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Одномерные лагранжианы порожденные квадратичной формой.

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???????? ?????. ??????????
2009, ? 5, c. 33?44
http://www.ksu.ru/journals/izv_vuz/
e-mail: izvuz.matem@ksu.ru
?.?. ????????
?????????? ???????????, ??????????? ????????????
??????
?????????. ? ?????? ?????? ???????????? ???????????? ?????????? ??????????? ? ???????????? ???????????? ????: ?????? ??????? ??? ?????? ??????????, ??????????? ??????
????????? ???????????? ?????? ??????????, ????????? ???????????????? ??????? ???????????? ???????????? ???????????? ????????? ???? x (t) ?? ??????? c (t) ? ????????????
?????? ????????? t = t (s).
???????? ?????: ???????????? ?????????? ?????????, ????????? ???????????????, ???????????? ???????????, ?????????? ????????????? ??????, ?????????? ???????????, ????????????? ??????, ????, ???????????? ?????, ????????? ????????????.
???: 519.3
Abstract. In this paper we apply the classical variational calculus to the Lagrangians of particular
type. Namely, we establish formulas for the 1st integrals and propose a technique for obtaining
invariant 1st integrals. We also deduce the di?erential homogeneity conditions for Lagrangians
with respect to multiplication of the path x (t) by the function c (t) and with respect to the change
of the parameter t = t (s).
Keywords: conformally connected manifold, Euler-Lagrange equation, integral functionals, isotropic
extreme curve, one-dimensional Lagrangians, orthogonal group, path, quadratic form, scalar product.
1. ????????
????????????? ?????? ?? ????????????? ?????????? ????????? ???????????? ??? ?????????? ????????? ???????????? ????????????, ? ??????? ??????????? ???????? ???
??????? ?? ????????? ???????????? (x(k) , x(l) ), ??? x(k) ???? k-? ??????????? ?? ????????? t ?? ?????? x(t) ?? ????????????. ????? ???????? ??????? ???????????, ???????? ?
??????????, ?????????? ????????? ??????????????? ??? ????? ???????????? ????????
???????????? ?????????, ???? ??????????? ??? ?????????? ????????????? ?????? (??. [1],
ДД 7, 8). ?? ????? ???????????, ????????, ??????????? ???????????? ????????????? ??????, ??????????? ???????? ????????? ?????????????, ??????? ???????? ??????? ?. ?????
????????? ??????????????? ????????? ?????? ????????? ? ??????????, ?????? ????? ?????????? ????????????? ??????. ?????????? ?????? ?????????? ??????????? ?????, ???
??? ? ??? ?????? ??????????? ??????? ????????. ???????????, ??? ??? ?????? ????????????? ?????? ????? ???? ????? ?????? ?????????, ??????? ??????????? ???????????? ????????????? ??????.
????????? 26.03.2007
33
34
?.?. ????????
? ?????? ?????? ???????????? ???????????? ?????????? ??????????? ? ????????????
?????????? ????: ?????? ??????? ??? ?????? ??????????, ??????????? ?????? ?????????
???????????? ?????? ??????????, ????????? ???????????????? ??????? ????????????
???????????? ???????????? ????????? ???? x(t) ?? ??????? c(t) ? ???????????? ??????
????????? t = t(s). ????? ????? ?????? ??????? ???? ???????????? ????? ? ??????????
????????????? ???????? ? ([2], ??. III). ??? ????????? ????????? ?????? ????????? ??
??????????? ? ??????? ????????????????? ?????????? ???????? ????????????? ??????
? ?????? ????? ?? ????????? ????? ? Rm . C???????????? ???? ??????? ????? ???????????
????????????? ??????????????? ?? ??????
dM
= [L, x];
dt
dE
= (L, x );
dt
dQ (k)
=
(x , Lx(k) ) ? (L, x),
dt
n
k=0
?????????? ????? ?????????? ??????????? ?? t.
2. ???????? ???????
????? ? ???????????? Rm ?????? ????????????? ???????????? ?????
(x, x) =
m
(1)
gij xi xj ,
i,j=1
??? xi ? ?????????? ??????? x. ????????? ???????? C ? (R, Rm ) ????????? ??????????
???????????????? ??????????? ?? R ? Rm . ?????????? ???????????? ??????? n ?????
