EE?ANUeEEE OI?AA?NEOAO ?IAI? OA?ANA OAA*AIEA

E?ioiaeue Iaaei Aiaoie?eiae/

OAeE 539.3

Ae?OAA INIIAIA A?AIE*IA CAAeA*A

OAI??? I?OAEIINO? AeEss OI?A

Niaoe?aeuei?noue: 01.02.04 – iaoai?ea aeaoi?i?aiiai oaa?aeiai o?ea

Aaoi?aoa?ao aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Ee?a – 1999

Aeena?oaoe??th ? ?oeiien

?iaioa aeeiiaia a Ee?anueeiio oi?aa?neoao? ?iai? Oa?ana Oaa/aiea

Iaoeiaee ea??aiee: aeieoi? o?ceei-iaoaiaoe/ieo iaoe,

/eai-ei?aniiiaeaio IAI Oe?a?ie, i?ioani?

Oe?oei Aiae??e Oaioaiiae/,

Ee?anueeee oi?aa?neoao ?iai? Oa?ana Oaa/aiea,

caa?aeoaa/ eaoaae?e oai?aoe/ii? oa i?eeeaaeii?

iaoai?ee

Io?oe?ei? iiiiaioe: aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

Iiae?eue/oe TH??e Ieeieaeiae/,

?inoeooo iaoai?ee ?i. N. I. Oeiioaiea IAI

Oe?a?ie, caa?aeoaa/ a?aeae?eo ?aieia??

eaiaeeaeao o?ceei-iaoaiaoe/ieo iaoe

Ai?aaiue Aieiaeeie? Ieaen?eiae/,

?inoeooo a?ae?iiaoai?ee IAI Oe?a?ie,

noa?oee iaoeiaee ni?a?ia?oiee

I?ia?aeia onoaiiaa: Aea?aeaaiee oi?aa?neoao “Euea?anueea iie?oaoi?ea”
(i. Euea?a)

Caoeno a?aeaoaeaoueny “ 8 ” aa?aniy 1999 ?. i “ 14 ” aiaeei? ia
can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee E 26.001.21 i?e Ee?anueeiio
oi?aa?neoao? ?iai? Oa?ana Oaa/aiea (252127, i. Ee?a — 127, i?iniaeo
Aeooeiaa 2, ei?ion 7, iaoai?ei-iaoaiaoe/iee oaeoeueoao).

C aeena?oaoe??th iiaeia iciaeiieoenue o iaoeia?e a?ae?ioaoe? Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea (252033, i. Ee?a — 33, aoe.
Aieiaeeie?nueea, 64).

Aaoi?aoa?ao ?ic?neaiee “ 20 ” eeiiy 1999 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee Eaie/ O. TH. CAAAEUeIA
OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue ? nooi?iue aeine?aeaeaiino? oaiaoeee aeena?oaoe??.
Iaeiei c ooiaeaiaioaeueieo iai?yie?a ?icaeoeo iaoaiaoe/ii? oai???
i?oaeiino? ? iiaoaeiaa oi/ieo ?ica’yce?a aaciaeo a?aie/ieo caaea/ aeey
eaiii?/ieo o?e. Ia i?ioyc? inoaiiueiai noi??//y oai??y i?oaeiino?
caaaaoeeany ?ica’yceaie iniiaieo a?aie/ieo caaea/ i?aeoe/ii aeey an?o
o?e, a?aeianaieo aei eii?aeeiaoieo nenoai, ui aeiioneathoue ?icae?eaiiy
ci?iieo a ??aiyii? Eaieana. Aea iaei aeine?aeaeaieie caeeoaeenue caaea/?
oai??? i?oaeiino? aeey o?e, a?aeianaieo aei o. ca. oeeee?aeieo
eii?aeeiao, oiaoi eii?aeeiaoieo nenoai, aea ia? i?noea iaiiaia
?icae?eaiiy ci?iieo a aa?iii?/iiio ??aiyii?. Aei iacaaieo nenoai
iaeaaeaoue oi?i?aeaeuei? oa a?noa?e/i? eii?aeeiaoe, eii?aeeiaoi?
iiaa?oi? yeeo ooai?ththoue a?aeiia?aeii oi? ? e?iciiiae?aia o?ei oa
a?-noa?e ? aa?aoaiiiiae?aia o?ei. Aeaia aeena?oaoe?eia ?iaioa i?enay/aia
iiaoaeia? oi/iiai aiae?oe/iiai ?ica’yceo ae?oai? iniiaii? a?aie/ii?
caaea/? oai??? i?oaeiino? aeey oi?a, yea, ca eeaneo?eaoe??th
I. ?. Ionoae?oa?e?, iieyaa? o aecia/aii? iai?oaeaii-aeaoi?iiaaiiai noaio
i?oaeiiai o?ea, yeui a?aeiieie ? ia?ai?uaiiy oi/ie eiai iiaa?oi?.
?ioa?an aei oe??? caaea/? oa ?? aeooaeuei?noue coiiaeththoueny, ye aaea
a?aecia/aeiny, ii??aiyii iaeith aeine?aeaeai?noth caaea/ oai???
i?oaeiino? aeey oi?a oa i?inoi?iaith iaiaeiica’yci?noth oi?i?aeaeueieo
o?e. I?ae iaaeinoaoiueith aea/ai?noth caaea/ i?oaeiino? aeey oi?a
?icoi??oueny a?aenooi?noue aiae?oe/ieo ?ica’yce?a aeinoaoiuei i?inoiai
aeaeo, ye? a aeicaieyee i?iaiaeeoe ye?niee aiae?c caaea/?. O a?aeiieo
aeine?aeaeaiiyo a?aie/i? caaea/? aeey oi?a, ye i?aaeei, caiaeeeenue aei
iane?i/aiieo e?i?eieo aeaaa?a?/ieo nenoai aeineoue neeaaeiiai aeaeyaeo,
ui aeiioneaee o?eueee /enaeueiee ?ica’ycie; iica oaaaith eeoaeeny oaeiae
iniaeeaino? ?ica’yce?a, ni?e/eiai? i?inoi?iaith aeaica’yci?noth
oi?i?aeaeueieo iaeanoae.

Oaia aeena?oaoe?eiiai aeine?aeaeaiiy ia? o?niee ca’ycie c i?ia?aiaie
iaoeiai-aeine?aeii? ?iaioe eaoaae?e oai?aoe/ii? oa i?eeeaaeii? iaoai?ee
Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea, a oiio /ene? c
eiiieaeniith iaoeiaith i?ia?aiith Ee?anueeiai oi?aa?neoaoo ?iai? Oa?ana
Oaa/aiea ia 1997–2000 ??. ca oaiith “Aeine?aeaeaiiy caeiiii??iinoae
aeaoi?ioaaiiy neeaaeieo iaoai?/ieo no?oeoo? c o?aooaaiiyi yaeu ? aoaeo?a
ca’ycaiino? iie?a ??cii? i?e?iaee ? ?ic?iaea iaoiae?a ?o e?euee?niiai
aiae?co”.

