u u IR n n = 2 3 t 0 <t T

IR n n =2 3 u() u u u u IR n n = 2 3 ξ A 0 A 0 0 0 < T

F IR n F A 0 A 0 A 0 A 0 F :IR n IR n A = F A 0 A 0 A 0 0 0 A F A 0 A F (, y) =0 a = T>0 b A 0 T 1 2 A IR n A A A F A 0 A 0 ξ A 0 = F (ξ) ε>0 δ ε > 0 ξ A 0 δ ε ξ F (ξ ) <ε = i i A 0 3 A 0 ξ A 0 {ξ k A 0} k=1,2,... ξ ε>0 M ε N k, h > M ε ξ k ξ h <ε >0 F (ξ k )= k k =1 2... { k } k=1,2,... F (ξ ) ξ A 0 (IR n, ) { k } k=1,2,... ε>0 M ε k, h > M ε k h <ε M ε k, h > M ε ξ k ξ h <δ ε F k h = F (ξ k ) h <ε { k } k=1,2,... C IR n n 2 [a, b] IR n C :[a, b] IR n σ (σ)

F A (ξ,) A 0 ξ F 1 A 0 A IR n n =2 3 0 F A 0 A ξ A 0 F (ξ,) A A ξ(,) A 0 F 1 A A 0 P 1 P 2 a F T Q 1 Q 2 Q 6 Q 7 Q 2 Q 3 Q 5 Q 6 Q 3 Q 4 Q 5 A T F A 0 A Q 2 Q 3 Q 5 Q 6 A T A T A 0 σ a = b =+ IR C C a b σ (σ 1)=(σ 2) σ 1 = σ 2 (a) =(b)

y y P 2 P 1 Q 7 Q 6 Q 2 Q 1 Q 5 Q 3 A 0 A Q 4 a b b P 1P 2 0 a a Q 2Q 3 Q 5Q 6 Q 1 Q 2... Q 7 T b A T F A 0 A F 1 F ξ F 1 A 0 ξ 0 U = F T ξ A V ξ A 0 F T (V ξ ) U F F U = (ξ,) A V ξ ξ F (V ξ ) U (ξ,) ξ 0 4 C IR n C U C n

F (V ξ ) A 0 ξ V ξ A (ξ,) a b a 0 b >0 ξ A 0 V ξ ξ A 0 F (V ξ ) F (V ξ ) U A (ξ,) V ξ ξ U 0 f A IR n n =2 3 0 f F 1 F 1 A ξ A 0 0 f f E () f f L (ξ,) f F 1 f E [(ξ,)] = f L (ξ,) F f E () k f L (ξ,) k ξ A 0 A F 1 ξ u L (ξ,) = (ξ,+ Δ) (ξ,) lim = (ξ,), Δ 0 Δ F 1

u E (,)=u L [ξ(,),]. L E u 0 (0) = ξ [0,T] d d () =u E [(),] ξ (ξ,) C ξ F [0,T] ξ A 0 r = / = δ u E () =ρ(δ) IR n n =2 3. ρ(δ) δ =0 δ 0 + ρ(0) = 0 A 0 =IR 3 F ξ = δ 0 r 0 ρ(δ) d d (δr) = δr + δṙ = ρ(δ)r δ(0)r(0) = δ 0 r 0. ṙ r r() r 0 δ = δ 0 δ 0 0 δ 0 =0 0 δ = ρ(δ) 5 f df /d f

δ δ 0 dη ρ(η) =. ρ(δ) = Uδ/R U R δ δ() = δ 0 ep(u/r) (ξ,)=ξep(u/r) U>0 U<0 R/U ρ(δ) = U[1 ep( δ/r)] δ() =R log[1 + (ep(δ 0 /R) 1) ep(u/r)] δ() δ 0 ep(u/r) δ 0 R δ() δ 0 + U δ 0 R ρ(0) = 0 ρ ρ(δ) =U(R/δ) n 1 n u() =UR n 1 n A 0 =IR n {0} U R q r q[ B r (0)] = ds() u() ν() B r(0) B 0 (r) ds() ν() S ν() =/r r B 0 (r) = 0 u 0 B 0 (r) q = χ n r n 1 ρ(r) χ n UR n 1, χ 2 =2π χ 3 =4π δ = ρ(δ) δ() =δ 0 [1 + n(r/δ 0 ) n 1 U/δ 0 ] 1/n.

