Вращение Земли. Анализ вариаций и их прогнозирование(Диссертация)

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Author: Зотов Л.В.

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%2@(%6 (&&%& &%@ &"@ (-B%$ '&@ '& '& % (& &"$ "&$ % &%&- "-%%$ d = L, H +w ×H VD!X dt 4( H = I w + h H 4&-& *&*%6 I H %&" %"#6 w H -&" 4&-& &"& -"J%$ 1*6 h H &%&%, 4&-& *&*%6 L H 4&-& *&*% ($ '"# %# *&*% & &"&%, &%#6 %, '% - &%&-%&* -%6 & ,6 &2--@J (-B% '&@ -%, % -4( "( FG6 '&-$J%%&* '"%&* %6 2( 2& '&("&2%& "*&"%, '"%, &-$@J (-B%$ '&@ &4%& &%-%#$* CFIE6 %%$ '"*"&- (-B%$ '&@ (,-@$ "+ &-$@J+ (xp , yp ) = (x, y)IERS + (∆x, ∆y)tidal + (∆x, ∆y)nutation ,

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p

W (t) = R3 (−s )R2 (xp )R1 (yp ),

p

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P N (t) = R3 (−E)R2 (−d)R3 (E)R3 (s),

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p

k=1

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V!!QX

4( η ) &7#% &2%$6 P (W ) ) %'"-% - *%&4&*"%&* '"&"%- '"*"6 - &* *&( %&"A4& ' 7& %'"-% &2"%& 4"(% #-& %# P (W ) = −∇E.

$ &'"(%$ '&'"-& '"*"* '"(,(J4& &$ *&B%& -&'&&-$ & B &"*&6 '"(-"%& &'"('&4"A%& % -,+&( 7&4& &$ 0"- '&4"A%& % '"(,(J "&-% *&B%&6 '&(- % 2,-A -,+&(6 '&*, '" -+&(6 &&%% -,+&(%&4& 4% & &B(*&4&6 *%- %# -# + '"&-&(%,* *&"* - - '"*" (-+&%@ - &(%&4& &-&4& 4%6 -,+&( &&"& &'"($$ -,"B%* yk = f

K

(2)

wki νi

=f

K

i=0

(2)

wki f

N

i=0

(1)

wij xj

j=0

r-$ %#$ ',-$ - -( E=

M 1

2

(yk − dk )2 =

k=1

4"(% &("B &*'&%%, ∂E (2)

∂wki

M 1

2

K 2 (2) f wki νi − di i=0

k=1

(2)

= (yk − dk )

df (uk )

2/

.

(2)

duk

(2)

νi = δk νi ,

4( u = w ν &'"($ '&'"- '"*"* w -,+&(%&4& &$ 0&'"- '"*"* w '"-&4& &$ &'"($@$ '&&-%* -,"B%$ (2) k

K j=0

∂E (1) ∂wij

(2) kj j

=

M k=1

(2) ki

(1) ij

(2)

(yk − dk )

df (uk ) (2) duk

(1) (2) df (ui ) wki xj (1) dui

(1)

= δi xj ,

4( u = w x * &2"&*6 ($ '&&-%$ 4&"* &2"%&4& "'"&"%%$ &A2 -B%,* &,-$ -,% 4"(% % B(&* "&-% $ 7&4& &,-$ (&2%,* '&&- &'"$B%%, 4", 6 %'"-%$ (4 - &&",+ *%%, % &2"%, +&(%,* %'"-%$* 6 * &'&-%, &7#%, H '"&-&(%, %# -# '& -+&(%,* 4%* &"$ 4"&- ( '"&, &"*, ($ -,%$ 4"(%&- &(%&"%& &'"(% '&'"-& '"*"* '"-&( *%A%@ &&%%$ -,+&(%,+ 4%&- & &-,+ 2,%& &2% '""J$ '" (&B% %&&"&4& (%%&4& "&-%$ '&4"A%& (%&6 &%& *&B %$ (%& -"*$6 *&4 -&% &B%& &,%* 4&2%&4& *%** #-& %# '" % 2&A&4& &%,+ *%**&- $ &"%$ &2%$ '"(&-"J%$ "-%$ &%,+ *%**+ '&@$ '#%, 4&"*,6 %'"*" '&'$#&%%, 4&"*, &'*# C!TE &B%& "&B #-@ %#@ - "$( (1) i

N j=0

(1) ij j

1 E(w + p) = E(w) + g t (w)p + pt H(w)p + . . . , 2

4( g(w) = ∇E H -&" 4"(%6 H(w) H *"# '"&-&(%,+ -&"&4& '&"$( 4&"* %&"A4& ' *&*%&* '&'"- '"*"* &'"($@$ *&(#"&-%%& &"*& ∆Wk = ηk Pk + α(Wk − Wk−1 ), 2

4( k H %&*" "#6 &7#% *&*% α '&(2"$ %"- [0, 1] 6 &2, - '&'(%$ - &%, *%**6 &4( P ≈ 06 (& %4& -,2"$6 '" 7&* '"&# &2%$ % (&B% &$ "+&($J*$