Iaoa ? caaea/? aeine?aeaeaiiy. Aaoi? noaaea ca iaoo iiaoaeiao oi/iiai
?ica’yceo ae?oai? iniiaii? caaea/? oai??? i?oaeiino? aeey oi?a a
aiae?oe/i?e oi?i?. Cie?aia, aaoi? noaaea ia?aae niaith oae? caaaeaiiy:

c’ynoaaiiy iniiaieo aeanoeainoae aaeoi?ieo a?aie/ieo caaea/, a yeeo
iiaa?oiath caaeaiiy a?aie/ieo oiia ? aeaica’ycia oi?i?aeaeueia iiaa?oiy,
ia i?eeeaae? caaea/? Noiena aeey oi?a, yea ? iaei?ino?oei ??ciiaeaeii
a?aie/ieo caaea/ oai??? i?oaeiino? a ia?ai?uaiiyo;

a?aeooiaooth/enue a?ae io?eiaiiai oi/iiai ?ica’yceo inaneiao?e/ii?
caaea/? Noiena aeey oi?a, aeaoe ?ica’ycie iiae?aii? caaea/? oai???
i?oaeiino?, ca?aoe aeo?aeio caaea/o aei iane?i/aiii? nenoaie e?i?eieo
aeaaa?a?/ieo ??aiyiue no??/eiai? no?oeoo?e c i?i?iaeueiei /eneii no??/ie
o iao?eoe? nenoaie;

iiaoaeoaaoe oi/iee aiae?oe/iee ?ica’ycie iane?i/aiieo e?i?eieo
aeaaa?a?/ieo nenoai c o?eae?aaiiaeueieie iao?eoeyie;

oeyoii ocaaaeueiaiiy io?eiaieo ?acoeueoao?a iiaoaeoaaoe ?ica’ycie
ae?oai? iniiaii? a?aie/ii? caaea/? oai??? i?oaeiino? aeey oi?a o
caaaeuei?e iainaneiao?e/i?e iinoaiiaoe?, i?ea?aoe ?? aei ?ica’ycaiiy
iane?i/aiieo o?eae?aaiiaeueieo aeaaa?a?/ieo nenoai iaeiioeiiiai aeaeo;

anoaiiaeoe oiiae, ca yeeo ae?oaa iniiaia a?aie/ia caaea/a oai???
i?oaeiino? aeey oi?a ia? iaeiicia/iee ?ica’ycie.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a. Aia?oa io?eiaii oi/iee
aiae?oe/iee ?ica’ycie caaea/? oai??? i?oaeiino? a ia?ai?uaiiyo aeey
oi?a, i?e/iio caaaeueiee i?aeo?ae, ui aeei?enoiaoaaany aaoi?ii,
aeicaiey? iaea?aeoaaoe oi/i? ?ica’ycee a?aeiia?aeiiai eeano caaea/ aeey
?ioeo o?e, a?aeianaieo aei oeeee?aeieo eii?aeeiaoieo nenoai. Ia iniia?
i?iaaaeaiiai aeine?aeaeaiiy ?ica’yce?a aaeoi?ieo a?aie/ieo caaea/
iaoaiaoe/ii? o?ceee a iaiaeiica’ycieo i?inoi?iaeo iaeanoyo iiaeaii
oiiae, ca yeeo a?aie/i? caaea/? oai??? i?oaeiino? a oi?i?aeaeueieo
iaeanoyo iaoeiooue iaeiicia/iee ?ica’ycie, ? iieacaii, ui iiaoaeiaaiee
aaoi?ii ?ica’ycie ? iaeiicia/iei. Cai?iiiiiaaii iaoiae oi/iiai
aiae?oe/iiai ?ica’ycaiiy iane?i/aiieo e?i?eieo aeaaa?a?/ieo nenoai
no??/eiaiai oeio, ca aeiiiiiaith yeiai io?eiaii ?ica’ycie iane?i/aiieo
nenoai c o?eae?aaiiaeueieie iao?eoeyie, aei yeeo caaeany ?icaeyaeoaaia
a?aie/ia caaea/a. Iaea?aeaii oi/i? ?ica’ycee iecee a?aie/ieo caaea/
oai??? i?oaeiino? oa noieniaeo oa/?e aeey oi?a ? iiaeaii aeaoaeueiee
aiae?c oeeo ?ica’yce?a.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a. Io?eiai? a aeena?oaoe??
caaaeuei? ?acoeueoaoe oa aeniiaee iathoue ia?aaeon?i oai?aoe/io
oe?ii?noue, i?ioa i?aaenoaaeai? ?ica’ycee /anoeiieo caaea/ iiaeooue aooe
aoaeoeaii aeei?enoai? a iecoe? i?eeeaaeieo oa ?iaeaia?ieo ?ic?iaie.
?ica’ycie caaea/? Noiena aeey oi?a iiaea aooe canoiniaaiee a
oaoiieia?/ieo i?ioeanao, aea oe?iei aeei?enoiao?oueny iiaeaeue a’ycei?
Noieniai? ??aeeie: o?i?/ieo oaoiieia?yo ci?ooaaiiy oa inaaeaeoaaiiy
/anoeiie o ?ic/eiao, i?e i?iaeooaaii? i?eno?i?a ia ??aeeeo e?enoaeao, o
a?ioaoiieia?yo. Iiaoaeiaai? ?ica’ycee caaea/ oai??? i?oaeiino? aeey oi?a
iiaeooue iaoe oni?oia canoinoaaiiy i?e ?ic?aooieo iiaaae?iee oa
e?eoa???a i?oeiino? eiino?oeoe?e, ui i?noyoue aei?noe? aeeth/aiiy
oi?i?aeaeueii? oi?ie.

Ai?iaaoe?y ? ioae?eaoe?y ?acoeueoao?a aeena?oaoe??. Iniiai?
?acoeueoaoe, io?eiai? a aeena?oaoe?ei?e ?iaio?, iaiaeii?aciai
aeiiia?aeaeeny ? iaaiai?thaaeeny ia iaoeiaeo nai?ia?ao “I?iaeaie
iaoai?ee” eaoaae?e oai?aoe/ii? oa i?eeeaaeii? iaoai?ee Ee?anueeiai
oi?aa?neoaoo ?iai? Oa?ana Oaa/aiea i?ae ea??aieoeoaii
/eaia-ei?aniiiaeaioa IAI Oe?a?ie A. O. Oe?oea, aeiiia?aeaeeny ia 3-io
I?aeia?iaeiiio neiiic?oi? oe?a?inueeeo ?iaeaia??a-iaoai?e?a o Eueaia?
(Oe?a?ia, i. Euea?a, 1997), ia I?aeia?iaei?e iaoeia?e eiioa?aioe??
“No/ani? i?iaeaie iaoai?ee ? iaoaiaoeee” (Oe?a?ia, i. Euea?a, 1998), ia
iaoeia?e eiioa?aioe?? Annual GAMM conference at the University of Metz
/GAMM-99/ (O?aioe?y, i. Iaooe, 1999). Oace aeiiia?aeae iioae?eiaaii.
Iniiai? ?acoeueoaoe aeine?aeieoeueei? ?iaioe ciaeoee a?aeia?aaeaiiy a
o?ueio ioae?eaoe?yo o oaoiaeo iaoeiaeo aeaeaiiyo.