U>0 U<0 IR {0} 0 (δ 0 ) B 0 (δ 0 ) 0 = R/(nU)(δ 0 /R) n U(R/δ 0 ) n 1 =0 = c = c (σ) d c dσ (σ) =u E [ c (σ),] c (0) = 0 0 σ 0 u u = 0 u u = 0 σ = = 0 u E (0) = c (0) (0) = c (0) σ

c (σ) = 1 2γ y c (σ) = 1 2γ { [(γ + χ ) 0 + χ y y 0 ]e γσ +[ (γ χ ) 0 χ y y 0 ]e γσ } { [χy 0 +(γ χ )y 0 ]e γσ +[ χ y 0 +(γ + χ )y 0 ]e γσ }. u u 0, v(y; ) =Ωy sin Ω, Ω =2π/ ( 0,y 0 ) d dσ = u 0 dy = Ω sin Ω y(σ) dσ (0) = 0, y(0) = y 0, σ = (σ) y = y(σ) y() =y 0 ep [ Ω u 0 sin Ω ( 0 ) ]. sin Ω u(, y; ) = Ω2 u 0 y sin Ω, v(y; ) = Ω2 2u 0 y 2 cos Ω, Ω =2π/ u 0 ( 0,y 0 ) d dσ = Ω2 sin Ω (σ)y(σ) u 0 dy dσ = Ω2 cos Ω y 2 (σ) 2u 0 (0) = 0, y(0) = y 0, σ Ω σ = Ωσ y(σ ) 0 y 0 = 0 y 0 0

y(σ )= y 0 1+β cos Ω σ β = Ωy 0 /(2u 0 ) y 0 =0 y(σ ) cos Ω σ 1/(βcos Ω) d = 2β sin Ω 1+βcos Ω σ dσ, β cos Ω σ 1 β cos Ω =0 2 β cos Ω 0 (σ )= 0 ep(2β sin Ω σ ) (σ )= 0 (1 + ) 2anΩ σ = (σ ) y = y(σ ) / 0 =(y 0 /y) 2anΩ y 0 u(, y; ) =Ω ( +2 Ω u 0 y 2 ) + u 0 sin Ω, v(, y) =u 0 2y, u 0 y Ω = 2π/ = η /u 0 η = y 2 dy dσ = u (σ) 0 2y(σ), η Ωη 2Ω 2 η = u 2 0 sin Ω, σ η(0) = y 2 0 η (0) = u 0 0 (σ) = 1 3Ωu 0 (αe 2Ωσ βe Ωσ ) y(σ) =± 1 6Ω [ (αe 2Ωσ +2βe Ωσ ) 3u 2 0 sin Ω ] 1/2, α() =2Ω 2 y0 2 +2Ωu 0 0 + u 2 0 sin Ω β() =2Ω 2 y0 2 Ωu 0 0 + u 2 0 sin Ω y 0 > 0 y 0 < 0

... 0 T 0 = T 0 = T 0 d f d ( τ) =u E [ f ( τ),] f (τ τ) = 0 f ( τ) τ f (T τ) 0 τ T τ< T σ = 0 = ξ y 4Δτ 3Δτ 2Δτ Δτ τ =0 5Δτ 0 0 T =6Δτ τ = kδτ k =0 1... 5 T =6Δτ 6

( 0,y 0 ) τ τ ( τ) = 0 + u 0 ( τ), y( τ) =y 0 ep(cos Ωτ cos Ω), = T (T τ) = 0 + u 0 (T τ), y(t τ) =y 0 ep(cos Ωτ cos ΩT), τ [0,T] T = /2 a T = b y 8 7 6 5 4 3 2 1 0 0 0.25 0.5 0.75 1 a y 8 7 6 5 4 3 2 1 0 0 0.25 0.5 0.75 1 b 0 =0 y 0 =1 T = /2 a T = b u 0 =1 =1 τ =0 a τ =5Δτ b 0 Δτ = /10 ( 0,y 0 ) y 0 =0 () 0 y() y 0 y 0 0 dy y 2 = Ω2 2u 0 cos Ω d d = Ω2 y sin Ω d u 0 y(τ) =y 0, (τ) = 0, Ω u 0 y 0 β = Ωy 0 /(2u 0 )

