К ТЕОРИИ СИММЕТРИЧЕСКИХ МЕХАНИЧЕСКИХ СИСТЕМ

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  • . . 2010. . 2. . 1

    18

    . 3, 99,12=t ( 0~

  • 19

    , , . - . . - , - -. , , , , - . [4], , . . . - -, () - . , - , . - - . , . .

    - , - . - - . - () . -- - , . - , . -, , [5-12].

    - - - . , - ,

  • . . 2010. . 2. . 1

    20

    - -- .

    - -, . - . [11]. -, - - .

    1. () - ,

    MyvutyxyxvutMMR xvut == ),,(),(),,,,(,: ),,(23

    ),,,,( RvutMyx - .

    )(,, ooooo vuvut

    yvutyx xooovut ooo ),,(),(),,( = , ., Myx - M = 3)(, RM , :

    a) x M - 0> ,xU

    zvut y),,( xU ,, xUxy ,0 Iuv ],[ ;uv < ) :

    ;,,),,(),,(),,( wuvuywutywuvvut xxx = (1) ;,),,(),,( vuxuvtyvut yx = (2)

    ,,),,( vuyytut x = (3) .,,,, RvutMyx

    . { } Itx yvut ),,( - , x ut = - y = tvt , .

    2. - M = 3)(, RM , M

    ,),,(),,( ywvwuwtyvut xx +++= (4)

  • 21

    .,;,,,, MyxRwvutvu 3. -

    , - - (.. - ).

    4. -, (.. - - , ).

    - ( ), :

    ( ) ( )x x xu t y ut y= , (5) ( )1x yt y t x= , (6)

    1x y y= , (7) ( )

    xx y t y xt u z u t z= , (8)

    1 1

    x z z xy y

    v v a v v av v

    =

    , (9)

    Rvtu ,,,1 ; Mazyxv ,,,,0 . . 1 3 (5) (9) -

    vut ,, .

    . 1. (5)

    - -

    (5) (7), -:

    ( ) ( ) ( ) ( )1 1 1 1t ta a = x yx y x y , (10)

    ( 1)( 1) ( 1)xx y y xt z t z = . (11)

  • . . 2010. . 2. . 1

    22

    . 2. (8) t = -1

    . 3. (9)

    . 4 ( (10)) 2=t .

  • 23

    . 4. (10) 1. -

    :

    ,,...,1,0),(),(2),( nltqCqtqBqqtqAq lilijil

    ijl ==+++ (12)

    ( RtMqMn = ,,dim ), .

    1. (12) , .., ,

    =

    =n

    i

    ili

    ili qtqBqtqB

    1),(),( .

    2. , (12), .

    2. -,

    nlqCqqBqqqAq lilijil

    ijl ,...,1,0)()(2)( ==+++ , (13)

    ( MqMn = ,dim ), .

  • . . 2010. . 2. . 1

    24

    3. ( QT ,, ), M -

    ( nqqqMq ,...,,, 21 -), T

    );,(),(),(21

    12 tqcqtqbqqtqaTTTTi

    iji

    ijo ++=++=

    ),...,,( 21 nQQQQ = , ),...,2,1(),(),(),( nktqqtqqqqtQ kiki

    jikij

    k =++= , , (13),

    ;~klkij

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    =

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    1;~

    );q

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    a

    qa(k

    jikij

    ijk

    jik

    +

    =

    21 ;~;

    1,1,

    1 n

    lk

    n

    jiklij aAaA ==

    ==

    ;~)(~)( 21

    21

    klji

    ik

    klki

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    qb

    qba

    taB

    +

    =

    ;~)( 21

    klk

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    tbC +

    = ( nlkji ,...,2,1,,, = ).

    4. , (12), -, , -:

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  • 25

    ;0)

    ()(

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    ,

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    ++

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    BABAB

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    BABABBABABAB

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    ;0)(2

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    =++

    +++

    ++

    +

    +

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    22(2)4

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    ++

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    22222

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    )222(2)22

    ,

    ,

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    +

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  • . . 2010. . 2. . 1

    26

    ;0)(2

    )()

    (2)(2

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    =++

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    +

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    3. , .

    4. .

    1. .. .

    .: , 1979. 416 . 2. .., .. -

    . .: , 1995. 448 .

    3. . -. .: , 1973. 188 .

    4. . . .: , 1985. 209 . 5. Matveyev O.A., Panshina A.V. Quasigroups on manifolds with trajec-

    tories // Webs and quasigroups. 1995. P. 88 105. 6. .., .. -

    // . , 1992. . 1. . 59 60.

    7. .., .. - // 34- - . .: , 1998. . 31 32.

    8. .., .. //

  • 27

    36 , . .: , 2000. . 2122.

    9. .., .. - // - . , 2001. . 62-68.

    10. .., .., .. // - - - . . 9. .: -, 2006. . 22 24.

    11. Matveyev O.A. On quasigroup theory of manifolds with trajectories / Webs and quasigroups. Tver, 2000. . 129 39.

    12. .. -// . --. . 3-4, ., 1998. . 10 15.

    O. Matveyev, A. Panshina o the theory of symmetric mechanical systems. The broad classes of mechanical systems c locally symmetric manifold and a flat af-

    fine connection, which as a result of the proposed construction of an accurate differential-geometric and algebraic descriptions are compared.

    Key words: mechanical systems, affine connection, quasigroup, symmetric space.

    07.04.10

    621.9 .. , ., .. , . . , ., .. , - . , ., . (4872) 33-24-38, preys@klax.tula.ru (, , )

    - - - .

    : , - , .

    -

    (), ,

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