4&"* '"*%%& *" &%&-% % "&B% #-& %# (& %&- -&"&4& '&"$( 0""-%-% %@ '"&-&(%& ( -,"B% p = −H −1 g,

BJ - &%&- %@&%&-&4& 4&"* &'*# 0"*%% 7&4& 4&"* "2 '&&B%& &'"(%%& 4% % B(&* A4 4&"* -*& 4% '&$ 4& '"2B% Y 2&%- '""& 4"(% % k A4 r = g − g 6 V = G *&B%& ' (&2%, "#&%%, &"*, k

2&

−1 k

k

k

k−1

rkT Vk−1 rk sk sTk sk rkT Vk−1 rk sTk Vk = Vk−1 + 1 + − , sTk rk sTk rk sTk rk T Vk−1 rk rkT Vk−1 sk sTk Vk = Vk−1 + T − . s k rk rkT Vk−1 rk

0" %%&* %% V = 1 *&B%& &2' '&&B%@ &'"(%%& ''"&*#&%%& *"#, 4% 2,"@ +&(*& 4&"* 4&"* -%2"4) "-"( 4% ''"&*"$6 '&&-%* &("BJ$ - 4"(% %&"*#6 &* "4$"#&%%&4& &" 2&%- e = y − d 0

⎛ ∂e

i

i

i

⎞ ∂e1 . . . e1 ∂w ∂wn ⎜ ∂e21 ∂e2 ⎟ ⎜ e2 ⎟ . . . ∂w ⎜ ∂w1 ⎟ ⎜ ⎟ n e = ⎝ ⎠,J = ⎜ ⎟, ... ⎝... ... ... ...⎠ ∂eM ∂eM ∂eM eM ∂w1 ∂w2 . . . ∂wn ⎛

⎞

1

∂e1 ∂w2 ∂e2 ∂w2

V!!RX

'&* -,"B%$ ($ 4"(% ''"&*"@J 4% 2

*"#, g = J T e,

V!!SX

G = J T J + R,

4( R ) &*'&%%, 4% 2& -,&&4& '&"$(6 &&", *%$@$ "4$"#&%%,* &"&* R = ν · 1 H '"*"&* -%2"4) "-"( ν 6 (&*%&B%%&* % *"# (%# `&* '"*" - % 4&"* '"($ %%6 *%&4& 2&A &2-%%&4& %%$ *"#, J J 6 *%*#$ -,'&%$$ %'"-% −g/ν 1* 4& %% *%A$ - 2& J-%%@ "& %% 4" '"-& 4*& - &"* ($ G6 % &'"(%%&* 7' 4&"* '"-"J$ - 4&"* ) @&% -("%& +&(*&@ &'*%&* "A%@ 0&(2&" &7#% &2%$ 4" -B%@ "& - 4&"*+ &2"%&4& "'"&"%%$ &A2 A&* *& %% '"-&( (%&* &2%@6 2&A& H 8'"'",4-%@9 *%** %&2+&(*& -&-" &*&B%& ('-%& *%% &7#% - '"&# &2%$ '"*"6 '&4"A%& √ ε = E % i A4 &2%$ &B$ 2&A6 * k ε 6 4( k ) &7#% (&'*&4& '""& '&4"A%&6 *&B%& -&'&&-$ -,"B%* T

i

i

w i−1

w

ηi+1 = ηi pd ,

4( p H &7#% *%A%$ η B '&4"A%& % '"-, k ε 6 *&B%& '&&- -,"B% d

w i−1

ηi+1 = ηi pi ,

4( p H &7#% -%$ η &B%& '&&- ($ B(&4& &$ %% 1 d

η ≤ min

ni

,

4( n H &-& -+&(&- i %"&% - & (%&%%&4& '&(+&( ( % J- i

2

W&A& -$% % -& "A%$ ( &,- & &- %"&%%& & %"&%&- - %+ (%&%%&4& '"- '&(2&" ( B % J- (%&6 ( ,- &6 & %(&&%& &-& %"&%&- % '&-& (& (&&%& '% &2&2J%$ ($ &B%&4& 4% A&* 2&A& & %"&%&-6 %&2&"&6 '"-( ('# A** 0&4"A%& &2%$ % &-,+ (%%,+ &'"($ '&4"A%& &2&2J%$ % "2&+ (%%,+ "&* 7&4&6 $6 & '&4"A%& &2&2J%$ - & &%&A%$ &- &2@J+ -,2&"& p %%@ %&&"&4& '"*" η6 &"B@J4& *" &B%& 6 %& -$%%&4& &* -$ - % 0"(&B%& &'"($ -"+%@@ %B%@@ 7& *", '& &"* 2

K N ≤ η ≤ 2Nw (1 + lgNn ), 2

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n

w

2%

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yn =

L

xn−i wi +

i=0

L

xn−j wij +

j=0

L

xn−k (wijk + . . .)

