
430:
427:
407:
402:
398:
394:
389:
385:
381:
370:
369:
365:
361:
349:
342:
327:
307:
303:
296:
292:
271:
270:
255:canonical flip
228:
198:
194:
188:
182:
141:
125:
122:
66:
62:
54:
36:tangent bundle
15:
13:
10:
9:
6:
4:
3:
2:
5249:
5238:
5235:
5233:
5230:
5229:
5227:
5215:
5210:
5205:
5200:
5199:
5193:
5187:
5184:
5178:
5176:
5174:
5173:
5168:
5163:
5161:
5160:
5155:
5154:another paper
5151:
5150:
5145:
5141:
5130:
5127:February 2013
5121:
5117:
5114:This section
5112:
5109:
5105:
5104:
5098:
5096:
5094:
5075:
5072:
5069:
5061:
5053:
5048:
5030:
5024:
5016:
5006:
5003:
4998:
4985:
4982:
4978:
4972:
4964:
4960:
4956:
4951:
4941:
4934:
4933:
4932:
4930:
4911:
4908:
4902:
4899:
4896:
4893:
4890:
4885:
4869:
4868:
4867:
4850:
4845:
4842:
4836:
4833:
4830:
4812:
4809:
4803:
4800:
4795:
4783:
4780:
4773:
4772:
4771:
4753:
4750:
4747:
4744:
4738:
4735:
4727:
4726:
4725:
4723:
4719:
4715:
4711:
4707:
4703:
4699:
4679:
4673:
4670:
4667:
4664:
4661:
4655:
4652:
4646:
4643:
4640:
4637:
4634:
4628:
4625:
4619:
4616:
4613:
4610:
4607:
4601:
4598:
4592:
4589:
4586:
4580:
4572:
4571:
4570:
4568:
4564:
4545:
4535:
4530:
4518:
4515:
4502:
4499:
4493:
4490:
4486:
4479:
4473:
4470:
4464:
4455:
4449:
4446:
4440:
4437:
4434:
4427:
4413:
4403:
4398:
4386:
4383:
4370:
4367:
4361:
4358:
4354:
4347:
4341:
4338:
4332:
4323:
4317:
4314:
4308:
4305:
4302:
4295:
4294:
4293:
4276:
4270:
4267:
4261:
4258:
4252:
4246:
4243:
4237:
4234:
4228:
4222:
4219:
4213:
4193:
4173:
4166:A semi-spray
4161:
4159:
4157:
4156:geodesic flow
4152:
4147:
4143:
4139:
4135:
4119:
4110:
4107:
4104:
4098:
4095:
4072:
4063:
4060:
4057:
4051:
4044:
4036:
4030:
4024:
4019:
4013:
4009:
4002:
3999:
3996:
3987:
3981:
3964:
3963:
3947:
3938:
3935:
3932:
3926:
3923:
3915:
3900:
3897:
3888:
3879:
3876:
3870:
3864:
3858:
3852:
3844:
3838:
3834:
3833:
3832:
3830:
3811:
3805:
3802:
3799:
3791:
3787:
3781:
3777:
3773:
3767:
3764:
3761:
3758:
3750:
3746:
3741:
3735:
3732:
3729:
3721:
3717:
3713:
3710:
3704:
3701:
3698:
3695:
3687:
3683:
3678:
3672:
3664:
3661:
3657:
3653:
3647:
3644:
3636:
3633:
3629:
3621:
3620:
3619:
3617:
3613:
3594:
3589:
3585:
3578:
3575:
3569:
3566:
3563:
3558:
3554:
3548:
3544:
3540:
3537:
3533:
3528:
3524:
3518:
3514:
3508:
3505:
3501:
3497:
3492:
3488:
3483:
3478:
3474:
3468:
3465:
3461:
3457:
3452:
3448:
3440:
3439:
3438:
3421:
3418:
3415:
3411:
3405:
3402:
3399:
3393:
3390:
3387:
3381:
3378:
3375:
3372:
3366:
3359:
3358:
3357:
3355:
3350:
3346:
3342:
3337:
3333:
3329:
3324:
3320:
3316:
3312:
3308:
3304:
3300:
3296:
3277:
3271:
3268:
3265:
3257:
3253:
3246:
3243:
3237:
3231:
3228:
3225:
3219:
3212:
3211:
3210:
3208:
3204:
3200:
3196:
3190:
3189:Geodesic flow
3184:
3176:
3174:
3172:
3168:
3164:
3161:and we see γ→
3145:
3142:
3139:
3134:
3130:
3121:
3117:
3113:
3110:
3105:
3098:
3095:
3083:
3080:
3076:
3070:
3065:
3061:
3057:
3052:
3048:
3042:
3038:
3032:
3027:
3015:
3007:
3003:
2987:
2984:
2981:
2966:
2963:
2959:
2950:
2949:
2948:
2931:
2916:
2912:
2903:
2894:
2889:
2885:
2876:
2872:
2863:
2859:
2850:
2845:
2825:
2820:
2817:
2813:
2807:
2801:
2798:
2795:
2787:
2783:
2775:
2774:
2773:
2771:
2767:
2746:
2742:
2738:
2730:
2726:
2717:
2713:
2704:
2699:
2686:
2682:
2678:
2673:
2669:
2665:
2650:
2646:
2637:
2628:
2623:
2619:
2610:
2606:
2597:
2593:
2584:
2579:
2561:
2558:
2555:
2548:
2547:
2546:
2527:
2523:
2519:
2511:
2507:
2498:
2494:
2485:
2480:
2467:
2463:
2459:
2454:
2450:
2446:
2438:
2434:
2425:
2421:
2412:
2407:
2394:
2390:
2386:
2383:
2380:
2375:
2371:
2363:
2362:
2361:
2359:
2355:
2351:
2347:
2326:
2323:
2318:
2314:
2310:
2307:
2304:
2301:
2293:
2292:
2291:
2289:
2285:
2281:
2278:
2274:
2270:
2266:
2242:
2236:
2233:
2227:
2219:
2215:
2211:
2205:
2199:
2190:
2186:
2167:
2164:
2161:
2153:
2150:
2146:
2120:
2117:
2114:
2106:
2103:
2099:
2077:
2072:
2068:
2064:
2061:
2038:
2034:
2024:
2020:
2005:
2000:
1996:
1992:
1989:
1981:
1962:
1959:
1956:
1944:
1940:
1931:
1927:
1918:
1913:
1901:
1895:
1892:
1889:
1881:
1878:
1874:
1851:
1836:
1832:
1828:
1825:
1820:
1816:
1812:
1803:
1800:
1797:
1789:
1786:
1782:
1778:
1773:
1769:
1760:
1757:
1741:
1738:
1733:
1728:
1724:
1720:
1712:
1709:
1706:
1691:
1687:
1683:
1677:
1674:
1671:
1663:
1659:
1655:
1650:
1646:
1622:
1619:
1611:
1603:
1600:
1592:
1577:
1572:
1568:
1564:
1561:
1553:
1534:
1531:
1528:
1516:
1512:
1503:
1493:
1487:
1484:
1481:
1473:
1469:
1448:
1443:
1438:
1434:
1430:
1425:
1413:
1409:
1405:
1399:
1396:
1393:
1385:
1381:
1377:
1372:
1368:
1360:The covector
1359:
1358:
1357:
1355:
1347:
1343:
1338:
1333:
1329:
1310:
1307:
1304:
1299:
1295:
1279:
1275:
1266:
1257:
1252:
1245:
1242:
1230:
1226:
1217:
1213:
1204:
1199:
1188:
1183:
1176:
1173:
1161:
1157:
1148:
1144:
1135:
1130:
1110:
1105:
1101:
1097:
1092:
1088:
1079:
1075:
1066:
1055:
1050:
1038:
1030:
1026:
1010:
1007:
1004:
989:
986:
982:
973:
972:
971:
969:
949:
946:
937:
928:
925:
919:
913:
907:
901:
896:
891:
887:
883:
877:
860:
859:
858:
856:
852:
848:
844:
840:
836:
833:
825:
808:
805:
802:
798:
792:
789:
786:
778:
774:
768:
764:
760:
754:
751:
748:
745:
737:
733:
725:
724:
723:
721:
718:
714:
710:
691:
683:
680:
677:
657:
653:
637:
634:
631:
623:
619:
615:
612:
604:
601:
598:
578:
574:
559:
555:
551:
546:
542:
534:
533:
532:
518:
498:
478:
456:
452:
428:
425:
400:
396:
392:
387:
383:
359:
355:
347:
343:
340:
336:
332:
328:
325:
321:
317:
313:
302:
291:
288:
287:
286:
284:
280:
276:
268:
264:
260:
256:
252:
248:
244:
240:
236:
232:
229:
226:
222:
218:
214:
210:
206:
202:
199:
193:
185:
180:
179:
178:
176:
172:
168:
165:
161:
157:
153:
149:
144:
139:
135:
131:
123:
121:
119:
115:
111:
107:
103:
99:
95:
91:
87:
83:
78:
76:
72:
60:
52:
48:
44:
40:
37:
33:
30:
26:
22:
5213:
5203:
5191:
5186:
5170:
5164:
5157:
5147:
5143:
5139:
5137:
5124:
5120:adding to it
5115:
5092:
5090:
4928:
4926:
4865:
4769:
4721:
4717:
4713:
4709:
4705:
4701:
4697:
4695:
4566:
4562:
4560:
4165:
4155:
4150:
4145:
4141:
4137:
4133:
4087:
3836:
3828:
3826:
3615:
3611:
3609:
3436:
3353:
3348:
3344:
3340:
3335:
3331:
3327:
3322:
3318:
3314:
3310:
3302:
3298:
3294:
3292:
3206:
3192:
3170:
3166:
3162:
3160:
2946:
2769:
2765:
2763:
2544:
2357:
2353:
2349:
2345:
2343:
2287:
2283:
2279:
2268:
2264:
2262:
2188:
2022:
1979:
1756:Hilbert-form
1755:
1551:
1353:
1345:
1341:
1336:
1331:
1327:
1325:
967:
965:
854:
850:
842:
838:
834:
829:
719:
716:
712:
708:
706:
371:
357:
353:
345:
338:
334:
330:
323:
319:
315:
311:
300:
289:
283:(full) spray
282:
278:
274:
273:A semispray
272:
266:
262:
258:
250:
246:
242:
238:
234:
230:
224:
220:
216:
208:
204:
200:
191:
183:
174:
170:
166:
158:(that is, a
155:
151:
147:
142:
137:
129:
127:
117:
113:
109:
105:
101:
79:
74:
73:is called a
70:
58:
50:
46:
38:
31:
29:vector field
24:
18:
3201:are called
337:satisfies =
43:quasilinear
5226:Categories
5179:References
3195:Riemannian
2290:such that
306:), where λ
171:semi-spray
82:Riemannian
75:semi-spray
5062:ξ
5045:Φ
5028:Θ
5025:τ
5017:∗
5004:−
4995:Φ
4983:−
4968:∞
4965:−
4961:∫
4952:ξ
4942:ϵ
4906:Θ
4903:τ
4897:ϵ
4891:ϵ
4843:−
4781:τ
4751:ϵ
4742:Θ
4653:−
4626:−
4477:∖
4462:→
4453:∖
4387:−
4345:∖
4330:→
4321:∖
4274:∖
4259:⊕
4250:∖
4226:∖
4117:→
4096:γ
4061:−
4037:γ
4031:ℓ
4010:λ
4000:−
3982:γ
3945:→
3924:γ
3901:λ
3880:˙
3877:γ
3859:γ
3806:ξ
3778:λ
3768:ξ
3765:λ
3736:ξ
3718:α
3714:λ
3705:ξ
3702:λ
3684:α
3673:ξ
3648:ξ
3645:λ
3564:−
3555:ξ
3545:α
3525:ξ
3515:ξ