No?oeoo?a oa ia’?i aeena?oaoe??. Aeena?oaoe?y no?oeoo?ii neeaaea?oueny
c? anooio, i’yoe ?icae?e?a (12 i?ae?icae?e?a), ui aeeth/athoue iaeyae
e?oa?aoo?e, aeniiae?a, nieneo aeei?enoaieo aeaea?ae oa aeiaeaoe?a.
Caaaeueiee ianya aeena?oaoe?? neeaaea? 126 noi??iie, aeiaeaoee A ? A
neeaaeathoue 20 noi??iie, nienie aeei?enoaieo aeaea?ae neeaaea? 6
noi??iie (76 iaeiaioaaiue).

CI?NO ?IAIOE

O anooi? iaa?oioiao?oueny aeooaeuei?noue ? ??aaiue aeine?aeaeaiino?
oaie aeena?oaoe??, aecia/ai? ?? iaoa oa caaaeaiiy, aea?oueny
oa?aeoa?enoeea ?iaioe, oi?ioeth?oueny iaoeiaa iiaecia, iaa?oioiaaii
oai?aoe/ia oa i?aeoe/ia cia/aiiy aeine?aeaeaiiy, ??aaiue eiai
ai?iaaoe??.

A ia?oiio ?icae?e? “Iaeyae e?oa?aoo?e” iaaiai?ththoueny ioae?eaoe??
a?o/eciyieo ? ca?oa?aeieo aaoi??a, i?enay/ai? iiaoaeia? ?ica’yce?a
caaea/ oai??? i?oaeiino? ? caaea/ noieniaeo oa/?e aeey o?e, a?aeianaieo
aei oeeee?aeieo eii?aeeiao ?, cie?aia, aeey o?e oi?i?aeaeueii? oi?ie.
Ia?oith ?iaioith a oeueiio iai?yieo ? noaooy A. Aaiaa??ia (1873), a ye?e
aeeeaaeaii caaaeueiee i?aeo?ae aei iiaoaeiae ?ica’yce?a caaea/ oai???
i?oaeiino? aeey o?e, a?aeianaieo aei ni?yaeaieo eii?aeeiaoieo nenoai
iaa?oaiiy. Ia aeaeue, naia a caaea/? aeey oi?a aaoi?ii aoei aeiiouaii
iiieeeo i?e aecia/aii? ?ica’yce?a aa?iii?/iiai ??aiyiiy aeey
oi?i?aeaeueieo iaeanoae. A i?aoeyo TH. ?. Nieiaeiaa,
TH. I. Iiae?eue/oea, A. N. Ee?eethea a?aie/i? caaea/? oai??? i?oaeiino?
aeey oi?a caiaeeeenue aei iane?i/aiieo e?i?eieo aeaaa?a?/ieo nenoai
no??/eiaiai aeaeo c /eneii no??/ie o iao?eoe? a?ae ainueie aei
/ioe?iaaeoeyoe, ?ica’ycee yeeo aoaeoaaeenue /enaeueii. A noaooyo
N. Aaee?, N. Iaaeaeoiaea?a, N. Ai?aia, I. I’Iaeea ?icaeyaeaeenue
inaneiao?e/i? oa iainaneiao?e/i? caaea/? noieniaeo oa/?e aeey oi?a.
Aeei?enoiaoth/e oaeoe/ii i?aeo?ae, cai?iiiiiaaiee I. Oaaeiia (1905),
a?aie/i? caaea/? noieniaeo oa/?e aeey oi?a a oeeo ?iaioao aoei caaaeaii
aei iane?i/aiieo o?eae?aaiiaeueieo nenoai, ?ica’ycee yeeo oaeiae
ooeaeenue /enaeueiei oeyoii. A?aoiaoth/e oea, a i?aaenoaaeai?e
aeena?oaoe?ei?e ?iaio? caaaeueiee i?aeo?ae aei iiaoaeiae oi/ieo
?ica’yce?a caaea/ oai??? i?oaeiino? nie?a?oueny ia ocaaaeueiaiiy
?ica’yce?a a?eueo i?inoeo nii??aeiaieo caaea/ noieniaeo oa/?e.

:

( SEQ equation \n \* MERGEFORMAT 1 )

ia? aeae

, ( SEQ equation \n \* MERGEFORMAT 2 )

a oi?i?aeaeueieo iaeanoyo caienothoueny o aeaeyae?

, ( SEQ equation \n \* MERGEFORMAT 3 )

, ui aeei?enoiaoaaeeny aaoi?ii i?e ?ica’ycaii? caaea/. O?ao?e
i?ae?icae?e “Caaea/? Ae??eoea oa Iaeiaia aeey oi?a” i?enay/aii iiaoaeia?
?ica’yce?a caaea/ a?aie/ieo Ae??eoea ? Iaeiaia aeey ciai?oiino? oi?a.
Iieacaii, ui caaea/a Iaeiaia aeey oi?a caiaeeoueny aei ?ica’ycaiiy
iane?i/aiieo o?eae?aaiiaeueieo e?i?eieo aeaaa?a?/ieo nenoai, i?e/iio
o?eueee a inaneiao?e/iiio aeiaaeeo aaea?oueny iaea?aeaoe i?inoee
caieiooee ?ica’ycie caaea/?.

) ni?aiaaeathoue c ?ica’yceaie ??aiyiue Noiena i?e aiaeia?/ieo
a?aie/ieo oiiaao.

A oeueiio ae i?ae?icae?e? iaaiai?th?oueny caaaeueia iaoiaeeea
?ica’ycaiiy ??aiyiue ??aiiaaae i?oaeiiai o?ea, ui canoiniaoaaeanue a
aeena?oaoe??. Ne?aeoth/e eeane/iiio i?aeoiaeo, cai?iiiiiaaiiio A. Eaia ?
?icaeiooiio A. Aaiaa??iei oa ?ioeie aaoi?aie, iiaoaeiaa ?ica’yceo
??aiyiiy Eaia

, ( SEQ equation \n \* MERGEFORMAT 4 )

, i?iii?oe?eii? aei ia’?iiiai ?icoe?aiiy

, ( SEQ equation \n \* MERGEFORMAT 5 )

caiaeeoueny aei iine?aeiaiiai ?ioaa?oaaiiy aeaio nenoai
ooiaeaiaioaeueieo ??aiyiue aaeoi?iiai iiey

( SEQ equation \n \* MERGEFORMAT 6 )

)

( SEQ equation \n \* MERGEFORMAT 7 )

.

-aiae?oe/ieo ooieoe?e:

( SEQ equation \n \* MERGEFORMAT 8 )

I?ae ?ica’yceii nenoaie REF CR_system \h (8) ?icoi??oueny ?ica’ycie
caaea/? ni?yaeaiiy, yea iieyaa? o ciaoiaeaeaii? iaei??? c ooieoe?e /a?ac
a?aeiio ni?yaeaio aei ia? ooieoe?th.

