A new geometrical look at Ostrogradsky’s procedure

Testo completo

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(106) j (j (V )) -

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(117)

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(121)

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(124) 3 -

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(126) L(t, q , q˙ , q¨ , . . .)

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(184) ---

(185)

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(192) n+1. N. n+1. N. 1. t. th. N −1. n+1. k. k. k. n+1.

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(204)

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(208) (n + 1)7

(209) t : V → R

(210) .

(211) t, q , . . . , q ;

(212) γ : R → V q = q (t)

(213)

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(219) n $

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(227) t:V → R ,

(228)

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(233) *

(234) j (V )

(235) *

(236) .

(237) t, q , q˙

(238) ;

(239) γ : R → V

(240)

(241) j (γ) : R → j (V ) dq . q = q (t), q˙ = dt 6 7

(242)

(243)

(244)

(245)

(246)

(247) i : A → j (V ) V

(248)

(249)

(250) 1. n+1. i. n+1. n. i. n+1. n+1. 1. i. n+1. n+1. i. i. i. 1. i. 1. i. n+1. 1. n+1. n+1. A ⏐ ⏐ π. i. −−−−→ j1 (Vn+1 ) ⏐ ⏐π . Vn+1. 2<3. Vn+1. 1 A

(251) .

(252) t, q , z 2A = 1, . . . , r < n3

(253) i : A → j (V )

(254) q˙ = ψ (t, q , . . . , q , z , . . . , z ) 2'3 ,

(255) rank ∂z∂ψ = r !

(256) γ : R → V

(257) .

(258)

(259) γˆ : R → A

(260) j (γ) = i · γˆ !

(261) γˆ : R → A i·ˆ γ = j (π·ˆ γ ) - .

(262) γˆ

(263) q = q (t), z = z (t)

(264)

(265)

(266) =

(267)

(268)

(269)

(270) 4; 1. i. A. n. 1. n+1. i. i. 1. r. i. A. n+1. 1. i. 1. i. A. A. 2>3 6

(271)

(272) A .

(273)

(274)

(275)

(276) .

(277) 7. 2

(278)

(279)

(280) 3 ?

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(282) ,

(283)

(284) π : C(A) → A

(285)

(286)

(287)

(288)

(289)

(290) T (A)

(291)

(292)

(293) 1 ω := dq − ψ (t, q , z ) dt 2%3 !

(294)

(295)

(296)

(297) V (V ) ⊂ T (V )

(298)

(299)

(300)

(301)

(302)

(303) V → R V (V − )

(304)

(305)

(306)

(307)

(308)

(309)

(310)

(311)

(312)

(313) A× V (V )

(314)

(315) dq i = ψ i (t, q 1 (t), . . . , q n (t), z 1 (t), . . . , z r (t)) dt. ∗. i. n+1. i. n+1. t. ∗. i. k. A. n+1. n+1. Vn+1. ∗. n+1. π. C(A) −−−−→ V ∗ (Vn+1 ) ⏐ ⏐ ⏐π ⏐ π A. −−−−→ π. @ C(A)

(316) .

(317) t, q , z σ = p (σ)ω ∀ σ ∈ C(A) i. i. i |π(σ). A. 2&3. Vn+1 , pi.

(318)

(319)

(320)

(321) .

(322) > !

(323)

(324)

(325)

(326)

(327)

(328)

(329)

(330)

(331)

(332) 1 Θ . Θ := p ω = p dq − ψ t, q , z dt 2A3 6

(333) ,

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(337)

(338)

(339)

(340)

(341)

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(343) 6

(344) ,

(345)

(346)

(347) . I [γ] := L dt = L(t, q (t), z (t)) dt 2B3 i. i. i. i. i. k. t1. γ ˆ. A. i. A. t0.

(348)

(349) γ : R → V 8

(350) 9 .

(351)

(352) L(t, q , z ) ∈ F(A)

(353)

(354) γˆ : R → A 6

(355)

(356) 2 3 .

(357)

(358)

(359) 2B3 ,

(360)

(361)

(362)

(363) . γ

(364)

(365) γ(t ), γ(t ) . 6

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(375)

(376)

(377)

(378) 6

(379)

(380)

(381)

(382)

(383)

(384) 1 2A3 L(t, q , z ) ∈ F (A)

(385)

(386) 1 ϑ C(A)

(387).

(388)

(389) ϑ := L dt + Θ = (L − p ψ ) dt + p dq := −H dt + p dq 2C3 6

(390) H(t, q , z , p ) = p ψ (t, q , z ) − L(t, q , z ) ∈ F (C(A)) ,

(391)

(392)

(393)

(394) D

(395) 17 2C3

(396)

(397) γ¯ : [t , t ] → C(A) . .