β < 1/2 y( τ) = y 0 1+β(sin Ω sin Ωτ). +2 β < 1/2 y() () 0 0 0 sin Ω sin Ωτ 0 = 0 2 d 0 =2Ωβ τ an(ωτ/2) d sin Ω 1+β(sin Ω sin Ωτ), ξ =anω /2 δ(τ) =β/(1 β sin Ωτ) < 1 [ ( an(ω/2) dξ an(ω/2) ( τ) = 0 ep 4 ξ 2 +1 = 0 ep { 4 [ Ω 2 ( τ) 1 1 δ 2 an(ωτ/2) dξ ξ 2 +2δξ +1 ( arcan an(ω/2) + δ arcan an(ωτ/2) + δ 1 δ 2 1 δ 2 ) ] ) ] }. y 0 0 y 0 0 η = y 2 20 15 y 10 5 0 0 5 10 15 20 0 =1 y 0 =1 T =10 3 10 20 30 40 u 0 =2

η = u 0 = η/u 0 η(τ) =y0 2 η(τ) =u 0 0 ( 0,y 0 ) τ α (τ) = 1 3 β (τ) = 1 3 [ y 2 0 + u 0 0 Ω + u2 ] 0 (cos Ωτ +2sinΩτ) 5Ω2 [ 2y 2 0 u 0 0 Ω + u2 0 2Ω 2 (sin Ωτ cos Ωτ) ], τ ẋ = η η y( τ) =± [ α e 2Ω( τ ) + β e Ω( τ ) + u2 0 10Ω 2 (cos Ω 3sinΩ) ] 1/2, = η/u 0 y 0 > 0 y 0 < 0 ( τ) = Ω u 0 [ 2α e 2Ω( τ ) β e Ω( τ ) u2 0 10Ω 2 (sin Ω+3 cosω) ]. [0,T] = T T T τ F IR 3 V F 1 :IR 3 IR 3 ξ V V = dv = dv 0 J(ξ,), V V 0 dv dv () dv (ξ) dv 0 J J = ( 1, 2, 3 ) (ξ 1,ξ 2,ξ 3 ) = ξ1 1 ξ2 1 ξ3 1 ξ1 2 ξ2 2 ξ3 2. ξ1 3 ξ2 3 ξ3 3

(ξ, 0) = ξ F 0 J(ξ, 0) 1. F J V d d V J = dv 0 J JdV 0 V 0 V 0 J = J dv V J, JdV 0 = dv V V u ν ds u ν = dv u, V V ds V dv u = d V d V J = dv V J, V dv ( J J u ) 0. V J J = u DJ/D D J J J J = ε ijk ξi 1 ξj 2 ξk 3, J

[ J = ε ijk ξi ( 1 ) ξj 2 ξk 3 + ξi 1 ξj ( 2 ) ξk 3 + + ξi 1 ξj 2 ξk ( 3 ) ], ξ l u ξ l = u l F u l = u l [(ξ,),] ξ m ξm u l = p u l ξm p. [ J = ε ijk p u 1 ξi p ξj 2 ξk 3 + ξi 1 p u 2 ξj p ξk 3 + ] + ξi 1 ξj 2 p u 3 ξk p = p u 1 ε ijk ξi p ξj 2 ξk 3 + p u 2 ε ijk ξi 1 ξj p ξk 3 + + p u 3 ε ijk ξi 1 ξj 2 ξk p. ε ijk ξi p ξj 2 ξk 3, ε ijk ξi 1 ξj p ξk 3, ε ijk ξi 1 ξj 2 ξk p, p p 1 p =2 p =3 p 2 p 3 p =1 p =2 p =3 J = J ( 1 u 1 + 2 u 2 + 3 u 3 ), { = e α ξ +(1 e α )η y = (1 e α )ξ + e α η, { = ξ + η sin Ω y = ξ sin Ω + η, α Ω J u