3475:ξ
3449:α
3416:λ
3406:ξ
3391:λ
3382:ξ
3379:λ
3272:ξ
3232:ξ
3203:geodesics
3099:¨
3096:γ
3062:∫
3058:−
3039:α
3004:γ
2909:∂
2901:∂
2895:−
2886:ξ
2869:∂
2860:ξ
2856:∂
2842:∂
2802:ξ
2743:ξ
2723:∂
2714:ξ
2710:∂
2696:∂
2683:ξ
2643:∂
2635:∂
2629:−
2620:ξ
2607:ξ
2603:∂
2590:∂
2576:∂
2524:ξ
2504:∂
2495:ξ
2491:∂
2477:∂
2460:−
2431:∂
2422:ξ
2418:∂
2404:∂
2384:α
2372:ι
2327:α
2315:ι
2311:−
2243:ξ
2234:−
2228:ξ
2220:ξ
2216:α
2206:ξ
2168:ξ
2121:ξ
2065:∈
2062:ξ
2039:ξ
1993:∈
1990:ξ
1963:ξ
1941:ξ
1937:∂
1928:ξ
1924:∂
1910:∂
1896:ξ
1826:⊗
1804:ξ
1774:ξ
1734:∗
1729:ξ
1721:∈
1713:ξ
1678:ξ
1660:α
1651:ξ
1647:α
1608:Ω
1604:∈
1601:α
1565:∈
1562:ξ
1535:ξ
1513:ξ
1509:∂
1501:∂
1488:ξ
1470:α
1444:∗
1431:∈
1400:ξ
1382:α
1373:ξ
1369:α
1344:around γ(
1272:∂
1264:∂
1258:−
1246:˙
1243:γ
1227:ξ
1223:∂
1210:∂
1196:∂
1177:¨
1174:γ
1158:ξ
1154:∂
1145:ξ
1141:∂
1127:∂
1102:∫
1098:−
1076:ξ
1072:∂
1064:∂
1027:γ
929:˙
926:γ
908:γ
888:∫
878:γ
803:λ
793:ξ
765:λ
755:ξ
752:λ
684:ξ
654:ξ
650:∂
646:∂
638:ξ
613:−
605:ξ
571:∂
567:∂
556:ξ
547:ξ
397:ξ
360:satisfy Φ
722:satisfy
253:is the
241:, where
207:, where
5167:Kosambi
3305:is the
2360:. Then
2271:) is a
1978:is the
1754:is the
1550:is the
845:on the
364:(λξ)=λΦ
211:is the
169:) is a
162:of the
160:section
88:as the
34:on the
3835:Since
3317:,ξ) =
3293:where
2189:Energy
1326:where
356:\0 of
92:whose
65:(λξ)=Φ
5152:. In
2344:Let (
1866:with
1638:with
1461:with
1348:) = γ
471:) on
281:is a
136:and (
132:be a
27:is a
25:spray
3419:>
3197:and
2545:and
2267:α∈Ω(
2187:The
806:>
372:Let
352:(ξ)∈
257:and
219:and
197:= ξ.
128:Let
84:and
57:(ξ)∈
23:, a
5122:.
4700:on
3209:by
2282:on
1554:of
316:TTM
312:TTM
295:= λ
277:on
267:TTM
251:TTM
247:TTM
227:\0.
215:on
173:on
167:TTM
154:on
19:In
5228::
5175:.
5162:.
4714:vH
4708:=Θ
4706:hH
4702:TM
4563:TM
4158:.
3349:ij
3336:ij
3323:ij
3299:TM
3207:TM
3167:TM
2358:TM
2284:TM
2269:TM
1340::→
1330::→
968:TM
884::=
839:TM
809:0.
709:TM
354:TM
348:→Φ
324:TM
320:TM
299:(λ
293:λξ
263:TM
225:TM
217:TM
201:JH
184:TM
181:(π
156:TM
143:TM
140:,π
138:TM
77:.
59:TM
53:→Φ
39:TM
5218:.
5208:.
5129:)
5125:(
5093:H
5076:.
5073:s
5070:d
5065:)
5059:(
5054:s
5049:V
5039:|
5034:)
5031:V
5022:(
5013:)
5007:s
4999:V
4991:(
4986:s
4979:e
4973:0
4957:=
4947:|
4929:H
4912:.
4909:V
4900:=
4894:+
4886:V
4880:L
4851:J
4846:H
4840:]
4837:H
4834:,
4831:V
4828:[
4822:L
4813:2
4810:1
4804:=
4801:v
4796:V
4790:L
4784:=
4754:.
4748:+
4745:V
4739:=
4736:H
4722:H
4710:V
4698:V
4680:.