(398) q = q (t), z = z (t), p = p (t)) ,

(399)

(400)

(401) .

(402) . . dq ¯ I [¯ γ ] := L(t, q (t), z (t)) + p (t) ϑ = 2E3 − ψ (t, q (t), z (t)) dt dt i. 0. i. A. L. i. i. k. n+1. 1. L. 0. A. A. k. i. i. k. A. t1. γ ¯. L. k. t0. A. i. k. i. 1. i. i. i. A. i. A. i. i. A. i. i. k. i. A. 6

(403)

(404)

(405)

(406)

(407)

(408)

(409) 2B3 .

(410)

(411)

(412) 2>3 -

(413)

(414) ν : C(A) → V

(415)

(416) *

(417) C(A) → A → V

(418)

(419)

(420)

(421)

(422) 8 9 .

(423)

(424)

(425) *

(426) γ = ν · γ¯

(427)

(428)

(429)

(430)

(431) 2E3

(432)

(433)

(434)

(435)

(436)

(437)

(438)

(439) 2E3

(440)

(441)

(442) *

(443) ν(¯γ (t )), ν(¯γ (t )) .

(444) 2n + r

(445) n+1. n+1. 0. 1. dq i ∂H = ψ i (t, q k , z A ) = dt ∂pi. 2<(3. ∂L dpi ∂ψ k ∂H = − pk = − i i dt ∂q ∂q i ∂q. 2<(3. 2<(3.

(446) , q (t), z (t), p (t)

(447)

(448)

(449)

(450)

(451)

(452)

(453)

(454) 2B3 -

(455) ,

(456)

(457)

(458)

(459)

(460)

(461) 2<(3

(462)

(463)

(464)

(465)

(466) ,

(467) 2<(3 6

(468)

(469)

(470)

(471) C(A)

(472)

(473) S 6

(474) H

(475)

(476) pi. i. A. ∂ψ i ∂L ∂H − = = 0 A A ∂z ∂z ∂z A. i.

(477) ∂ 2H det = 0 ∂z A ∂z B σ. 2<<3.

(478) %. σ ∈ S @

(479)

(480) 2<(3

(481) z

(482)

(483)

(484)

(485) z = z (t, q , p ) 2<'3 F

(486)

(487)

(488)

(489)

(490)

(491) S

(492) (2n+1)7 i : S → C(A) G

(493)

(494) V (V ) 6 7 H := i (H)

(495)

(496)

(497) . .

(498) H(t, q , p ) = p ψ (t, q , z (t, q , p )) − L(t, q , z (t, q , p )) 2<>3

(499)

(500) S 6

(501)

(502)

(503) 2<(3 2<(3 ,

(504)

(505)

(506)

(507)

(508) 2<(3 ,

(509)

(510)

(511)

(512) ∂H ∂H = = ψ 2<%3 ∂p ∂p A. A. A. i. i. ∗. n+1. ∗. i. i. h. h. i. A. k. i. k. A. k. k. i. i. i. 2<%3. ∂H ∂ψ k ∂H ∂L = = p − k i i i ∂q ∂q ∂q ∂q i. ,

(513)

(514) 2<(3 2<(3

(515)

(516) . 2<&3. ∂H dq i = dt ∂pi. 2<&3 6

(517)

(518)

(519)

(520)

(521)

(522) S ,

(523)

(524) H(t, q , p )

(525)

(526)

(527) 7 H = i (H) ∂H dpi = − i dt ∂q. i. ∗. . i.

(528)

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(531)

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(535)

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(558) ,

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(578)

(579) 6 ,

(580) .

(581)

(582). . !!. 0

(583) L(t, q , q˙ , q¨ )

(584) ; 7

(585) ∂L d ∂L d ∂L − + = 0, i = 1, . . . , n 2<A3 ∂q dt ∂ q˙ dt ∂ q¨ i. i. i. 2. i. i. 2. i. . . !

(586)

(587)

(588) det ∂ q∂¨ ∂Lq¨ = 0

(589)

(590)

(591)

(592)

(593) ∂L d ∂L ∂L q , q˙ , p := − , p := 2<B3 ∂ q˙ dt ∂ q¨ ∂ q¨ 2. i. i. i. 0 i. i. i. 1 i. j. i.

(594) & .

(595) (4n + 1)7 ,

(596) p p

(597) .

(598)

(599) *

(600)

(601)

(602) q q˙ F

(603)

(604)

(605)

(606)

(607)

(608)

(609)

(610) 2<B3 .