Aei ?ica’ycaiiy caaea/? ni?yaeaiiy REF CR_system \h (8) i?eaiaeeoue
ieoaiiy i?i iaeiicia/i?noue ?ica’yce?a aaeoi?ieo caaea/ iaoaiaoe/ii?
o?ceee o aaaaoica’ycieo iaeanoyo, aea aaciaith caaea/ath ? ocaaaeueiaia
caaea/a Ae??eoea aeey aiae?oe/ieo ooieoe?e i?i a?aeooeaiiy aa?iii?/ii? a
iaeano? ooieoe?? ca caaeaieie a?aie/ieie cia/aiiyie, i?e oiia?, ui
ni?yaeaia aei ia? ca oiiaaie Eio?-??iaia ooieoe?y ia? aooe
iaeiicia/iith. A?aeiii, ui aeey aa?aiooaaiiy iaeiicia/iino? ?ica’yceo
iienaii? caaea/? a aaaaoica’ycieo ieineeo iaeanoyo iio??aii iaeeaaeaoe
ia a?aie/i? cia/aiiy ooeaii? ooieoe?? aeaye? iaia?aae caaeai?
iaiaaeaiiy. Aaoi?ii iieacaii, ui oeae ?acoeueoao caa??aa?oueny ? aeey
i?inoi?iaeo oi?i?aeaeueieo iaeanoae. Iaiao?aei?noue aeine?aeaeaiiy
?ica’yce?a nenoaie ni?yaeaiiy REF CR_system \h (8) a caaea/? aeey
oi?a aeeeoo?oueny oei, ui aeoi? ? oene o a’yce?e ianoeneea?e ??aeei?
iia’ycai? ??aiyiiyie oeio Eio?-??iaia, ? i?inoi?iaa aeaica’yci?noue oi?a
aeiaaa? anoaiiaeaiiy oiia, ca yeeo oe? ?ica’ycee aoaeooue iaeiicia/ieie.

, ye ?ica’ycee caaea/? ni?yaeaiiy REF CR_system \h (8) , iathoue
caaeiaieueiyoe ??aiyiiyi

, ( SEQ equation \n \* MERGEFORMAT 9 )

aecia/a?oueny c REF k_harmonic_operator \h (2) . Aeine?aeaeaiiy
?ica’yce?a ??aiyiue REF CR_system \h (8) ?, a?aeiia?aeii, REF
omega_and_theta_equations \h (9) , a iaeanoyo, iaiaaeaieo iiaa?oiath
oi?a, i?iaiaeeoueny a oi?i?aeaeueieo eii?aeeiaoao REF
toroidal_coordinates \h (1) . I?aaenoaaeaiiy aeey ooieoe?e REF
omega_and_theta_equations \h (9) a i?inoi?? c oi?i?aeaeueiith
ii?iaeieiith caienothoueny o aeaeyae?

( SEQ equation \n \* MERGEFORMAT 10 )

, ui caaeiaieueiythoue nenoai? REF CR_system \h (8) :

( SEQ equation \n \* MERGEFORMAT 11 )

aeiaaa? aeeiiaiiy oiiae

. ( SEQ equation \n \* MERGEFORMAT 12 )

-aiae?oe/ieo ooieoe?e a i?inoi?iaeo oi?i?aeaeueieo iaeanoyo.

oi?ioeththoueny c oiia i?eeeiaiiy /anoeiie ??aeeie aei iiaa?oi? o?ea,
ui ? iane?aeeii ?? a’yceino?, ? noaeino? a?ae?iaeeiai?/iiai oeneo a
??aeei? ia iane?i/aiiino?:

, ( SEQ equation \n \* MERGEFORMAT 13 )

, ( SEQ equation \n \* MERGEFORMAT 14 )

, yea oa?aeoa?eco? oene ia iane?i/aiiino?, aac iaiaaeaiiy caaaeueiino?
aaaaea?oueny ??aiith ioeaa?. Oi/iee ?ica’ycie caaea/? iiaeaii aeey aeaio
aeiaaee?a aeai?o caaaeueiiai ?ica’yceo ??aiyiue Noiena, ui aeicaieeei
i?iaanoe aeaoaeueiee aiae?c i?iaeaie.

– ae?oaiai ??aiyiiy nenoaie REF Stokes_equations \h (7) :

, ( SEQ equation \n \* MERGEFORMAT 15 )

a?aie/i? oiiae REF Stokes_boundary_condition_for_u \h (13)
iaaoaathoue aeaeyaeo

, ( SEQ equation \n \* MERGEFORMAT 16 )

.

ia? aeae

, ( SEQ equation \n \* MERGEFORMAT 17 )

. ?ica’ycie oeeo nenoai ia? aeae

, ( SEQ equation \n \* MERGEFORMAT 18 )

caienothoueny o aeaeyae?

( SEQ equation \n \* MERGEFORMAT 19 )

, i?ney /iai io?eio?ii

. ( SEQ equation \n \* MERGEFORMAT 20 )

.

?ioee niin?a iiaoaeiae ?ica’yceo caaea/? Noiena aeey oi?a nie?a?oueny
ia i?aaenoaaeaiiy aaeoi?a oaeaeeino? o oi?i?, yea oioiaeii caaeiaieueiy?
ia?oa ??aiyiiy nenoaie REF Stokes_equations \h (7) :

, ( SEQ equation \n \* MERGEFORMAT 21 )

iathoue caaeiaieueiyoe aeiaeaoeiaiio ni?aa?aeiioaiith

, ( SEQ equation \n \* MERGEFORMAT 22 )

yea aea?aaeaioia ??aiyiith ia?ic?eaiino?. I?e oaeiio aeai?? caaaeueiiai
?ica’yceo ??aiyiue Noiena caaea/a i?eaiaeeoueny aei ?ica’ycaiiy
iane?i/aiii? nenoaie e?i?eieo aeaaa?a?/ieo ??aiyiue c o?eae?aaiiaeueiith
iao?eoeath:

( SEQ equation \n \* MERGEFORMAT 23 )

( SEQ equation \n \* MERGEFORMAT 24 )

ye? ? eiao?oe??ioaie Oo?ue? ?icaeiaiue ooieoe?e REF
Stokes_solution_through_three_functions \h (21) a oi?i?aeaeueieo
eii?aeeiaoao:

( SEQ equation \n \* MERGEFORMAT 25 )

A caaaeueiiio aeiaaeeo ia aaea?oueny iiaoaeoaaoe ?ica’ycie
o?eae?aaiiaeueieo nenoai oeio REF System_3x_Stokes \h (23) , yeee ae
ia i?noea ?aeo?aioieo caeaaeiinoae. Aeey aeaii? caaea/? aaoi?ii
ciaeaeaii oi/iee caieiooee ?ica’ycie nenoaie REF System_3x_Stokes \h
(23) , i?e oeueiio aeei?enoiaoaaany iaea?aeaiee ?ai?oa oi/iee ?ica’ycie
REF cn_dn_with_lambda \h (18) – REF lambda_value \h (20) a?aie/ii?
caaea/? REF boundary_conditions_for_stream_function \h (16) aeey
ooieoe?? oieo:

. ( SEQ equation \n \* MERGEFORMAT 26 )

A aeena?oaoe?ei?e ?iaio? i?iaaaeaii iiaiee aiae?c oa/?? a ieie? oi?a.
Oeyoii ?ioaa?oaaiiy iai?oaeaiue ii iiaa?oi? oi?a ia/eneaii iniiaio
a?ae?iaeeiai?/io oa?aeoa?enoeeo iioieo – neeo iii?o ?ooia? oi?a:

, ( SEQ equation \n \* MERGEFORMAT 27 )

yea a oa?i?iao aa?iii?/ieo ooieoe?e REF
Three_Stokes_functions_in_toroidal_coord \h (25) ia? aeae