???????? ??????? (??????????? ????? ??? ????????????????) ??????????? L : C ? (R, Rm )?
C ? (R, R), ????????? ?? ?????? ?? ???? x(t), ?? ? ?? ??? ??????????? ?? ??????? n ????????????: x (t), x (t), . . . , x(n) (t). ???????, ??? x(k) (t) ? C ? (R, Rm ).
? ?????? ?????? ?? ?????????? ?????????? ??????????? ????? ?????? ??????, ???????????? ? ???? ??????? ??????? ?? ????????? ???????????? (x(k) , x(l) ), ???????????
m
(k) (l)
gij xi xj .
???????????? ?????? (1), ??? k, l ? {0, 1, . . . , n}, ?. ?. (x(k) , x(l) ) =
i,j=0
m
????????? ????? Lx(k) ??????, j-? ?????????? ???????? ?????
?L ij
(k) g ,
?x
i
i=1
??? (gij ) ?
???????, ???????? ??????? (gij ) ???????????? ????? (1). ??????????? ???????
Lx(k) =
l=k
?L
?L
(l)
(k)
x
x
+
2
=
Akl x(l) .
?(x(k) , x(l) )
?(x(k) , x(k) )
n
???????, ??? s-? ?????????? ??????? Lx(k) , ??????????? ?? ???????
s-? ?????????? ??????? Lx(k) , ??????????? ?? ??????? (2), ?. ?.
m
?L
p=1
l=k
gps =
(k)
?xp
?L
x(l)
(k)
?(x , x(l) ) s
+2
?????????????,
m
?L
g
(k)
p=1 ?xp
ps
=
n
m p=1 l=0
(2)
l=0
?(x(l) , x(k) ) ps
?L
и
g =
(k)
?(x(l) , x(k) )
?xp
?L
x(k) .
(k)
?(x , x(k) ) s
m
?L ps
,
(k) g
?x
p
p=1
?????
?????????? ???????????
=
n
m
?L
?(x(l) , x(k) )
l=0
?
(l) (k)
m
i,j=1 gij xi xj
35
gps =
(k)
?xp
p=1
=
n
(l) (k)
m
m ?(xi xj )
?L
gij gps =
?(x(l) , x(k) ) p=1 i,j=1 ?x(k)
p
l=0
(l) (k)
(????????, ??? ??????? ???????????
=
n
l=0
?(xi xj )
(k)
?xp
?? ????? ???? ?????? ??? p = j, ???????)
(l) (k)
m
?(xi xj )
?L
gij gjs =
?(x(l) , x(k) ) i,j=1 ?x(k)
j
(???????? ??? ????? ?? ??? ?????????)
=
l=k
(l) (k)
(k) (k)
m
m
?(xi xj )
?(xi xj )
?L
?L
js
gij g +
gij gjs .
?(x(l) , x(k) ) i,j=1 ?x(k)
?(x(k) , x(k) ) i,j=1 ?x(k)
j
j
???????
l=k
=
l=k
m
m
m
m
?L
?L
(l)
(k)
js
x
gij g +
2x
gij gjs =
?(x(l) , x(k) ) i=1 i j=1
?(x(k) , x(k) ) i=1 i j=1
(l)
(k)
?L
?L
?L
?L
s
s
(l)
x
x(k) ,
x
?
+
2x
?
=
+2
i
i
s
i
i
(l) , x(k) )
(k) , x(k) ) s
?(x(l) , x(k) ) i=1
?(x(k) , x(k) ) i=1
?(x
?(x
l=k
m
m
??? ? ??????????? ????????.
? ???? ????????? ?????????? ???????????? (x, y) ? ??????? (2) ???????
Akl = Alk .
? ?????? (2) ?????? ??????????? ?? t ?? ??????????? L ???????????? ? ????
d
L=
(Lx(k) , x(k+1) ).
dt
n
k=0
?????????????,
n
d(x(l) , x(k) )
?L
d
и
L=
=
(l) , x(k) )
dt
dt
?(x
k,l=0
=
n k=0 l=k
?L
?(x(l) , x(k) )
и (x(l) , x(k+1) ) + 2
=
n n
k=0
??? ? ??????????? ????????.