,

k=0

4( B($ ** &&-- "# -+&(%&4& 4% x L &*'&%% %%,* "&* '"-&4& '&"$( *'%& +""&6 &&"@ ( " (@J4& "&-%$ ( &"* *&B 2, '"'% - -( "&B%$ &"" yn =

L i=1

wi xn−i +

L L

wij xn−i xn−j +

i=1 j=0

L L L

wijk xn−i xn−j xn−k +. . .

i=1 j=0 k=0

+

L i1 =1

...

L ik =1

wi1 i2 ...ik xn−i1 . . . xn−ik .

0&(&2%$ *&B 2, '&&-% ($ (%# @2&4& %%%&4& &26 "# A*&-6 '"&4%&"&-%$ 4% (" 2& """%%,+ &%&-% % '&( -,+&(%&4& 4% ("B& % -+&( p&$6 '& - -(*&6 &4& "&( 2

% -"@$ - '""&(6 + '&&-% '& &2%$ '& '#%,* 4&"** &,-$ '&%,* %&6 + '&*&J@ *&B%& &'"( *"#, ("&-%%& (%*& *, '& %2@(%$* &"*"&-%%&4& @ 4% - "%&* -"*% 0& - -(*&6 - -%%,+ %"&%%,+ $+ -$ *B( %"&%* -@$ '" + &(%&-"*%%& -# `& %A& &"B% - *&( p226 &&", '"(&B '"-& &2%$6 - &&"&* - %"&% *%$@$ '"&'&"#&%%& -+&(%&* -,+&(%&* 4% % *&B%& '

∆wij = ηyi yj .

∆W = ηyX = ηX T W X = CW,

4( C ) &"$#&%%$ *"# -+&(%&4& 4% $ 2# &2%$ %B%& '&&- &7#% 2,-%$ γ %&"*"&- -&-& -&" % B(&* A46 &2, ||w|| = 1 wij (t + 1) = wij (t)(1 − γ) + ∆wij

p22&-6 &"$#&%%, %+&($ A"&& '"*%% '" "A% ( % 4-%,+ &*'&%% 4% "(%$ 4% % %-*, &*'&%%, 2% 2 $ '"*%$$ ($ *&&"4%# % &%&- &%"%# &&$ 2&A&4& &2"2,-@J+ -+&(%& 4% %"&%&- & %,* %%,* '"*"* W "&%6 ($ &&"&4& "&$% ||X − W || - *, -,2"%%& *" &,-$ *%*%,*6 %&-$ '&2(* % '&(-"4$ &2%@ -* %"&%* %&&"& 4& &"%& 4&"* &+&%% &2% i %"&% '"&-&($ '& '"- i

i

Wi (k + 1) = Wi (k) + ηi (k)G(i, X)[X − Wi (k)],

4( G(i, X) ) *" &(-6 %'"*" &-$ d2

− 2λi2

G(i, X) = e 2)

,

( λ ) "&-% &(-6 d ) "&$% & %"&%)'&2($6 &&"& *&B -,2"$ '& " 4&"* %"&%%&4& 4 2& 7-%&*6 * 4&"* &+%%6 &(-& W&#*%&-& di

G(i, X) = e− λ ,

d - & -%, "&$%$ ||X −W || *&&"4%# '"*%$@$ ($ &("&-%$ &2"B% J 2& (&2%,* &,-@$ *&&"4%@J$ %"&%%, 6 '&"&%%, % 2 &" %+ *%&B-6 &&", *&4 '"&-&%& &%&@ ''"&*"&- @2@ %%%@ %#@ *%&4+ '"*%%,+6 "A ( # B$ (%%,+ C!IE i

i

, -$ '&"$ 3 ,44 - ' -

0"-(* %&&", &%&-%, -(%$ &" ("%#%,+ "-%% (%*+ *6 '&-(% &&",+ &',-$ 7* "-%%$* CRTE6C!DE6CFFE %* "*&"% '"&A4& %%&4& ("%#%&4& "-%%$ '"-&4& '&"$(6 *@J4& -( dx V!FTX + p(t)x = f (t). dt W(* p(x) f (t) %'"",-%,* %#$* t - & &26 - &&"& "2$ '"&%4""&- %%& "-%% f (t) ≡ 06 "-%% V!FTX %,-$ %%,* &(%&"&(%,* %%& "-%% * "A% − p(t)dt

x = ce

,

c = 0.

A% %&(%&"&(%&4& "-%%$ *&B%& '& *&(&* -"# '&&$%%& %& 2( (,-$ %&&"&4& %&4& "A%$ 2.

%&(%&"&(%&4& "-%%$ &2J4& "A%$ &(%&"&(%&4&6 (&*%&B%%&4& % &%% c 1

−

x=e

f (t)e

p(t)dt

p(t)dt

−

dt + c1 e

p(t)dt

.