(611) , q¨

(612)

(613)

(614)

(615) q¨ = q¨ t, q , q˙ , p 2<C3 -

(616) ,

(617)

(618) H := p q˙ + p q¨ − L, 2<E3 .

(619)

(620) q , q˙ , p , p

(621) 2<C3

(622)

(623) ,

(624)

(625)

(626) 0 i. i. 1 i. i. i. i. i. k. 0 i i. i. ∂H = q˙ i , ∂p0i. i. 1 k. 1 i i. 0 i. ∂H = qï , ∂p1i. k. 1 i. ∂H ∂L =− i, ∂q i ∂q. ∂H ∂L = p0i − ∂ q˙ i ∂ q˙ i. - ,

(627)

(628)

(629)

(630) 2<A3 2<B3

(631)

(632)

(633)

(634) dq i ∂H = q˙ i = , dt ∂p0i. dq˙ i ∂H = qï = dt ∂p1i. 2'(3. d ∂L dp0i d2 ∂L ∂L ∂H = − 2 i = =− i, i dt dt ∂ q˙ dt ∂ q¨ ∂q i ∂q. d ∂L dp1i ∂L ∂H = = − p0i = − i dt dt ∂ qï ∂ q˙ i ∂ q˙. 2'(3. 6

(635)

(636)

(637)

(638) ,

(639) V → R

(640)

(641)

(642) *

(643) ?

(644)

(645) , . ,

(646)

(647)

(648) j (V )

(649) Q q

(650) q q˙

(651) q

(652) t : Q → R

(653) .

(654) t, q , α = 0, 1 ,

(655) D

(656)

(657)

(658) *

(659) j (V )

(660)

(661). 5

(662)

(663)

(664) *

(665) j (Q) .

(666)

(667)

(668) 2

(669)

(670) 7

(671) 3 n+1. 1. i. n+1. i 0. i α. 2. i. i 1. n+1. 1. i. j2 (Vn+1 ) −−−−→ j1 (Q) ⏐ ⏐ ⏐π ⏐ π Q. 2'<3. Q. 1 j (Q)

(672) *

(673) .

(674) t, q , q˙

(675) i(j (V )) ⊂ j (Q)

(676)

(677) A

(678)

(679) q˙ = q @

(680) A

(681) .

(682) t, q , z

(683)

(684) A → j (Q)

(685)

(686)

(687) 2

(688) 2'33 q˙ = ψ (t, q , q , z ), 2''3 ,

(689) ψ = q ψ = z !

(690)

(691) , A V

(692)

(693) j (V )

(694) (t, q , q , z ) ←→ (t, q , q˙ , q¨ ) # ,

(695)

(696)

(697) V L(t, q , q˙ , q¨ ) ∈ F (j (V ))

(698)

(699)

(700)

(701) Q ,

(702)

(703)

(704) A → j (Q) 2''3 L(t, q , z ) ∈ F (A) -

(705)

(706)

(707)

(708) .

(709) , ,

(710)

(711) #

(712) -- 6

(713) ,

(714)

(715)

(716) C(A) A

(717) ν : C(A) → Q

(718)

(719) *

(720) C(A) → A → Q t, q , z , p

(721) .

(722) C(A)

(723)

(724) σ = p (σ) dq − ψ dt ∀σ ∈ C(A) 2'>3 i α. 1. i α. i 1 2. i 1. i. i α. i. n+1. 1. 1. i α. i 0. i 1. i. i. i 0. n+1. i. 2 i 1. i 0. i. i α. i 0. i α. i 1. i. i. i. n+1. i. n+1. i. 2. n+1. 1. i α. α i. i α. i α. |π(σ). i. α i.

(725) A

(726)

(727)

(728)

(729) .

(730)

(731)

(732) #

(733)

(734) ,

(735)

(736) L(t, q , z ) ∈ F (A) ,

(737)

(738)

(739)

(740) 1 i α. i. i i α i ϑL = Ldt + pα i dqα − ψα dt = −H dt + pi dqα ∈ C(A). ,

(741) . α i i i 0 i 1 i i i H(t, qαi , z i , pα i ) = pi ψα − L(t, qα , z ) = pi q1 + pi z − L(t, qα , z ).

(742)

(743)

(744)

(745) ;

(746) ,

(747)

(748) γ¯ : [t , t ] → C(A)

(749)

(750)

(751) 0. I¯ [¯ γ ] :=. 2'%3. 1.

(752) t1 i i i α α dqα −H(t, qα , z , pi ) + pi ϑL = dt dt t0. γ ¯. 6

(753)

(754)

(755)

(756)

(757)

(758)

(759)

(760)

(761)

(762)

(763) .