. ( SEQ equation \n \* MERGEFORMAT 28 )

o a?aie/ieo aeiaaeeao “cae?eoiai” oi?a (oi?a aac ioai?o) ? “oiieiai”
oi?a. Iieacaii, ui a?aeiioaiiy nee iii?o oi?a oa iienaii? iaaeiei iueiai
noa?e a ia?oiio a?aie/iiio aeiaaeeo ni?aiaaea? c a?aeiia?aeiei
a?aeiioaiiyi aeey cae?eoiai oi?a, io?eiaiei c oi/iiai ?ica’yceo
a?aeiia?aeii? caaea/?. Iaea?aeaii aneiioioe/iee caeii niaaeaiiy neee
iii?o i?e ciaioaii? oiaueie oi?a aei ioey. Iieacaii, ui aneiioioe/i?
ae?ace nee iii?o aeey oiieiai oi?a ? oiieiai ieineiai e?eueoey
ni?aiaaeathoue. Iiaoaeiaaii a?ao?e caeaaeiino? neee iii?o a?ae aaiiao???
oi?a, e?i?? oieo i?e iao?eaii? oi?a iaeii??aeiei ia iane?i/aiiino?
iioieii ??aeeie oa ?ciaa?e.

:

( SEQ equation \n \* MERGEFORMAT 29 )

iathoue caaeiaieueiyoe aeiaeaoeiaiio ni?aa?aeiioaiith

. ( SEQ equation \n \* MERGEFORMAT 30 )

A?aie/i? oiiae caaea/? oi?ioeththoueny o aeaeyae? REF
Stokes_boundary_condition_for_u \h (13) . Caaea/a caiaeeoueny aei
o?eae?aaiiaeueii? nenoaie e?i?eieo aeaaa?a?/ieo ??aiyiue oeyoii
i?aaenoaaeaiiy a?aie/ieo oiia o aeaeyae?

, ( SEQ equation \n \* MERGEFORMAT 31 )

a ??aiyiiy REF elastic_axisymmetric_solution_condition \h (30)
i?eeaeaii aei o?e/eaiii? nenoaie

, ( SEQ equation \n \* MERGEFORMAT 32 )

ni?aiaaea? c nenoaiith REF System_3x_Stokes \h (23) a caaea/?
Noiena.

Aiae?c iaea?aeaiiai oi/iiai ?ica’yceo REF System3x_Stokes_solution \h
(26) o?eae?aaiiaeueii? nenoaie REF System_3x_Stokes \h (23)
aeicaieea aaoi?o cai?iiiioaaoe a ae?oaiio i?ae?icae?e? “Iaoiae iiaoaeiae
oi/iiai ?ica’yceo iane?i/aiieo e?i?eieo aeaaa?a?/ieo nenoai no??/eiaiai
aeaeo” caaaeueiee i?aeo?ae aei ?ica’ycaiiy iane?i/aiieo aeaaa?a?/ieo
nenoai no??/eiaiai oeio c aeia?eueiith e?euee?noth ae?aaiiaeae. ?aeay
iaoiaeo iieyaa? o i?aaenoaaeaii? ?ica’yceo ae?aaiiaeueii? nenoaie oeio
REF elastic_axisymmetric_System3x \h (32) o aeaeyae?

. ( SEQ equation \n \* MERGEFORMAT 33 )

. O aeiaaeeo o?eae?aaiiaeueii? nenoaie ??aiyiue REF
elastic_axisymmetric_System3x \h (32) oe? aea? nenoaie iaoeiooue aeae

, ( SEQ equation \n \* MERGEFORMAT 34 )

. ( SEQ equation \n \* MERGEFORMAT 35 )

Nenoaia REF System3x_equation_for_y_n \h (35) ia? oi/iee ?ica’ycie,
yeee caieno?oueny o aeaeyae?

. ( SEQ equation \n \* MERGEFORMAT 36 )

Aoaeue-yeee ?ica’ycie iaeii??aeii? nenoaie oeio REF
System3x_equation_for_alpha_n \h (34) ia? caeaaeaoe a?ae iaei??? aai
a?eueoa aeia?eueieo eiinoaio. Aeey oiai, uia io?eiaoe ?ica’ycie
iaeii??aeii? nenoaie, yeee aoaea i?noeoe eeoa iaeio noaeo, iaiao?aeii
i?aaenoaaeoe iaeii??aeio nenoaio a iaiaeii??aei?e oi?i?. Oae, nenoaia
REF System3x_equation_for_alpha_n \h (34) caieoaoueny

aecia/athoueny c inoaii?o ??aiyiue ?aeo?aioii:

( SEQ equation \n \* MERGEFORMAT 37 )

nei?ioeoueny.

aeaea?oiaeo eii?aeeiao a?aie/i? oiiae caaea/? a oeee?iae?e/ieo
eii?aeeiaoao iathoue caienoaaoenue

. ( SEQ equation \n \* MERGEFORMAT 38 )

O a?aeiia?aeiino? aei REF elastic_asymmetric_boundary_conditions \h
(38) ia?ai?uaiiy a i?oaeiiio i?inoi?? i?aaenoaaeythoueny o aeaeyae?

. ( SEQ equation \n \* MERGEFORMAT 39 )

-aa?iii?/ieo ooieoe?e

, ( SEQ equation \n \* MERGEFORMAT 40 )

ia? caaa?ooaaoenue ??aiyiiyi

. ( SEQ equation \n \* MERGEFORMAT 41 )

Canoiniaoth/e i?aeo?ae, aiaeia?/iee aei iienaiiai a iiia?aaeiueiio
i?ae?icae?e?, a?aie/io caaea/o aaea?oueny caanoe oaae aei o?e/eaiii?
aeaaa?a?/ii? nenoaie.

A iaio caaea/ao ia/eneaii iaeio c ?ioaa?aeueieo oa?aeoa?enoee
iai?oaeaiiai noaio a ieie? aeeth/aiiy – neeo cnoao, iaiao?aeio aeey
ci?uaiiy aeeth/aiiy ia caaeaio a?aenoaiue. Iiaoaeiaaii a?ao?ee
caeaaeiino? nee cnoao a?ae aaiiao??? oi?a aeey ??cieo cia/aiue /enea
Ioaniia. Ii??aiyiiy nee cnoao aeey aeiaaee?a inaneiao?e/iiai oa
iainaneiao?e/iiai ci?uaiue iieaco?, ui neea cnoao a inaneiao?e/iiio
aeiaaeeo caaaeaee a?eueoa a?ae a?aeiia?aeii? oa?aeoa?enoeee aeey
iainaneiao?e/iiai aeiaaeeo.

A i’yoiio ?icae?e? “Ae?oaa iniiaia a?aie/ia caaea/a oai??? i?oaeiino?
aeey oi?a” ocaaaeueiaii ?acoeueoaoe, io?eiai? a iiia?aaei?o /anoeiao
aeena?oaoe??. I?aenoaaith aeey iiaeeeaino? cae?eniaiiy oaeiai
ocaaaeueiaiiy noaa oie oaeo, ui caaea/? oai??? i?oaeiino? aeey oi?a ye a
inaneiao?e/i?e, oae ? a /anoeii?e iainaneiao?e/i?e iinoaiiaeao a ?aieao
iaeiiai i?aeoiaeo caaeeny aei ?ica’ycaiiy ??aiyiue iaeiiai oeio.