* &2"&*6 &2J "A%6 &',-@J "&"@ x(t) (%*& *, '"-&4& '&"$( - &(%&*"%&* &-&* '"&"%-6 % -+&( &&"& '&' -&(- f (t)6 'A$ - -( t

x(t) = x0 (t) +

h(ξ)(t − ξ)dξ. 0

$ *'& &-& (%*& *, -$% %%,+ &- %* -"*% 2( *%A$ lim x0 (t) = 0, t→0

%&--A$$ "&"$ 2( &',-$ %,* "A%* %4"&*)-"& t x(t) =

h(ξ)f (t − ξ)dξ, −∞

4( h(ξ) ) *'%$ +"" -&-$ %#$ B% '"( %4""&-%$ -$ "-%,* %@6 '&&* & ($ "*,+ %,+ * h(ξ) &% & %$ A ($ '&&B%&4& %%$ "4*%6 % "&"@ &23 % &,-@ -$% %%$ f (t) - 2(J* 0 h(ξ) = h(ξ)

ξ < 0, ξ > t . 0≤t<ξ

%%& ("%#%& "-%% n)4& '&"$( ',-$ -( a (t)x + a (t)x + ... + a (t)x˙ + a (t)x = u(t), V!FDX 0

(n)

1

(n−1)

(n−1)

2&

n

x˙ = x , x¨ = x , . . . , x = x %-%,* %#$*6 V!FDX -&($ * n "-%% '"-&4& '&"$( ⎧ 1

(n−1)

2

⎪ x˙ = x1 ⎪ ⎪ ⎪ ⎨x˙ = x 1

⎪ ... ⎪ ⎪ ⎪ ⎩x˙

n−1

0"'A* V!FDX - -(

n−1

2

= f (t, x, x1 , . . . , xn−1 )

V!F!X

f (t) ≡ 06 "-%% 2( &(%&"&(%,* %%,* ("%#%,* &'"&"&* %&-* x(n) + p1 (t)x(n−1) + . . . + p(n−1) (t)x˙ + pn (t)x = f (t),

L[x] = x(n) + p1 (t)x(n−1) + . . . + p(n−1) (t)x˙ + pn (t)x.

% &2( -&-&* %%&

L[αx + βy] ≡ αL[x] + βL[x],

4( α β H &%%, A%* &(%&"&(%&4& "-%%$

L[x] = 0

%'"",-%,* % [a, b] &7#%* p (t) 2( %%$ &*2%#$ x = c x n %%& %-*,+ % 7&* &" "A% x '"&-&%,* '&&$%%,* &7#%*

- &7#%, p (t), i = 1, 2, . . . , n '&&$%%,6 %, "A%$ &(%&"&(%&4& "-%%$ *&4 2, %(%, - -( x = e 6 4( k H '&&$%%$ 0&(%&- ( +""& "-%% i

n

i i

i=1

i

i

kt

a0 (t)k (n) + a1 (t)k (n−1) + . . . + a(n−1) (t)k + an (t) = 0,

&&"&4&6 '" &- & - &"% p (t), i = 1, 2, . . . , n "%,6 %+&($$ n %%& %-*,+ "A% &4( &2J "A% * -( i

x = c1 ek1 t + c2 ek2 t + . . . + cn ekn t . 22

B &"% k 2( * "%& α 6 * 2( &&--&- &*2%#$ "A% i

α i −1

i

cji eki t = (c0i + c1i t + c2i t2 + . . . + cαi −1 tαi −1 )eki t .

j=0

A%* %&(%&"&(%&4& "-%%$ L[y] = f (t)

2( ** 4& %&&"&4& %&4& "A%$ x˜ &2J4& "A%$ &(%&"&(%&4& "-%%$ x = x˜ +

n

ci xi .

i=1

A% x˜ *&B%& '&(&2"6 -&'&&-$ *&(&* -"# '&&$%%& *&(&* &A 0&(% @$ - &*6 & -%& -$J & &(%&4& '"*" "A% &(%&"&(%&4& "-%%$ K(t, s)6 (&--&"$@J &-$* ˙ K(s, s) = K(s, s) = . . . = K (n−2) (s, s) = 0,

K (n−1) (s, s) = 1,

& - 7&* %,* "A%* %&(%&"&(%&4& "-%%$ 2( t x˜ = x(t) =

K(t, s)f (s)ds, t0

(&--&"$@J %%,* &-$* x(t0 ) = x(t ˙ 0 ) = . . . = x(n−1) (t0 ) = 0.

$ %"'"#$ %# K(t, s) *&B 2, &-5 ' (& *&*% -"*% s (%*$ * %+&( - &&$% '&&$6 *J% && -,-%& & f (t) &%& & %$ A '"&*B s < t < s + ε6 '"* *' 7& , "-% D ε

s+ε

fε (τ )dτ = 1.

s

//

&4(6 '" ε → 06 - '"('&&B%6 & f (t) H @2$ %&'&&$%%$ %#$6 *&B%& '&6 & ε

s+ε lim xε (t) = lim K(t, τ )fε (τ )dτ = K(t, s).