?L
(x(k) , x(k+1) ) =
?(x(k) , x(k) )
l=0
(l)
(k+1)
Akl x , x
n
(Lx(k) , x(k+1) ),
=
k=0
36
?.?. ????????
????? ????? ?????????? ?????????? L, ????????? ?? ????????? ???????????? (x(k) , x(l) ),
??????????? ???????????? ?????? (1). ????? ??? ??????? ???????? ???? x(t) ????? ????????? ????????
t1
(3)
L(t)dt.
I=
t0
???? t0 ???????????, ? t1 ???????? ????????????, ?? (3) ????? ????????????? ??? ??????????, ????????? ?? ???? x(t) ? ??? ???????????, ? ????? ?? ???????? ????? x1 = x(t1 ).
?? ????? ???????? ??? ???????????? ????????. ???????????? ????? ??????????? (????????? ? ????????) ????? ???
t1 n
(Lx(k) , ?x(k) )dt.
?I = L(t1 )?t1 +
t0 k=0
(k)
??????? ??? ????? ????????????? ??????????? ?????????? ?x ? ?x1 , k ? {0, . . . , n ?
1}. ???????? ? ?????????? ? ??????? k ? 1 ? ?????? ????? ????? ????????? ???????
?????????????? ?? ?????? k ??? ? ??????????, ??? ?x(i) |t0 = 0. ??????? ?????????????? ??
?????? ?????????? ???????????? ????? ???
t1 t1 dv
du
t1
dt = (u, v)|t0 ?
, v dt.
u,
dt
dt
t0
t0
???????
t1
dv
u = L (k) ;
= ?x(k) ; (k)
x
dt
(Lx(k) , ?x )dt = du
d
(k?1) =
t0
dt = dt Lx(k) ; v = ?x
t1 d
(k?1)
(k?1)
L (k) , ?x
, Lx(k) )|t1 + (?1)
dt =
= (?x
dt x
t0
t1 2
d
(k?1)
(k?2) d
2
(k?2)
, Lx(k) )|t1 ? ?x
, Lx(k) + (?1)
L (k) , ?x
dt = и и и
= (?x
dt
dt2 x
t1
t0
t1 k
k?1
p
d
p
(k?p?1) d
k
(?1) ?x
, p Lx(k) + (?1)
L (k) , ?x dt.
иии =
dt
dtk x
t1
t0
p=0
(k)
???????? ? ?????? ?????????? ??????? ????????????????? ??????????? x1
(k)
(k+1)
?t1 ,
(k+1)
?t1 .
?x1 = ?x(k) |t1 + x1
?????????? ? ????
(k)
?x(k) |t1 = ?x1 ? x1
????? ????? ?????
k?1
k?1
p
dp
(k?p?1)
(k?p)
p
(k?p?1) d
p
(?1) ?x
, p Lx(k) =
(?1) ?x1
? x1
?t1 , p Lx(k) =
dt
dt
t1
t1
p=0
=
k?1
p=0
p=0
p
(?1)
?x
(k?p?1)
k?1
p
dp
(k?p) d
p
, p Lx(k) ?
(?1) x1
, p Lx(k) ?t1 .
dt
t1
dt
t1
p=0
???????? ?????????????? ????????? ??????? ???????? ??????????? ?I ?? k ?? 0 ?? n
? ????????????? ?????, ?????????? ?t1 . ????? ???????
?????????? ???????????
37
k?1
n p
(k?p) d
p
?I = L(t1 ) ?
(?1) x1
, p Lx(k) ?t1 +
dt
t1
k=1 p=0
t1 k
k?1
n n
p
d
(k?p?1) d
p
k
(?1) ?x1
, p Lx(k) (?1)
L (k) , ?x dt =
+
+
dt
dtk x
t1
t0
k=1 p=0
k=0
(??????? ??????? ????? ????????????)
n?p
n?1
(k) dp
p
(?1)
x1 , p Lx(k+p) ?t1 +
= L(t1 ) ?
dt
t1
p=0
+
k=1
n?1
p
(?1)
n?p?1
p=0
(k)
?x1 ,
k=0
dp
L
(k+p+1) dtp x
t1
t1
+
n
t0
k=0
dk
(?1) k Lx(k) , ?x dt,
dt
k
???????????? ???????