A ia?oiio i?ae?icae?e? “Oi/iee ?ica’ycie ae?oai? iniiaii? a?aie/ii?
caaea/? oai??? i?oaeiino? aeey oi?a” iaea?aeaii caaaeueiee ?ica’ycie
??aiyiiy Eaia, yeee ia? aeae eiia?iaoe?? aaeoi?ii? oa neaey?ii?
aa?iii?/ieo ooieoe?e

, ( SEQ equation \n \* MERGEFORMAT 42 )

ye? ca’ycai? i?ae niaith ni?aa?aeiioaiiyi

. ( SEQ equation \n \* MERGEFORMAT 43 )

. ( SEQ equation \n \* MERGEFORMAT 44 )

Aiaeia?/iei /eiii iiaeaaoe aaeoi? B oa i?aenoaaeaoe oe? i?aaenoaaeaiiy a
REF elastic_vectorial_solution \h (42) , i?ney iacia/ieo ia?aoai?aiue
iaea?aeeii:

( SEQ equation \n \* MERGEFORMAT 45 )

i?e/iio ooieoe?? a e?aeo /anoeiao ??aiinoae iathoue caaeiaieueiyoe
??aiyiiyi

. ( SEQ equation \n \* MERGEFORMAT 46 )

Oiiaa REF elastic_vectorial_solution_condition \h (43) iaaoaa?
aeaeyaeo

. ( SEQ equation \n \* MERGEFORMAT 47 )

A?aie/i? oiiae caienothoueny o aeaeyae?

( SEQ equation \n \* MERGEFORMAT 48 )

. Aiiny/e oe? ?icaeiaiiy a ??aiyiiy REF
elastic_thetak_phik_psik_chik_condition \h (47) , i?eoiaeeii aei
o?e/eaiii? nenoaie

( SEQ equation \n \* MERGEFORMAT 49 )

. Oi/iee ?ica’ycie nenoaie REF elastic_general_system3x \h (49)
aoaeo?oueny ca iaoiaeii REF System3x_general_solution \h (33) – REF
System3x_expression_for_alpha_n \h (37) .

.

O?ao?e i?ae?icae?e “I?i ?ica’yci?noue ocaaaeueiaieo ??aiyiue oeio
Eio?-??iaia a oi?i?aeaeueieo iaeanoyo” i?enay/aii aeine?aeaeaiith
??aiyiue, ye? aeieeaathoue c ia?oi? c nenoai REF Fundamental_equations
\h (6) o iainaneiao?e/ieo caaea/ao. Oe? ??aiyiiy ? ocaaaeueiaiiyie
??aiyiue oeio Eio?-??iaia REF CR_system \h (8) ? iathoue aeae

( SEQ equation \n \* MERGEFORMAT 50 )

) nenoaie REF generalized_CR_systems \h (50) ia?aoai?ththoueny ia
eeane/i? ??aiyiiy Eio?-??iaia. Aaoi?ii iiaoaeiaaii ?ica’ycee nenoai
REF generalized_CR_systems \h (50) a oi?i?aeaeueieo iaeanoyo, a oaeiae
anoaiiaeaii oiiae oeio REF CR_solvability_condition \h (12) , ca yeeo
oe? ?ica’ycee aoaeooue iaeiicia/ieie. Iieacaii, ui iaiaeiicia/i?noue
?ica’yce?a nenoai REF generalized_CR_systems \h (50) iia’ycaia c
aeaica’yci?noth ?icaeyaeoaaieo iaeanoae – ciai?oiino? aai aioo??oiino?
oi?a. Aeiaaaeaii, ui i?e aeei?enoaii? cai?iiiiiaaii? oi?ie caaaeueiiai
?ica’yceo ??aiyiiy Eaia REF elastic_vectorial_solution \h (42) – REF
elastic_vectorial_solution_condition \h (43) ??aiyiiy REF
generalized_CR_systems \h (50) caaaeaee iathoue iaeiicia/iee
?ica’ycie, oiaoi oiiae ?ica’yciino? aeeiiothoueny aaoiiaoe/ii.

A aeiaeaoeo A iaaiaeyoueny iniiai? iaoiaee oa ?acoeueoaoe noiniaii
?ica’ycaiiy iane?i/aiieo nenoai e?i?eieo aeaaa?a?/ieo ??aiyiue, ui
canoiniaoaaeenue aaoi?ii i?e ?iaio? iaae aeena?oaoe??th.

A aeiaeaoeo A i?iaaaeaii aneiioioe/ia ia/eneaiiy noieniai? neee iii?o
aeey oiieiai ieineiai e?eueoey, ui ?ooa?oueny o a’yce?e ianoeneea?e
??aeei?.

Aeniiaee. A aeena?oaoe?? iiaoaeiaaii oi/iee ?ica’ycie ae?oai? iniiaii?
a?aie/ii? caaea/? oai??? i?oaeiino? aeey oi?a. Aiae?c caaea/? ?icii/aoi
c ?ica’yceo inaneiao?e/ii? caaea/? Noiena aeey oi?a – i?inoi?
iiaeaeueii? caaea/?, ui aeicaieeea aeine?aeeoe iaea?eueo aaaeeea?
iniaeeaino? iiaaae?iee ?ica’yce?a aaeoi?ieo caaea/ a oi?i?aeaeueieo
iaeanoyo. Oeaio?aeueia i?noea a ?ica’yceo oe??? caaea/? caeia?
aeine?aeaeaiiy nenoaie ??aiyiue oeio Eio?-??iaia, ye?e caaeiaieueiythoue
ooieoe?? aeoi?o ? oeneo a ??aeei?. Iaea?aeaii oi/iee ?ica’ycie caaea/?
Noiena aeey oi?a o aeaio aeiaaeeao aeai?o caaaeueiiai ?ica’yceo ??aiyiue
Noiena – o aeaeyae? ooieoe?? oieo oa o aeaeyae? eiia?iaoe?? o?ueio
aa?iii?/ieo ooieoe?e. A ae?oaiio aeiaaeeo a?aie/ia caaea/a caaeany aei
?ica’ycaiiy iane?i/aiii? o?eae?aaiiaeueii? e?i?eii? aeaaa?a?/ii?
nenoaie. I?e ?ica’ycaii? oe??? nenoaie aoei aeei?enoaii ciaeaeaiee
ca’ycie i?ae caaaeueieie ?ica’yceaie ??aiyiue Noiena aeacaieo aeae?a, ui
aeicaieeei iiaoaeoaaoe ?ica’ycie o?eae?aaiiaeueii? nenoaie, yeee ia
i?noeoue ?aeo?aioieo ni?aa?aeiioaiue. Ia/eneaii neeo iii?o, ui ae?? ia
oi?, iiaoaeiaaii e?i?? oieo oa ?ciaa?e aeey aeiaaeeo iao?eaiiy oi?a
iaeii??aeiei ia iane?i/aiiino? iioieii ??aeeie.