ε→0

ε→0

V!FFX

s

$ %#$ %& %-% *'%& +"" 0"(-%#@ f (t) - -( **, %# f (t)6 &%,+ & %$ A % *,+ '"&*B+ ∆t 6 *&B%& "*"- -$% %'"",-%& (-@J ,6 %&B% -$% *4%&-%%,+ *'&i

i

t x(t) =

K(t, s)f (s)ds ∼ =

N

K(t, si )f (si ).

i=1

t0

0&& "-%% V!F!X ( %%@ (%*@ * '&"$( n6 #&%"%$ "&"$ %&4&6 *'& &-&4& &23 -%&- &',-$ %4"&*)-"& t x(t) =

h(ξ)f (t − ξ)dξ. 0

B%, +"" (%*& *, '&@$ '" '&'% % -+&( %&&",+ &-,+ %# 6 % -+&( '&' f (t) = δ(t) H δ)%#$ "6 '&* *'%@ +"" *,6 %&4%@ V!FFX t x(t) =

h(ξ)δ(t − ξ)dξ = h(t). 0

0&'% % -+&( %# p-( V!!!X ( +"" *, t

x(t) =

h(ξ)H(t − ξ)dξ = Hp(t), 0

/

%,-*@ '"+&(%& %# 0&'% % -+&( &*'%& 7'&%%, f (t) = e ( % -,+&( B 7'&%%6 (&*%&B%%@ % '"(&%@ %#@ *, pt

t x(t) =

h(ξ)ep(t−ξ) dξ = W (p)ept . 0

0" "2& ("%#%,* "-%%$* 2,- (&2%& '&&- &'"&"%, '&(+&( 2&%* dk x D x= k, dt k

&4( "-%% V!FDX *&B 2, '%& - -(

(n) (n−1) + . . . + a(n−1) (t)D + an (t) x = F (D)x = u(t), a0 (t)D + a1 (t)D

- &&"&* -,"B% F (D) = a0 (t)D(n) + a1 (t)D(n−1) + . . . + a(n−1) (t)D + an (t)

%,-$ &'"&"%,* *%&4&%&* &B%& *6 & F (D)ekt = ekt F (d), F (D2 ) sin at = sin atF (−a2 ), F (D2 ) cos at = cos atF (−a2 ),

V!FGX

F (D2 )ekt v(t) = ekt F (D + k)v(t).

&* (-$ &'"&" "A% "-%%$ x=

1 F (D)

% %'"",-%@

1 u(t). F (D)

&* -&(-$ F (D) % x $-$$

1 u(t) = u(t). F (D)x = F (D) F (D)

/

u(t)

$-$$

&*& -,A6 % -+&( (%*& *, '&($ 7'&%% u(t) = e 6 & *&B%& ' pt

x=

1 1 pt u(t) = e = W (D)ept , F (D) F (D)

4( W (D) H '"(&%$ %#$ *, 1',-$ %& "A% " -" '&* t

x(t) =

t

h(ξ)ep(t−ξ) dξ = ept 0

h(ξ)e−pξ dξ = ept L [h(ξ)],

0

4( L [f ] H &(%&&"&%% '"&2"&-% ' @( *&B%& @6 & '"(&%$ %#$ (%*& *, W (D) =

1 =L [h(t)] F (D)

%'&"(-%%& -$% " '"&2"&-% ' *'%& +""& *, 0" -,-&( 7&4& " '"*" p *%$$ % &'"&" D B '"('&4$6 & a = 16 * *'& &- $-$$ %& * % $-$$ %&6 "2$ -,'&%$ (-+&"&%% '"&2"&-% ' 1

L b [f ] = L[f (t), p] + L[f (−t), −p].

p""* (%*& *, $-$$ B &*'%$ &%$ +"" W (iω)6 "($@J$$ % *'(%@ A &-@ ψ &%, +"" A(ω) = |W (iω)|,

ψ(ω) = arg[W (iω)].

% -+&( (%*& *, '&' 4"*&% 4% u(t) = a cos(ωt + ϕ)6 & &&* 2( x(t) = aA(ω) cos(ωt + ϕ + ψ(ω)),

&( -(%&6 & % -+&( %%& (%*& *, '&' 4% & '"&* S(ω)6 '" -,+&(%&4& 4% 2( * -( W (ω)S(ω)6 '" 7&* B(& &-$@J *&B '&$-$ &-& B% *%$ *'( /

3 , %

&"$ -"J%$ 1* -&"4 #%" * "&$ % &%&- "-%%$ VD!X6 &&"& *&B 2, '"'%& '&&*'&%%%& &%&@ (& &-$@J+ '"-&4& '&"$( - -( "-%% `") -$6 AΩm ˙ 1 + (C − A)Ω2 m2 + Ωc˙13 − Ω2 c23 + h˙ 1 − Ωh2 = L1 , AΩm ˙ 2 − (C − A)Ω2 m1 + Ωc˙23 − Ω2 c13 + h˙ 2 + Ωh1 = L2 , CΩm ˙ 3 + Ωc˙2 3 + h˙ 3 = L3 ,