(764)

(765)

(766)

(767) . ν(¯ γ (t0 )) ν(¯ γ (t1 )). ∂H dqαi = , dt ∂pα i. 2'&3. ∂H dpα i =− i dt ∂qα. 2'&3.

(768) , q (t), z (t), p (t) 6

(769) 2<(3

(770)

(771) -

(772) 2'&3

(773)

(774)

(775)

(776)

(777)

(778) 2<B3

(779) (t, q , q , z ) ←→ (t, q , q˙ , q¨ ) 4

(780) S

(781)

(782) C(A) 2'&3

(783) 2'%3

(784)

(785) ∂ L

(786)

(787)

(788)

(789) 7

(790) det ∂ q¨ ∂ q¨ = 0 ∂ L det ∂z ∂z = 0

(791)

(792)

(793)

(794)

(795)

(796)

(797) 2'%3 @

(798) 2'&3

(799) z

(800)

(801) .

(802) z = z (t, q , p ) 2'A3.

(803)

(804) 2<C3 ;.

(805)

(806) #

(807) -- 2'A3 ,

(808) S S− → C(A) G

(809)

(810) V (Q) - , 2<B3 2'&3

(811)

(812)

(813)

(814) 2'%3

(815)

(816) S

(817)

(818) ∂H ∂L = p1i − =0 ∂z i ∂z i. i α. i 0. i. i 1. i. α i. i. i. i. 2. i. j. 2. i. j. i. i. i. i. k α. 1 k. ∗. 0 i 1 i k 1 i i k 1 H (t, qαi , pα i ) := pi q1 + pi z (t, q α , pk ) − L(t, qα , z (t, q α , pk )).

(819)

(820)

(821)

(822)

(823) 2<E3

(824)

(825)

(826) ∂H ∂H = α , ∂pα ∂p i i. ∂H ∂H = ∂qαi ∂qαi. ∂H dq0i = , dt ∂p0i. ∂H dq1i = dt ∂p1i. ∂H dp0i =− i, dt ∂q0. ∂H dp1i =− i dt ∂q1.

(827)

(828)

(829)

(830) 2'&3

(831)

(832)

(833)

(834) .

(835)

(836)

(837)

(838)

(839)

(840) 2'(3.

(841) B ". !!. 6

(842)

(843)

(844) .

(845)

(846) .

(847) 6

(848)

(849) j (V )

(850)

(851) N *

(852) 7

(853)

(854)

(855) .

(856) t, q , q , . . . , q #

(857)

(858) q = q

(859) ; 7

(860)

(861) ,

(862) L(t, q , q , . . . , q )

(863)

(864) ,

(865)

(866) i. N . (−1)α. α=0. N i 1. n+1 i. th. i N i 1. i 0. i N. d α ∂L = 0, dtα ∂qαi. i. 2'B3. i = 1, . . . , n. ? α = 0, . . . , N − 1

(867)

(868)

(869) p *

(870)

(871)

(872) .

(873) q

(874)

(875)

(876) α i. pα i :=. N −1 . (−1)β−α. β=α. ,

(877) . i α. 2'C3. d β−α ∂L i dtβ−α ∂qβ+1. 2'C3 6 t, q , p , α = 0, . . . , N − 1 .

(878) (2nN + 1)7 F

(879)

(880) det ∂q∂ ∂qL = 0 2'C3 q

(881) .

(882) q = q (t, q , . . . , q ,p ) 2'E3 6

(883)

(884)

(885) −1 pN := i. i α. ∂L i i i i (t, q , q1 , . . . , qN ) ∂qN. α i. 2. i N. i N. i N. H(t, q0i ,. i 0 . . . , qN −1 , pi ,. i N. −1 . . . , pN ) i. k 0. :=. j N. N −1 k. k N −1. N −1 . i i i i pα i qα+1 − L(t, q0 , . . . , qN −1 , qN ). α=0. 2>(3. ,

(886) q 2'E3 -

(887) ,

(888) 2'C3

(889)

(890)

(891)

(892)

(893)

(894)

(895)

(896) 2>(3 =

(897) ∂H dq = =q (α = 0, . . . , N − 2), 2><3 dt ∂p i N. i α. α i. i α+1. i k dqN ∂H −1 N −1 i k = t, q0 , . . . , qN = qN −1 , pk N −1 dt ∂pi. 2><3. 2><3 "

(898)

(899) , ,

(900)

(901) 2><3

(902)

(903) ,

(904) 2'C3

(905)

(906)

(907) ; 7

(908) 2'B3 !