ni?aiaaea? c a?aeiia?aeiith nenoaiith a caaea/? Noiena aeey oi?a.
A?aeooiaooth/enue a?ae aeaeo iaea?aeaiiai a caaea/? Noiena oi/iiai
?ica’yceo o?eae?aaiiaeueii? nenoaie, cai?iiiiiaaii iaoiae iiaoaeiae
oi/iiai aiae?oe/iiai ?ica’yceo iane?i/aiieo aeaaa?a?/ieo nenoai
no??/eiaiai aeaeo c aeia?eueiei /eneii ae?aaiiaeae. Aeei?enoiaoth/e oeae
iaoiae, io?eiaii oi/i? ?ica’ycee nenoai aeey inaneiao?e/ii? oa
iainaneiao?e/ii? caaea/. Ia/eneaii neee cnoao, iaiao?aei? aeey ci?uaiiy
oi?i?aeaeueiiai aeeth/aiiy ia caaeaio a?aenoaiue acaeiaae oa
ia?iaiaeeeoey?ii aei a?n? neiao??? oi?a.

sse ocaaaeueiaiiy iaea?aeaieo ?acoeueoao?a, iiaoaeiaaii oi/iee
?ica’ycie ae?oai? iniiaii? a?aie/ii? caaea/? oai??? i?oaeiino? aeey
oi?a. Caaaeueiee ?ica’ycie ??aiyiiy Eaia aeae?aany o aeaeyae?
eiia?iaoe?? aa?iii?/ieo aaeoi?ii? oa neaey?ii? ooieoe?e. A?aie/io
caaea/o caaaeaii aei iane?i/aiii? iiiaeeie o?eae?aaiiaeueieo
aeaaa?a?/ieo nenoai, eiao?oe??ioe yeeo iathoue iaeiioeiiee aeae. A
yeino? i?eeeaaeo canoinoaaiiy io?eiaiiai caaaeueiiai ?ica’yceo
?icaeyiooi caaea/o i?i iai?oaeaiee noai i?oaeiiai i?inoi?o c aei?noeei
oi?i?aeaeueiei aeeth/aiiyi i?ae ae??th aa?oeeaeueieo coneeue,
i?eeeaaeaieo ia iane?i/aiiino?. Iiaoaeiaaii aith?e ii?iaeueieo oa
aeioe/ieo iai?oaeaiue ia iiaa?oi? aeeth/aiiy. Ia caaa?oaiiy ?ica’yceo
ae?oai? iniiaii? a?aie/ii? caaea/? oai??? i?oaeiino? aeey oi?a
i?iaaaeaii aiae?c ocaaaeueiaieo nenoai oeio Eio?-??iaia, ui aeieeathoue
a iainaneiao?e/iiio aeiaaeeo. Oeei nenoaiai caaeiaieueiythoue
Oo?ue?-eiiiiiaioe ia’?iiiai ?icoe?aiiy oa i?iaeoe?e aaeoi?a aeoi?o.
Io?eiaii ?ica’ycee aeacaieo nenoai, aeey iaeiicia/iino? yeeo iathoue
aeeiioaaoeny iaai? oiiae ia eiao?oe??ioe ooieoe?e, ui aoiaeyoue aei
aeaieo ??aiyiue. Aeiaaaeaii, ui cai?iiiiiaaiee aeae caaaeueiiai
?ica’yceo ??aiyiiy Eaia caaacia/o? aaoiiaoe/ia aeeiiaiiy oeeo oiia.

Iniiai?  iieiaeaiiy aeena?oaoe?eiiai aeine?aeaeaiiy ciaeoee nai?
a?aeia?aaeaiiy a ianooiieo ioae?eaoe?yo o oaoiaeo iaoeiaeo aeaeaiiyo:

E?ioiaeue I. A., Oe?oei A. O. Oi/iee ?ica’ycie caaea/? Noiena aeey oi?a
// A?niee Ee?anueeiai oi-oo, Na?. iao. ? iao. – 1998. – Aei. 2. – N.
48–55.

E?ioiaeue I. A. Inaneiao?e/ia caaea/a oai??? i?oaeiino? a ia?ai?uaiiyo
aeey oi?a // A?niee Ee?anueeiai oi-oo, Na?. o?c.–iao. iaoee. – 1999. –
Aei. 1. – N. 30–35.

E?ioiaeue I. A. Aea? caaea/? oai??? i?oaeiino? aeey i?inoi?o c aei?noeei
oi?i?aeaeueiei aeeth/aiiyi // Iaoeiiciaanoai. – 1999. – ? 3. – N. 32–38.

E?ioiaeue I. A. Ae?oaa iniiaia a?aie/ia caaea/a oai??? i?oaeiino?
aeey oi?a. – ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa
o?ceei-iaoaiaoe/ieo iaoe ca niaoe?aeuei?noth 01.02.04 – iaoai?ea
aeaoi?i?aiiai oaa?aeiai o?ea. – Ee?anueeee oi?aa?neoao ?iai? Oa?ana
Oaa/aiea, Ee?a, 1999.

Aeena?oaoe?eia ?iaioa i?enay/aia iiaoaeia? oi/iiai ?ica’yceo ae?oai?
iniiaii? caaea/? oai??? i?oaeiino? aeey oi?a. Ocaaaeueithth/e
?acoeueoaoe iiia?aaei?o aeine?aeaeaiue, aaoi?ii iiaoaeiaaii caaaeueiee
?ica’ycie ??aiyiue oai??? i?oaeiino?, yeee aeicaiey? aoaeoeaii
?ica’ycoaaoe oe?ieee eean a?aie/ieo caaea/ oai??? i?oaeiino? a
eii?aeeiaoieo nenoaiao c iaiiaiei ?icae?eaiiyi ci?iieo. Aeaiee i?aeo?ae
aeicaieea caanoe ae?oao iniiaio a?aie/io caaea/o aeey oi?a aei
iane?i/aiieo nenoai aeaaa?a?/ieo ??aiyiue c o?eae?aaiiaeueieie
iao?eoeyie. Cai?iiiiiaaii caaaeueiee iaoiae iiaoaeiae oi/iiai
aiae?oe/iiai ?ica’yceo iane?i/aiieo nenoai e?i?eieo aeaaa?a?/ieo
??aiyiue no??/eiaiai oeio, ia aac? yeiai io?eiaii ?ica’ycie ciaeaeaieo
o?eae?aaiiaeueieo nenoai. I?iaaaeaii aeine?aeaeaiiy ?ica’yciino?
a?aie/ieo caaea/ oai??? i?oaeiino? ye aaeoi?ieo a?aie/ieo caaea/
iaoaiaoe/ii? o?ceee a iaiaeiica’ycieo oi?i?aeaeueieo iaeanoyo.

Eeth/ia? neiaa: oi?, oeeee?aei? eii?aeeiaoe, oi/iee ?ica’ycie, caaea/a
a ia?ai?uaiiyo, caaea/a Noiena, iaiaeiica’yci? iaeano?.

E?ioiaeue I. A. Aoi?ay iniiaiay e?aaaay caaea/a oai?ee oi?oainoe aeey
oi?a. – ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.02.04 – iaoaieea
aeaoi?ie?oaiiai oaa?aeiai oaea. – Eeaaneee oieaa?neoao eiaie Oa?ana
Oaa/aiei, Eeaa, 1999.