V!F X

- &&",+ Ω = 7.292 · 10 "(e H &"& -"J%$ 1*6 C H &-&6 A H 7-&"%, *&*%, %"# 1*6 *, -%, m (@ '&'"- -&" 4&-& &"& −5

i

ω1 = Ωm1 ,

ω2 = Ωm2 ,

ω3 = Ω(1 + m3 )

&*'&%%, c &'"($@ -&*J%$ ∆I %&" %"# i

⎞ ⎛ I12 = − xyρdV I = − zxρdV I11 = (y 2 + z 2 )ρdV 13 2 2 ⎠ ⎝ I= I22 = (z + x )ρdV I23 = − I21 = I12 2 yzρdV I32 = I23 I33 = (x + y 2 )ρdV I31 = I13

% &%&%&4& *&*% &- (-B%$ &'"($$ h=

ρ[r × v]dV

'"-& "-%% !F &$ &*'&%%, -%A%+ L "-%%$ `")-$ *&4 2, '"'%, - -( im ˙ + m = ϕ, σc dϕ3 dm3 = , dt dt

/%

V!FIX

( σ H %("&- &6 m = m -&2B(@J %#6 ϕ = ϕ + iϕ 6 c

1

+ im2

1

6 - '"-& &$

2

1 dc23 dh2 2 c + Ω + (Ω + Ωh − L2 ), 13 1 Ω2 (C − A) dt dt 1 dc13 dh1 (Ω2 c23 − Ω + Ωh2 − + L1 ), ϕ2 = 2 Ω (C − A) dt dt t 1 ϕ3 = 2 (−Ω2 c33 − Ωh3 + Ω L3 dt). ΩC ϕ1 =

V!FQX

0

0" -+&(%&* -&(- &*'&%%* ϕ(t)6 ϕ (t) "A%$ '"-&4& -&"&4& "-%% !FI *@ -( z

⎡

m(t) = eiσ0 t ⎣m0 − iσ0

t

⎤

ϕ(τ )e−iσ0 τ dτ ⎦ ,

−∞

V!FRX

m3 (t) = ϕ3 (t) + m30 ,

4( m m ) %&&", &%%,6 &'"($@J$ %%,* &-$* * &2"&*6 &"& -"J%$ 1* -$% &*'&%%& -&2B(%$ ϕ (t) &-(%,* &2"&*6 ( -&"&4& "-%%$ V!FRX A% ($ (-B%$ '&@ "*&"* 2& '&("&2%& %& *&B 2, '"'%& - -( 0

30

3

t m(t) = eiσ0 t m0 − iσ0

ϕ(τ )eiσ0 (t−τ ) dτ,

−∞

-$% %%,+ &- ($ &-& *, & -"*%* +&( % %6 - &'"($$ %,* "A%* %&(%&"&(%& *,6 *@J* -( %4")-" ]%("&-$ & & '"(-$$ - -( i , σc = 2πf0 1 + 2Q /

4( f H & &2%6 Q H (&2"&%& &4%& &#%*6 (%%,* - CRDE f = 0.843 # - 4&(6 Q = 175 (% &%&-%,+ (6 -&%@J+ '" (&-% -"J%$ 1* H &#% -&2B(@J+ %# 0"(&%& %# ($ "-%%$ (-B%$ '&@&- $-$$ σ W (D) = , V!FSX iD + σ *'%& +""& h(t) = −iσ e . V!GTX 0

0

c

c

c

iσc t

3 5 6

*%&* W@ ""2&%, *&(, ($ &#% -&" &&$%$ (%*& *, C!DE6 CFGE " *% '&-&$ ""%%&6 % &%&- '"&"%,+ (%%,+ & "'"(% %,+ -%6 &%$ '& %2@(%$* &&$% *%&4&*"%& (%*& *, &-$* '"*%*& *&( $-$@$ %2@(*& (%*& *,6 %&""&-%& 4% A*&-6 *"&-& &( &#% B*&4& 4% &%&-% % "A% "-%%$ "4" *B( %& '&4"A%&@ &#%-%$ B*&4& 4% %-$* %2@(% 0 (%*$ * (% - *"%&)-&"%&* -( x(t) ˙ = G(t)x(t) + F (t)v(t),

4( v(t) H -+&(%& 4%6 J -4& A* 2@(*$ -% z(t) = C(t)x(t) + r(t)

&$4% '&4"A%&$* r(t) 0&&B* %2@(%$ '&'@ ("%, *&*%, -"*% j = 1N¯ $ ("%& *, &"*"@J " *&B 2, (% - -( x(j + 1) = A(j)x(j) + B(j)v(j), /)

($ %'"",-%& &% *&B 2, ("&-% tj+1 x(j + 1) = Φ(tj+1 , tj )x(j) + Φ(tj+1 , τ )F (τ )v(τ )dτ, tj

( Φ(t , t ) H %(*%%$ *"# *, 1%$ &-"#&%%@ *"# A*&- R =< rr > *$ - %&&", *&*% -"*% '"&"%, &#% &&$%$ x &-"#&%%& *"#, '&4"A%& P 6 B*, 4% *&B%& &#% '& +* i

j

T

− j

− x

x+ (tj ) = x− (tj ) + K(tj )(z(tj ) − C(tj )x− (tj )), K(tj ) = Px− (tj )C T (tj )[C(tj )Px− (tj )C T (tj ) + R(tj )]−1 , P + (tj ) = P − (tj ) − K(tj )C(tj )Px− (tj ), x− (tj+1 ) = Φ(tj+1 , tj )x+ (tj ).