?I = E?t1 +
n?1
p
(?1)
n?p?1
p=0
(k)
?x1 ,
k=0
???
def
E = L?
t1
dp
L (k+p+1) (L, ?x)dt,
+
dtp x
t1
t0
n?1
n?p
p=0
k=1
(?1)p
x(k) ,
dp
L (k+p)
dtp x
(4)
(??? ????????? ? ????????? ????????? ??????????? ? ????? t1 ), ?
n
dk
L=
(?1)k k Lx(k) .
dt
(5)
k=0
???? t1 ???????????, ?? ??????? (4) ??????????
t1
(L, ?x)dt.
?I =
t0
3. ?????????? ?? ???????????
????? x, y ? Rm ????? ?????????? xi ? yi . ????????? ???????? [x, y] ????? ?????????
????????????
[x, y] = (xi yj ? xj yi ).
??? ????????????????? ?????????????? ???????
n?p?1
n?1
p
def
p
(k) d
(?1)
x , p Lx(k+p+1)
M =
dt
p=0
k=0
???????? ?????? ??????????? ?? t
n?1
n?p?1
n?p?1
n?1
p
p+1
dM
p
(k+1) d
p
(k) d
=
(?1)
, p Lx(k+p+1) +
(?1)
x
x , p+1 Lx(k+p+1) =
dt
dt
dt
p=0
k=0
p=0
k=0
38
?.?. ????????
(??? ??????? ?????????? ?????? ???????????? ? k = 1, ? ??? ??????? ? ? p = 1, ?????
???????)
n?p
n?p
n?1
n
p
p
p
(k) d
p
(k) d
(?1)
(?1)
x , p Lx(k+p) ?
x , p Lx(k+p) =
=
dt
dt
p=0
p=1
k=1
k=0
(??????? ?? ?????? ????? ?????????, ???????????? ??? p = 0, ? ?? ?????? ????? ?
?????????, ???????????? ??? p = n)
n?p
n
n?1
p
(k)
p
(k) d
[x , Lx(k) ] +
(?1)
x , p Lx(k+p) ?
=
dt
p=1
k=1
k=1
n?1
p
(?1)
p=1
n?p
(k)
x
k=0
dp
dn
n
, p Lx(k+p) ? (?1) x, n Lx(n) =
dt
dt
(???????????? ????????? ????? ????????? ??????? ????????????, ????????? ?????? ????????? ? ??????? k = 0)
n
n?1
dp
dn
(k)
p
n
[x , Lx(k) ] ?
(?1) x, p Lx(p) ? (?1) x, n Lx(n) =
=
dt
dt
p=1
k=1
(???????????? ????????? ????? ???????? ??? ?????? ?????? ????????????)
n
n
p
(k)
p d
[x , Lx(k) ] ? x,
(?1) p Lx(p) =
=
dt
p=1
k=1
(? ???????????? ? ??????????? ??????? ???? ? ?? ?? ????????? [x, Lx ])
n
n
p
(k)
p d
[x , Lx(k) ] ? x,
(?1) p Lx(p) .
=
dt
p=0
k=0
? ???? ????????? (2), ? ????? ?????????????? Akl ? ?????????????????? [x(k) , x(l) ] ?????
n
n
(k)
[x , Lx(k) ] =
Akl [x(k) , x(l) ] = 0.
k=0
?????
k,l=0
n
dp
dM
= ? x,
(?1)p p Lx(p) .
dt
dt
p=0
????????? ??????? (5), ???????????? ????????
dM
= [L, x].
dt
??????????? ?????????? ??????????
dE
= (L, x ).
dt
???? ?? ?? ???????? ? ????????? M ????????? ????????? ????????
n?p?1
n?1
p
def
p
(k) d
(?1)
x , p Lx(k+p+1) ,
Q =
dt
p=0
k=0
(6)
(7)
?????????? ???????????
39
?? ?? ??????????? ?? t ????? (???? ??????????) ??????????? ?? ???????
dQ (k)
=
(x , Lx(k) ) ? (L, x).
dt
n
k=0
??? ????????, ???????????? ??????????? L ???????? ??????? ????????? ???????????????, ??????? ? ?????? (5) ????????? ? ????
L = 0.
???????, ??? ? ([3], ?. 39) ??? ?????????? ?????????? ???????????????. ?? ??????????