4677:]
4674:Y
4671:J
4668:,
4665:X
4662:h
4659:[
4656:v
4650:]
4647:Y
4644:h
4641:,
4638:X
4635:J
4632:[
4629:v
4623:]
4620:Y
4617:h
4614:,
4611:X
4608:h
4605:[
4602:J
4599:=
4596:)
4593:Y
4590:,
4587:X
4584:(
4581:T
4567:T
4546:.
4541:)
4536:J
4531:H
4525:L
4519:+
4516:I
4511:(
4503:2
4500:1
4494:=
4491:v
4487:;
4483:)
4480:0
4474:M
4471:T
4468:(
4465:T
4459:)
4456:0
4450:M
4447:T
4444:(
4441:T
4438::
4435:v
4414:,
4409:)
4404:J
4399:H
4393:L
4384:I
4379:(
4371:2
4368:1
4362:=
4359:h
4355:;
4351:)
4348:0
4342:M
4339:T
4336:(
4333:T
4327:)
4324:0
4318:M
4315:T
4312:(
4309:T
4306::
4303:h
4280:)
4277:0
4271:M
4268:T
4265:(
4262:V
4256:)
4253:0
4247:M
4244:T
4241:(
4238:H
4235:=
4232:)
4229:0
4223:M
4220:T
4217:(
4214:T
4194:M
4174:H
4151:H
4146:F
4144:,
4142:M
4134:H
4120:M
4114:]
4111:b
4108:,
4105:a
4102:[
4099::
4073:.
4067:)
4064:a
4058:b
4055:(
4052:2
4045:2
4041:)
4034:(
4025:=
4020:2
4014:2
4006:)
4003:a
3997:b
3994:(
3988:=
3985:)
3979:(
3974:S
3948:M
3942:]
3939:b
3936:,
3933:a
3930:[
3927::
3898:=
3895:)
3892:)
3889:t
3886:(
3871:,
3868:)
3865:t
3862:(
3856:(
3853:F
3840:ξ
3837:g
3829:H
3812:,
3809:)
3803:,
3800:x
3797:(
3792:i
3788:G
3782:2
3774:=
3771:)
3762:,
3759:x
3756:(
3751:i
3747:G
3742:,
3739:)
3733:,
3730:x
3727:(
3722:i
3711:=
3708:)
3699:,
3696:x
3693:(
3688:i
3679:,
3676:)
3670:(
3665:j
3662:i
3658:g
3654:=
3651:)
3642:(
3637:j
3634:i
3630:g
3616:L
3614:,
3612:M
3595:.
3590:2
3586:F
3579:2
3576:1
3570:=
3567:L
3559:i
3549:i
3541:=
3538:E
3534:,
3529:j
3519:i
3509:j
3506:i
3502:g
3498:=
3493:2
3489:F
3484:,
3479:i
3469:j
3466:i
3462:g
3458:=
3453:i
3422:0
3412:,
3409:)
3403:,
3400:x
3397:(
3394:F
3388:=
3385:)
3376:,
3373:x
3370:(
3367:F
3354:x
3352:(
3345:g
3341:x
3339:(
3332:g
3328:x
3326:(
3319:g
3315:x
3313:(
3311:F
3303:R
3301:→
3297::
3295:F
3278:,
3275:)
3269:,
3266:x
3263:(
3258:2
3254:F
3247:2
3244:1
3238:=
3235:)
3229:,
3226:x
3223:(
3220:L
3171:H
3163:M
3146:,
3143:t
3140:d
3135:i
3131:X
3127:)
3122:k
3118:G
3114:2
3111:+
3106:k
3089:(
3084:k
3081:i
3077:g
3071:b
3066:a
3053:i
3049:X
3043:i
3033:b
3028:a
3022:|
3016:=
3013:)
3008:s
3000:(
2995:S
2988:0
2985:=
2982:s
2976:|
2967:s
2964:d
2960:d
2932:.