V!GDX

"# K(t ) %& %-% *"#, %$ 1'A* (%*@ *&( -"J@J$ 1* - *"%&) -&"%&* -( - "*%+ '"&"%- &&$% m ˙ = Gm + F χ, V!G!X j

4( m =

m1 m2

6

χ1 χ= χ2

6 *"#, 4& '& - -(

F =

−β −σ0 σ0 −β

= −G,

4( σ β H (-%$ *%*$ %("&-& &, 0

σc = σ0 + iβ = 2πFc + i

A% "-%%$ *, V!G!X

|λI − G| = 0

( +""

λ1,2 = −β ± iσ0 . /.

πFc . Q

%(*%%$ *"# *, *&B 2, %(% - -( Φ(t, t0 ) = Θ(t)Θ−1 (t0 ),

4( Θ(t) = Hϕ(t)c 0 &%% D6 &4( *"# &2-%%,+ -&"&- *&B 2, '&% - -( *"# ϕ(t) * -(

1 1 H= , i −i

λt e 1 0 . ϕ(t) = 0 eλ2 t

,%$ (@ ($ %(*%%& *"#, V!G!X -,"B% Φ(t, t0 ) = e−β(t−t0 )

cos σ0 (t − t0 ) − sin σ0 (t − t0 ) . sin σ0 (t − t0 ) cos σ0 (t − t0 )

'"6 '&&-%* &"* V!GDX6 +* *%&-& "# &* -+&(%&4& -&2B(%$ *&B%& ' - -( m+ (tj ) = m− (tj ) + K(tj )(z(tj ) − C(tj )m− (tj )), K(tj ) = Pm− (tj )C T (tj )[C(tj )Pm− (tj )C T (tj ) + R(tj )]−1 , P + (tj ) = P − (tj ) − K(tj )C(tj )Pm− (tj ), tj+1 ξ(tj+1 , τ )v(τ )dτ. m− (tj+1 ) = Φ(tj+1 , tj )m+ (tj ) + F

V!GFX

tj

0" 7&* ($ %2@(% '&&- *&( z = Cm + r,

4( C H %-,"&B(%%$ (4&%%$ *"#6 r ) A* %2@(% &-"#&%%& *"# R(t) -(%&6 * %2@(* %2@(@$ &2 &-$@J (-B%$ '&@ /&

33 &$'#

0" &2"2& %2@(%,+ (%%,+ %"(& -&%@ (6 '&-%%, %&""%& ( 2& %&-, &%&%& &A2& %2@(%6 2& *@ %(%-%%& "A%6 %,* &-*6 &% % (&--&"$@ &-$* &""%& '&

(*"5 DX "A% (&B%& J-&- !X "A% (%-%%& FX "A% %'"",-%& - & +&(%,+ (%%,+ &4& -"*$ '&(&2%, ( % %+&( "A%$ GT) + 4&(+ UU - ($ "A%$ + ( +&%&-,* 2, '"(&B% *&( "4$"# CR!E6CRFE6CRGE % '"4&(% ($ "A%$ (6 ($ &&",+ -%&6 & "A% '"%(B %&&"&* *%&* *%&B- ( *&4 2, &%%, &-%&)&""%,+ '& +&%&- ( 0 '"&2"&-% ($ %%,* &'"&"&* A x = Au.

&""%$ ( -&%&-%$ *%*# %#&%

u

*&B 2, "A%

V!GGX ( u ) '"&"%, '"('&&B%$ &%&%& "A%$6 α H '"*" "4$"# %%, %#&% &$ & ""$ )4"%B *%*# -(" %-$ %* %6 +""@J4& '"&"%, '"('&&B%$ &2 &*&* "A%6 *%&B%%&4& % '"*" "4$"# '"&"%, '"('&&B%$ & "A% *&4 -&($ '"('&&B%$* & %#6 &&"&* &%& '"%(B 6 '& '"%(BJ4& L "A%$6 *@J4& *%*%@ %&"*6 '&&2%&4& '"- %2@(%$* x6 '"-&( % - '"-& V!GGX -( α||u|| 0& "A%$6 '"%(BJ4& W *%*%& %&"*& '"-&( 2&" α(||u|| + ||u|| ˙ ) A% %+&($ '* *%*# V!GGX - '"&# "# *%A@J*$ %%* '"*" "4$"#6 &4&-%%,* &2&2J%%& %-$& Φ = ||x − Au||2 + α||u − ua ||2 ,

a

2

2

2

½&

2

2 1

!#'#./ &* & / ! ""/ ($#0!' " 1

/2

$ &'"&"6 (%%&4& '"2B%%& *"# C 6 %2@(% l t '&4"A%&$* ε δ &&--%%&6 '& &'&-*&4& '& &%& "A%$ z -($ '& "#&%%& +* z α = (C T C + α)−1 C T l,

4( z H "A%6 &&--@J %%@ '"*" "4$"# α *%A@J$ '&(&-%& 1%% α '&(2"$ &&-- ""* &2&2J%%& %-$ α

||l − Cz α || = δ + ε||z α ||.