?????????, ? ????? ?? ?????? (6) ? (7) ???????
??????????? 1. ???????? M ? E ????????? ????? ???????????.
???????? Q ???? ????????? ??? ?????????????? ??????? ???????????? ??????????? L
n
(x(k) , Lx(k) ) = 0).
???????????? ????????? ???? x(t) ?? ?????????? c (? ???? ??????
k=0
????????? ???????? E ? Q ??????????? ???????????? ???????? ????????????? ??????.
???????????? ??????????????????? ??????????
?(?) = det(Mij ? ?gij ),
??? Mij ? ?????????? ??????? M , ????? ????? ????????????? ?????????? ??????????,
??????????? ?? ???????????.
4. ???????? ????????????, ?????????? ???????????? ????????? ??
???????
????? ?????????? ???????? ?????????? ???????????? ????????? ????????? x(t) ?? ???????????? ??????? c(t):
L[c(t)x, (c(t)x) , (c(t)x) , . . . , (c(t)x)(n) ] = L(x, x , x , . . . , x(n) ).
(8)
????????? y = c(t)x ? ?????????????????? ????? ????? ????????? (8) ?? c(p) , ????? ?????
n
n
n
? ki=0 Cik c(i) x(k?i)
?y (k)
(Ly(k) , (p) ) =
Cpk (Ly(k) , x(k?p) ),
Ly(k) ,
=
(p)
?c
?c
k=p
k=p
k=p
??? ??? ?? ?????? ???? ????????? ?????????? ?????? ??? i = p. ????????, ???????????
?????? ????? ????????? (8) ?? c(p) ????? ????. ???? ?? ???????? c(t) ? 1, ???????
n
Cpk (Lx(k) , x(k?p) ) = 0, p ? {0, 1, . . . , n}.
(9)
k=p
????????? ????? ????? ????????? (9) ????? Sp , ????? ??? ??????????? ? ????
Sp = 0, p ? {0, 1, . . . , n}.
(10)
????? ???????, ????? ?????
??????????? 2. ???? ?????????? ????????????? ??????? ???????????? (8), ?? ??????????? ????????? (10).
40
?.?. ????????
??? ???????????? ????????????? ??????????? ?????????
k
i
(?1)p Cpk Cp?i
p?1 = (?1) ,
p=i
??????? ???????????? ??????? ?????????????? ????????.
? ??????? ????? ????????? ????? ???????? ???????
n
dp?1
(?1)p?1 p?1 Sp = Q,
dt
p=1
??????? ??????????? ? ??? ??????????????? ??????? (8).
?????
n
n
n
p?1
p?1 p?1 d
p?1 d
(?1)
Sp =
(?1)
Cpk (Lx(k) , x(k?p) ) =
p?1
p?1
dt
dt
p=1
p=1
k=p
=
n
n (?1)p?1 Cpk
p=1 k=p
dp?1
(L (k) , x(k?p) ) =
dtp?1 x
(????? ??????? ????? ???????????? ? ???????? ??????? ???????? ?????????????????
????????????, ???????)
=
k
n p?1
(?1)
Cpk
k=1 p=1
p?1
Cp?1?i
p?1
i=0
di
L (k) , x(k?i?1)
dti x
=
(????? ?????? ??????? ????? ????????????)
k?1 i
n k
d
(k?i?1)
Lx(k) , x
(?1)p?1 Cpk Cp?1?i
=
=
p?1
i
dt
k=1 i=0
p=i+1
(???????? ????????????? ?????????)
=
n k?1
k=1 i=0
i
(?1)
di
L (k) , x(k?i?1)
dti x
=
(????? ?????? ??????? ????? ????????????)
i
n?1
n?i?1
n?1
n
di
d
(k?i?1)
i
(k)
(?1)i
L
,
x
(?1)
L
,
x
=
.
=
(k)
(k+i+1)
dti x
dti x
i=0 k=i+1
i=0
k=0
?? ??????????? ????????? ????????? ????? Q, ??? ? ??????????? ????????.
?? ??????????? ????????? ? ??????????? 2 ???????
??????????? 3. ???? ?????????? ????????????? ??????? (8), ?? ???????? Q ???????????? ????? ????.
???? ?????????? L ???????? ?????????? ???????? ??????? ?, ?. ?.