2927:)
2917:i
2913:x
2904:L
2890:j
2877:j
2873:x
2864:i
2851:L
2846:2
2833:(
2826:2
2821:i
2818:k
2814:g
2808:=
2805:)
2799:,
2796:x
2793:(
2788:k
2784:G
2770:M
2766:H
2747:i
2739:d
2731:j
2727:x
2718:i
2705:L
2700:2
2687:j
2679:+
2674:i
2670:x
2666:d
2661:)
2651:i
2647:x
2638:L
2624:j
2611:j
2598:i
2594:x
2585:L
2580:2
2567:(
2562:=
2559:E
2556:d
2528:j
2520:d
2512:j
2508:x
2499:i
2486:L
2481:2
2468:i
2464:X
2455:j
2451:x
2447:d
2439:j
2435:x
2426:i
2413:L
2408:2
2395:i
2391:Y
2387:=
2381:d
2376:H
2354:H
2350:Y
2348:,
2346:X
2340:.
2324:d
2319:H
2308:=
2305:E
2302:d
2288:E
2280:H
2265:d
2259:.
2246:)
2240:(
2237:L
2231:)
2225:(
2212:=
2209:)
2203:(
2200:E
2184:.
2171:)
2165:,
2162:x
2159:(
2154:j
2151:i
2147:g
2124:)
2118:,
2115:x
2112:(
2107:j
2104:i
2100:g
2078:M
2073:x
2069:T
2035:g
2018:.
2006:M
2001:x
1997:T
1966:)
1960:,
1957:x
1954:(
1945:j
1932:i
1919:L
1914:2
1902:=
1899:)
1893:,
1890:x
1887:(
1882:j
1879:i
1875:g
1852:x
1847:|
1842:)
1837:j
1833:x
1829:d
1821:i
1817:x
1813:d
1810:(
1807:)
1801:,
1798:x
1795:(
1790:j
1787:i
1783:g
1779:=
1770:g
1742:M
1739:T
1725:T
1716:)
1710:,
1707:x
1704:(
1699:|
1692:i
1688:x
1684:d
1681:)
1675:,
1672:x
1669:(
1664:i
1656:=
1626:)
1623:M
1620:T
1617:(
1612:1
1590:.
1578:M
1573:x
1569:T
1538:)
1532:,
1529:x
1526:(
1517:i
1504:L
1494:=
1491:)
1485:,
1482:x
1479:(
1474:i
1449:M
1439:x
1435:T
1426:x
1421:|
1414:i
1410:x
1406:d
1403:)
1397:,
1394:x
1391:(
1386:i
1378:=
1354:t
1352:(
1350:0
1346:t
1342:M
1337:s
1332:R
1328:X
1311:,
1308:t
1305:d
1300:i
1296:X
1290:)
1280:i
1276:x
1267:L
1253:j
1231:i
1218:j
1214:x
1205:L
1200:2
1189:+
1184:j
1162:i
1149:j
1136:L
1131:2
1118:(
1111:b
1106:a
1093:i
1089:X
1080:i
1067:L
1056:b
1051:a
1045:|
1039:=
1036:)
1031:s
1023:(
1018:S
1011:0
1008:=
1005:s
999:|
990:s
987:d
983:d
962:.
950:t
947:d
944:)
941:)
938:t
935:(
920:,
917:)
914:t
911:(
905:(
902:L
897:b
892:a
881:)
875:(
870:S
855:M
851:M
843:R
841:→
837::
835:L
799:,
796:)
790:,
787:x
784:(
779:i
775:G
769:2
761:=
758:)
749:,
746:x
743:(
738:i
734:G
720:G
713:H
692:.
687:)
681:,
678:x
675:(
669:|
658:i
641:)
635:,
632:x
629:(
624:i
620:G
616:2
608:)
602:,
599:x
596:(
590:|
579:i
575:x
560:i
552:=
543:H
519:M
499:H
479:M
457:i
453:x
449:(
429:M
426:T
406:)
401:i
393:,
388:i
384:x
380:(
366:H
362:H
358:H
350:H
346:t
341:.
339:H
335:V
331:H
322:→
314:→
310::
308:*
304:ξ
301:H
297:*
290:H
279:M
275:H
269:.
265:→
259:H
249:→
245::
243:j
239:H
237:=
235:H
233:∘
231:j
221:V
209:J
205:V
203:=
195:ξ
192:H
189:*
187:)
175:M
152:H
148:M
146:,
130:M
118:H
114:M
110:H
106:H
102:M
71:H
67:H
63:H
55:H
51:t
47:M
32:H
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