Aeenna?oaoeeiiiay ?aaioa iinayuaia iino?iaieth oi/iiai ?aoaiey aoi?ie
iniiaiie e?aaaie caaea/e oai?ee oi?oainoe aeey oi?a. Iaiauay ?acoeueoaou
i?aaeuaeoueo enneaaeiaaiee, aaoi?ii iino?iaii iauaa ?aoaiea o?aaiaiee
oai?ee oi?oainoe, iicaieythuaa yooaeoeaii ?aoaoue oe?ieee eeann e?aaauo
caaea/ oai?ee oi?oainoe caaea/ a eii?aeeiaoiuo nenoaiao n iaiieiui
?acaeaeaieai ia?aiaiiuo. I?aaenoaaeaiiue iiaeoiae iicaieee naanoe
aoi?oth iniiaioth e?aaaoth caaea/o aeey oi?a e aaneiia/iui nenoaiai
eeiaeiuo aeaaa?ae/aneeo o?aaiaiee n o?aoaeeaaiiaeueiuie iao?eoeaie.
I?aaeeiaeai iauee iaoiae iino?iaiey oi/iiai aiaeeoe/aneiai ?aoaiey
aaneiia/iuo nenoai eeiaeiuo aeaaa?ae/aneeo o?aaiaiee eaioi/iiai oeia, ia
iniiaaiee eioi?iai iieo/aii ?aoaiea iaeaeaiiuo o?aoaeeaaiiaeueiuo
nenoai. I?iaaaeaii enneaaeiaaiea ?ac?aoeiinoe e?aaauo caaea/ oai?ee
oi?oainoe eae aaeoi?iuo caaea/ iaoaiaoe/aneie oeceee a iaiaeiinayciuo
oi?ieaeaeueiuo iaeanoyo.

Eeth/aaua neiaa: oi?, oeeeeeaeiua eii?aeeiaou, oi/iia ?aoaiea, caaea/a
a ia?aiauaieyo, caaea/a Noiena, iaiaeiinayciua iaeanoe.

Krokhmal P. A. The Second Fundamental Boundary-Value Problem of
Elasticity for a Torus. – Manuscript.

Dissertation for the Candidate Degree in Physics and Mathematics by
speciality 01.02.04 – Mechanics of Solids. – Kyiv Taras Shevchenko
University, Kyiv, 1999.

The dissertation is devoted to the construction of exact solution of
the second fundamental boundary-value problem of elasticity for a torus.

The structure of the dissertation is the following: introduction, five
chapters, which include the bibliography review, conclusion, references
and two appendixes.

In the introduction to the dissertation the author stresses out the
reasons of choosing the theme for dissertation, the innovations
produced, the approbation of the researcher’s results (conferences,
publications etc.).

The First Chapter contains the bibliography review, where a number of
papers on elasticity and Stokes flow problems for a torus are discussed.
By critical analysis of known publications the author accents the
problems and questions that weren’t resolved by his predecessors.

In the Second Chapter the general properties of the toroidal coordinate
system and of the Legendre functions of semi-integer indices are
considered, and the exact solutions of the basic potential problems for
a torus are presented. It is shown that the Dirichlet problem for a
torus has simple and trivial solution, while the Neumann problem
requires solving of infinite three-diagonal linear algebraic systems,
and only in axisymmetrical case the solution can be written in a closed
form.

The construction of solution of the second fundamental boundary-value
problem of elasticity for a torus begins in the Third Chapter with
addressing to the axisymmetric Stokes flow problem for a torus as to the
simplest problem of elasticity. This allows one to omit some complicated
calculations and investigate the most important features of the problem
as a vectorial problem of mathematical physics using the simplified
model statement of a problem. One of the key points in solution of the
problem is the analysis of system of equations of Cauchy-Riemann type,
which is satisfied by the vorticity and the pressure in viscous fluid.
The obtained solutions of this system for the exterior and interior of a
torus are non-single-valued due to the double-connectedness of these
regions. In order to ensure the uniqueness of solutions of the system of
equations of Cauchy-Riemann type one has to impose some preassigned
restrictions on values of the contained functions. This result is a
generalization of the related planar potential problem known as the
generalized Dirichlet problem.

The existence of such preassigned restrictions for the functions of
vorticity and pressure in a fluid must be taken into account when
solving the axisymmetric Stokes problem for a torus by introduction of a
stream-function. In this way the original boundary-value problem is
reduced to solving of an infinite set of algebraic systems of the second
order, each of which contains in its right-hand members an unknown
constant. This constant enters the boundary conditions for the
stream-function and determines the non-simple-connectedness of the
body’s surface. The value of the constant must be chosen in the way that
the aforementioned restrictions for pressure and vorticity functions to
be satisfied. Another approach to construction of solutions of Stokes
flow problems is based on representation of the velocity vector by
combination of three harmonic functions, one of which is the pressure in
fluid. In this case the problem may be reduced to an infinite algebraic
system with a three-diagonal matrix. Using the connection between
solutions of both types, an exact solution of this system was obtained.
This solution is based on constructed stream-function for a torus and
does not contain recursive relations. A detailed analysis of the Stokes
problem for a torus is performed. The drag force is calculated, and the
streamlines and izobars patterns are built. An asymptotical analysis of
the drag force in the limiting cases of closed torus and slender torus
is carried out.

In the Fourth Chapter the constructed solution of Stokes problem for a
torus is generalized for the elastic medium instance. Two problems
concerning the displacement of a rigid toroidal inclusion along and
perpendicular to the symmetry axis in an elastic space are considered.
When constructing the general solutions of the Lame equation in the
axisymmetric and asymmetric cases, the technique by O. Tedone was used.
In accordance to this method, the displacement vector is represented by
combination of axisymmetric harmonic functions, where one of which is
the dilatation. But the number of unknown functions exceeds the number
of the boundary conditions for their determination, and as the last
necessary condition the definitive equation for dilatation has to be
used. By use of this approach the aforesaid axisymmetric and partial
asymmetric boundary-value problems were reduced to infinite algebraic
systems with three-diagonal matrices. When the Poisson number m is equal
to 2, the system of the axisymmetric elastic problem is equal to the
system of the Stokes flow problem for which an exact solution was
obtained. As a generalization of that solution the method for solving of
infinite n-diagonal systems for arbitrary n is suggested. Using this
method, an exact solution for the three-diagonal system is given. For
both problems the force needed to shift the inclusion on a prescribed
distance is calculated, and the asymptotical analysis of the forces is
performed.

The Fifth Chapter is devoted to generalization of obtained results.
Exact solution of the second fundamental boundary-value problem for a
torus is presented. The general solution of Lame equation is represented
by combination of a harmonic vector and the dilatation. The displacement
boundary-value problem is reduced to an infinite set of three-diagonal
algebraic systems with uniform coefficients. As an example the problem
of tension of an elastic space with toroidal inclusion by vertical
loading applied at infinity is solved. The analysis of stress field in
the vicinity of the inclusion is carried out. In conclusion of the
investigation of the second fundamental boundary-value problem for a
torus the analysis is given and the solutions are constructed for the
generalized systems of equations of Cauchy-Riemann type. These equations
arise when solving the asymmetric elastostatics equations. They bind the
Fourier components of vorticity and dilatation. The conditions under
which these equations have unique solutions, and the exact solutions are
obtained. It is shown that these conditions are automatically satisfied
if the suggested general solution of Lame equation is chosen.

Key words: torus, cyclide coordinates, exact solution, displacement
boundary-value problem of elasticity, Stokes problem,
non-simple-connected domains.

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