= ξ + η = ξ +(1 e α )ζ y = ξ + η + ζ y =(1 e α )ξ + e α η z = η + ζ, z = (1 e α )η + e α ζ, = ξ + η sin Ω = ξ + 2 ζ y = η cos Ω +(1 cos Ω)ζ y = ξ + η z = ξ sin Ω + ζ cos Ω, z = ξ +(1+ 2 )ζ. u F u = u(,) i =1, 2, 3 : u i () u i ( )+ k u i ( ) ( k k), j 1 u 1 2 u 1 3 u 1 u = 1 u 2 2 u 2 3 u 2. 1 u 3 2 u 3 3 u 3 (i, j) ( u) ij i j u i S Ω S = S Ω = Ω u 1 [ u +( u) ] + 1 [ u ( u) ] = S + Ω, } 2 {{}} 2 {{} S Ω 7 j u i ju i

u() u( )+S( ) ( )+Ω( ) ( ), i S ( ) S ik ( k k ) e i e i S u = S ii u = ε ijk Ω kj e i Ω y Ω y y ω = ω i e i = u = ε ilm l u m e i, u Ω jk ω ε ikj ω i i ε ikj ω i = ε ikj ε ilm l u m =(δ kl δ jm δ km δ jl ) l u m = k u j j u k =2Ω jk. Ω y = Ω jk y k e j = 1 2 ε ikjω i y k e j = 1 2 ε jikω i y k e j = 1 2 ω y, u() u( )+ 1 2 ω( ) ( ) + S( ) ( ) }{{}}{{} S( ) ( ) u( ) ω( )/2 S {ε i,i=1, 2, 3} χ i i =1, 2 3 R {e i } {ε j } R 1 = R S a S {ε i } S a = R 1 SR = χ 1 0 0 0 χ 2 0, 0 0 χ 3

S a y χ S y a = R 1 y {ε i } ai i S S a S y S a {ε j } S {e i } S a R S R 1 χ i > 0 χ i < 0 i S ( ) S ( ) S i u i = u S a χ 1 + χ 2 + χ 3 = u, ω S S Ω u = u() ϕ = ϕ() u = ϕ ϕ ϕ u u d u

dϕ C[ 0, ] 0 d u( ) C[ 0,] C 0 0 ϕ C[ 0, ] ϕ() = d u( ), C[ 0,] ϕ( 0 )=0 C C C 0 d u( ) C C d u( )+ d u( ), C C C C 0 C = C C S S = C C d u( )= da( ) ω( ) ν( ). C S C C S = C 0 S C C d u( )=Γ, C Γ 0 u C

Γ 0 C ϕ 0 C ω = u 0 ( ) ( y 2 ) ( ) ( ) y u() = y + z, y 2, y, z, zy z 2 z ( ) ( y 2 + z 2 y 2 ) ( ) 2 z 2 z 2 2 + y 2, + y + z yz 2, y yz. ϕ() F = (ξ,) ξ a L (ξ,)= u L (ξ,)= 2 (ξ,). a u L (ξ,)=u E [(ξ,),], a L (ξ,)= u E [(ξ,),]+u E i [(ξ,),] i u E [(ξ,),]. a L F 1 Du/D

Du D = u(,)+u(,) u(,). D/D D u u u() u D u ( ) ξ cos Ω η sin Ω ( ξ + 2 η ) (ξ,)= ξ sin Ω + η, (1 )η, (ξ + η)sinω + ζ (ξ + η)+ζ ( ) ( ) ξ ζ sin Ω ξ + 2 ζ η, η cos Ω + ζ sin Ω, η sin Ω + ζ ζ(sin Ω)/(Ω) ( ξ 2 ) ( η + ζ ξ(sin Ω)/(Ω) ) 3 ξ + η ξ sin Ω +(1 )ζ, η cos Ω + 2 ζ (ξ + η)sinω + ζ 2

k ω u i i u k u i ( i u k k u i )+u i k u i. ε jik ω j = ε jik ε jlm l u m =(δ il δ km δ im δ kl ) l u m = i u k k u i, u i i u k e k = ε jik ω j u i e k + k u i u i 2 e k = ω u + u 2 2. Du D = u + ω u + u 2 2. D u 2 3 4 6 8 3 6 7 9 12 13