&""%$ &'"#$ ("%#"&-%$ B *&B 2, "4$"&-% 0 A H %4"%, &'"&"6 $("&* K b x = Au =

K(ξ − s)u(s)ds = K ∗ u. a

0" '& "A%$ L "" *%*# 2

dΦ =0 du

'"-&( &'"&"%&* -,"B%@

A x = Wreg x, u= AA+α

( A"+&* &2&%%& &'"$B%6 W ) "4$""@J &'"&" &%& &2 "A% ',-$ - -( reg

ˆ xˆ K = F (Wreg )F (x), uˆ = ˆ K ˆ +α K

4( F (f ) fˆ ) ")'"&2"&-% V! X 0" '& "A%$ W "" *%*# '"&2" -( dΦ = (A A + α)u − A x + α¨ u = 0, du /

2 1

- &%& &2 "A% ',-$ - -( uˆ =

ˆ xˆ K ˆ K ˆ + α(u)(1 + ω 2 ) K

.

0"+&($ - &%@ &2 &2"%& '&&-%* &"*, & -" V!QX6 '&* u = F −1 (F (Wreg )F (u)) = F −1 (F (Wreg )) ∗ u = Kreg ∗ u,

&( -(%&6 & "A% *&B 2, '&%& -"& %2@(% "4$""@J* $("&*6 &&"& ($ $ '& "A% L %+&($ '& &"*5 2

Kreg (ξ) = F

−1

ˆ K ˆ K ˆ +α K

1 = 2π

ˆ K eiωξ dω, ˆ ˆ −∞ K K + α ∞

($ $ '& "A% W H '& &"*5 2 1

Kreg (ξ) = F

−1

1 (Wreg ) = 2π

∞

−∞

ˆ K ˆ K ˆ + α(1 + ω 2 ) K

eiωξ dω.

0"*" "4$"# -%&- "#&%%& '&(2"$ &4&-%%& &2&2J%%& %-$& p(α) = ||Auα − x|| − δ − ||uα ||,

4( δ H '&4"A%& %2@(%6 h H '&4"A%& (%$ &'"&"

4 ? , ' ' 8! ! ! ' @ 7 A- +7.

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^ 3 ? <# + 0; - c E - 1 ^+ + +

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? *+ + - # + 0 1 BJ -BJ* 0+1

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tidal model

сек

сек

code tidal model code-tides

? BJ -BJ` + 9:; ! + 0 1 * 8 0+1

"$(&- %2@(% W '"(-$$ "(&* &J-*,* ) + %&(%&"&(%& "",-%& (%& -,'&%% &&%%&4& '"&2"&-%$ " V* !!!X '&-&$ '" @%@ &2 &- 8'&&2"%&4&9 76 *@J4&$ - (%%,+ xsfK V* '"&B% :X 3 ; <44

&'"&&* '"&+&B(%$ 4"*&% - '" xsfK)&#%& '"&(* F6G6 6I6Q6R6S6DT6DD #&- - %"&- *%&4 (&- CRIE6 CFQE6 CSIE W,& (%& @%6 & &*6 & 7& H "6 '"&+&B(% &&"&4& *&B 2, -$%& *&(* &2"2&6 '"*%$-A*$ #%"&* xsfK 7&* 0&& &(%* &%&-%,+ 7'&- &2"2&6 2, %"'&$#$ &"26 *, '"('&&B6 & " *&4 2, (-* '#%& '"&-(%%& (&-% '&&6 & %"'&$#$ t &

A4&* ∆t (%%,+6 %%& (%%,+ A4&* ∆T > ∆t6 &&",+ '"- 4"*&%$ &-$@J$ '"&(&* T '"-&( '"&-%@ 7%"4 7& 4"*&% % &+ ν 6 BJ+ - %"- *B( &* %- V!RX +&(%&4& '"&%"'&"&-%%&4& "$( ]&, ν (@$ &&%&A%* k

∆T 1 = νk k

k

∆T 1± T

,

4( k '"%* %"%, %%$ `& 7 V* " FIX -&% '" '&&-% "%,+ -(&- %"'&$#6 %&& *%A$ - '%)%"'&$# % &2"% -%&* 7 V* "( !!DX6 &&", -&% '" '"*%% 4%6 &("BJ* 7%"4@ - '" % -,&+ &+6 &'"# -%$ %"- *B( &*

)? e55 c + +3

* $& &$$ $5. 6377* 0777 0-1 $&)'$ ; '

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