(909)

(910)

(911)

(912) 2'C3 2>(3

(913) Q := j (V )

(914) (N −1) *

(915)

(916)

(917) t : V → R t : Q → R j (Q)

(918)

(919) *

(920) 6 N *

(921) j (V )

(922)

(923) 5 . j (Q) =

(924)

(925)

(926)

(927) ∂H dp0i ∂L =− i = , dt ∂q0 ∂q0i. N −1. th. ∂H dpα ∂L i = − i = −pα−1 + i i dt ∂qα ∂qα. th. n+1. n+1. 1. N. n+1. 1. i. jN (Vn+1 ) −−−−→ j1 (Q) ⏐ ⏐ ⏐ ⏐π π Q. Q. (α = 2, . . . , N ).

(928) C !

(929) t, q , (α = 0, . . . , N − 1) .

(930) Q j (Q)

(931) *

(932) .

(933) t, q , q˙

(934) A := i(j (V )) ⊂ j (Q)

(935)

(936) q˙ = q α = 0, . . . , N − 2 @

(937) A

(938) .

(939) t, q , . . . , q , z

(940)

(941) A → j (Q)

(942)

(943)

(944) q˙ = ψ (t, q , . . . , q ,z ), α = 0, . . . , N − 1 2>'3 ,

(945) ψ = q α = 0, . . . , N − 2 ψ = z !

(946)

(947) , A V

(948) j (V )

(949)

(950) (t, q , . . . , q , z ) ←→ (t, q , . . . , q , q )

(951)

(952) ,

(953)

(954)

(955) V L(t, q , , . . . , q ) ∈ F (j (V ))

(956)

(957)

(958)

(959) Q ,

(960)

(961)

(962) A → j (Q) 2>'3 L(t, q , . . . , q , z ) ∈ F(A) 6

(963) Q

(964)

(965)

(966)

(967)

(968)

(969)

(970) C(A)

(971) .

(972) t, q , z , p α = 0, . . . , N − 1 6

(973)

(974)

(975) .

(976) #

(977) ---!

(978)

(979)

(980)

(981) .

(982)

(983)

(984) i α i α. i α i N −1. i 0. 1. i α. N. i α+1 i. 1. 1. i α. i α. n+1. i α. i 0. i N −1. i α+1. i. i N −1. i 0. i N −1. i 0. n+1. i. i. i N. i 0. i N −1. N. N −1 . i N −1. N. i N. n+1. n+1. 1. i. i α. H(t, qαi , z i , pα i ) =. n+1. i 0. i. α i. i i i pα i ψα − L(t, qα , z ) =. α=0. =. N −2 α=0. N −1 i i i i pα z − L(t, q i , . . . , qN i qα+1 + pi −1 , z ). 2>>3.

(985)

(986)

(987) C(A) −→ Q

(988)

(989) C(A) → A → Q

(990)

(991) γ¯ : [t , t ] → C(A) , ,

(992)

(993)

(994) ν. 0. 1. I¯ [¯ γ ] :=. γ ¯. −H dt +. N −1 . pα i. dqαi. α=0. =. t1 . t0. −H +. N −1 . pα i. α=0.

(995) dqαi dt dt. -

(996)

(997)

(998)

(999)

(1000)

(1001)

(1002)

(1003)

(1004) *

(1005) .

(1006)

(1007)

(1008)

(1009) . ν(¯ γ (t0 )) ν(¯ γ (t1 )). ∂H dqαi = , dt ∂pα i. 2>%3. ∂H dpα i =− i dt ∂qα. 2>%3.

(1010) , q (t), z (t), p (t) ; 2>%3

(1011)

(1012)

(1013) 2'C3

(1014) (t, q , . . . , q , z ) ←→ (t, q , . . . , q , q ) 4

(1015) S

(1016)

(1017). C(A) 2>%3

(1018)

(1019)

(1020)

(1021) 7 ∂ L ∂ L

(1022) ∂q ∂q = 0 det ∂z ∂z = 0

(1023)

(1024)

(1025)

(1026)

(1027)

(1028)

(1029) 2>>3 @

(1030) 2>%3.

(1031) z

(1032)

(1033)

(1034) . z = z (t, q , . . . , q ,p ) 2>&3.

(1035)

(1036) 2'E3 ; 2>&3

(1037)

(1038)

(1039)

(1040)

(1041)

(1042) S ⊂ C(A)

(1043) i : S → C(A) G

(1044)

(1045) V (Q) !

(1046)

(1047) , ,

(1048)

(1049)

(1050) H := i (H)

(1051)

(1052)

(1053) 2>>3

(1054)