16 19 4 5 8 10 16 1.2 13 14 15 12 1932 3 2 ϕ ϕ = u ϕ 31 33 2 2 72 79 81 2.3 84 88 u u u e u v = ϕ ϕ u v 2.7 100 102 2.8

1.1 1.2 1.3 1.4 1.4.1 63 64 1.5.3 2.3 4.8 248 1.2 1.3 14 1.9 2 6 10 2 5 5 8 15 20 1.8 4.1 1.7 4.7

4 5 6 1 49 50 2 23 3

2D 2D (, y) e z u(,)= ( u(,) v(,) ) = M () = ( χ() + χ y() y χ y() χ () y ) χ χ y z u = u + yv u = e z u =( yu + v) e z χ χ y M M λ (1,2) = ±γ = ± χ 2 + χ 2 y v (1) = v (2) = ( ) cos α sin α ( ) cos α sin α = = 1 2γ(γ χ) 1 2γ(γ + χ) ( ) χy γ + χ ( ) χy, γ + χ v (2) R = α = α ±π/2 ( cos α cos α sin α sin α ), M v (1) R 1 = R T R T M R = ( γ 0 0 γ ), R M (ξ,η) α (, y) ξ ξ η η ξ ξ η η γ α M

χ χ y γ α χ = γ cos 2α, χ y = γ sin 2α. Ṙ T R R ( ξ η ) = R T ( y ) = ( cos α + y sin α cos α + y sin α ) ẋ = M ξ = RT ẋ =(R T M R) ξ { ξ =+γξ η = γ η, ξ(0) = ξ 0 η(0) = η 0 χ χ y R ξ = R T ẋ +(ṘT R) Ṙ T Ṙ T R ξ() =ξ 0 ep(+γ) η() =η 0 ep( γ) () ( ) ( ξ0 ep(+γ) 0 cos 2α + y 0 sin 2α () =R = η 0 ep( γ) 0 cosh γ + 0 sin 2α y 0 cos 2α ) sinh γ. χ χ y χ χ y ẋ yẏ 2 + y = 2 χ, ẏ + yẋ 2 + y 2 = χy, m θ ṁ m cos 2θ θ sin 2θ = γ cos 2α ṁ m sin 2θ + θ cos 2θ = γ sin 2α, χ χ y ε = θ α ε = (γ sin 2ε + α), z =anε =an(θ α) ż = [2γz + α(1 + z 2 )]. II γ α III γ α I α =0 γ

α =0 2D z() =z(0) e 2 Γ () Γ () = 0 d γ( ) > 0. Γ + + z 0 θ α 0 θ α π ξ ξ Γ Γ + γ 0 z z(0) ep( 2Γ ) ṁ/m = γ cos 2ε γ ṁ m = 1 2 ( 1 z + 2z z 2 +1 m ) ż, z() m() =m(0) 2cos 2 ε(0) sinh 2Γ ()+e 2Γ (). m ε(0) = π/2 3π/2 η m Γ + m 0 Γ + m + Γ Γ γ α α 0 dz z 2 +2μz +1 = αd, μ = γ/ α 1 μ > 1 γ> α 2 μ < 1 γ< α μ = ±1 z 1,2 = μ ± ω ω = μ 2 1 z α >0 α <0 1 μ > 1 ż ż/ α = z 2 +2μz +1, ż α z z z 1 z 2

ż ż z 2 z 1 z z 2 z 1 z α <0 α>0 z ż z 1 z ż α <0 z z 2 z(0) z(0) >z 1 z + ε =+π/2 z 2 α >0 z z 1 + z(0) <z 2 z ε = π/2 + α <0 z(0) <z 1 z z 2 z() z(0) dz z 2 +2μz +1 = α, z(0) >z 1 + ( + dz + z(0) z() dz ) 1 z 2 +2μz +1 = α. z() = [z(0) z1]z2e 2ω α [z(0) z 2]z 1 [z(0) z 1]e 2ω α [z(0) z 2]. α >0 z(0) >z 2 z() dz z 2 +2μz +1 = α, z(0) z(0) <z 2 ( dz + z(0) z() + dz ) 1 z 2 +2μz +1 = α. z() = [z(0) z2]z1e2ω α [z(0) z 1]z 2 [z(0) z 2]e 2ω α [z(0) z 1],

z 1 + μ < 1 ω = 1 μ 2 z 1,2 z 2 +2μz +1 z = ω ζ μ ω 2 (ζ 2 +1) ż α T z T = 1 + α dz z 2 +2μz +1 = π α2 γ 2. α <0 α >0 z(0) T 0 = 1 α2 γ 2 ( π 2 arcan z(0) + μ ω ), α <0 α >0 <T 0 z() = μ + ω an ( arcan z(0) + μ ω ω α ). T 0 T 0 T z() = μ + ω an ( ω α π 2 ), γ ṁ m = γ cos 2ε = 1 4 ( 2z 2z +2μ z 2 +1 z 2 +2μz +1 ) ż, ṁ/m = γ cos 2ε m() =m(0) [ z 2 ()+1 z 2 (0) + 1 ] 1/4 [ z 2 (0) + 2μz(0) + 1 z 2 ()+2μz()+1 ] 1/4, μ > 1 m + z 1,2 μ < 1 T A r IR 3 B(r) = { y IR 3 y <r }, B

A A : B(r) B(r) A. IR 3 αa α A α αa α A α A α B B A α αa α IR 1 A k =(0, 1+ 1/k) k=1a k X C B = B = C C B IR 3 C, B B B B IR 3 X τ = {A α} 1) X τ 2) τ τ τ 3) τ τ (X, τ) X τ V X τ U τ τ B U V U B V B B V V f f : X Y, (X, τ X) (Y,τ Y ) f X U Y τ Y f() V X τ X f(v X) U Y f X f A Y τ Y : f 1 (A Y ) τ X. y = f() X = Y = IR 1 τ

1,2 E E K K 1 E K L E F α 1,2 K L(α 1 1 + α 2 2)=α 1L( 1)+α 2L( 2). L E K F E w K E E E {e k,k =1,...,n} E = i e i h =1,...,n θ h (e k )=δ h k, δk h K 1 h = k 0 h k E n {e k } {θ h } w = y i θ i θ h {e j} e j = R p j ep, E {e i} n n R =(R p j ) p j {e j} {e j} E {e i} {e j} R e i = R q i e q, e i = R q i R p qe p, e j = R p j R q pe q, ( δ p i R q i R p q ) ep = 0, ( δ q j Rp j R q p ) e q = 0. E R q i R p q = δ p i, Rp j R q p = δ q j, R R {e i}

= k e k = k (R 1 ) p k e p = p e p, = k e k = k R p k ep = p e p. n i i R 1 R R E δ h k = θ h (e k)=θ h (R q k eq) =Rq k θ h (e q), θ h θ h = R h mθ m, R {θ i } {θ j } e q θ h (e q)=r h mθ m (e q)=r h mδ m q = R h q. δ h k = R q k R h q. δ h k = θ h (e k )=θ h (R q ke q)=r q kθ h (e q), R = R 1 θ h R E θ h =(R 1 ) h mθ m. e q E {e k} θ h (e q)=(r 1 ) h mθ m (e q)=(r 1 ) h mδ m q =(R 1 ) h q. δ h k = R q k (R 1 ) h q. R = R 1 R E R 1 E E E E n E (R 1 ) 1 = R

E R 1 E E E E E E K T E F K E F K T E F (, y) E y F {e i,i =1,...,n} E {f k,k =1,...,p} F T (, y) E F n p (e i, f k ) T T (, y) = i T (e i, y) = i y k T (e i, f k ). T n p T (e i, f k ) T E F K E F E F E F w E v F K w v (, y) =w () v (y). {θ h } {γ k } E F {e i} {f j } E F θ h γ k E F n p E F R E R F E F T = T ijθ i γ j T ij E F K (θ h,θ k ) T T hk =(R E) l h(r F ) m k T lm, E F T E F E F E F θh γk e h f k E F T hk =(R 1 E )h l (R 1 F )k m T lm,

E r p q p + q = r ( (E,...,E,E,...,E }{{}}{{} p q ) E K R T i 1,...,i q j 1,...,j p = R h 1 j 1... R hp j p (R 1 ) i 1 k1... (R 1 ) iq k q T k 1,...,k q h 1,...,h p.