L[c(t)x, (c(t)x) , . . . , (c(t)x)(n) ] = (c(t))? и L(x, x , . . . , x(n) ),
(11)
?????????? ???????????
41
?? ?????????????, ???????????? ???, ??? ???? ????????? ??? ?????????????? ???????
(9), ????? ????????, ??? ?????? ????????? (9) ??? p = 0 ????? ??????????? ???????
n
(Lx(k) , x(k) ) = ?L,
k=0
?????? ????? ????? ????? ????????? ???????????? ????? S0 , ?. ?. S0 = ?L. ??????? (10)
????? ??????????? ?????? ??? p ? {1, . . . , n}.
5. ???????? ????????????, ???????????? ???????????? ?????? ?????????
?????? ?????????????? ????????, ??????? ????? ???????? ??????????? ???????????,
??????? ? ?????????????? ???????????? ?????? ????????? t = t(s):
L[x(t), x (t), . . . , x(n) (t)]dt = L[x(s), x (s), . . . , x(n) (s)]ds.
(12)
? ???? ?????? ?????????? ???????? (3) ?? ???????? ??? ?????? ?????????, ?????????????,
??? ???????? ????? ????. ?? ????????? (4) ???????, ??? ??? ??????? (12) E = 0. ???
?????????? (12) ????? ??????????? ??????? ?? ?????????? ????????? ???????. ?????
????? ??????? ??? k-? ??????????? ?? ??????????
k
dk x dt k i dk?i+1 x dt k?i di t
dk x
= k
+
Ck k?i+1
+ (и и и ).
dsk
dt
ds
dt
ds
dsi
i=2
????? ???????? (и и и ) ?????????? ?????????, ?????????? ?????????????, ?? ??????? ????,
d2 t d3 t
??? ??????????? ?? t ??????? ???? ??????? (????????, ds
2 и ds3 ). ??? ??????? ???????????? ?? ????????. ?? ???????? ???????? ??????? ?????????????? ????? ???????????, ???
??? ??? ?? ???????????? ??????? ??????????????? ????????.
??????? ????????? (12) ? ????
L[x(t), x (t), . . . , x(n) (t)]t (s) = L[x(s), x (s), . . . , x(n) (s)]
? ????????????????? ??? ?? ????????? t (s). ????????, ??? ?x = 0, ????? ????? ?????
????????????????? ????? ????? L?t . ?????? ????? ???????? ??????? ??????????? ?? ?????????? ????? ????? ???
n k
(k) k
i (k?i+1) k?i (i)
Ck x
(t ) t + (и и и )
=
Lx(k) , d x (t ) +
i=2
k=1
n
(Lx(k) , x(k) k(t )k?1 ?t +
=
+
n k=1
k=1
Lx(k) ,
k
Cik x(k?i+1) ((k ? i)(t )k?i?1 t(i) ?t + (t )k?i ?t(i) ) + (и и и ) =
i=2
(?????????, ?????????? ??????????? ??????? ?????? ???????: t , t , . . . , ??????? ? (и и и ),
????? ???????)
n k
(k) k?1 i (k?i+1) k?i (i)
Ck x
(t ) ?t + (и и и ) =
Lx(k) , kx (t ) ?t +
=
k=1
i=2
42
?.?. ????????
=
n Lx(k) ,
k=1
k
Cik x(k?i+1) (t )k?i ?t(i)
+ (и и и ) =
i=1
(???????? ??????? ????? ????????????)
n n
(Lx(k) , Cik x(k?i+1) (t )k?i ?t(i) + (. . . )) =
=
i=1 k=i
=
n n?i+1
(Lx(k+i?1) , Cik+i?1 x(k) (t )k?1 ?t(i) + (и и и )).
i=1 k=1
??????????? ????? ? ?????? ????? ?????????????????????? ?????????, ???????
n n?i+1
(Lx(k+i?1) , Cik+i?1 x(k) (t )k?1 ?t(i) + (и и и )).
L?t =
i=1 k=1
???? ? ???????????? ????????? ???????? t(s) ? s, ?? ??? ?????????, ??????? ? (и и и ),
????????? ? ????. ? ???? ????????????? ?????????????? ?t(i) ????? ????? ??????? ???????
?? ?????????? L
n?i
(k)
Ci+1
k+i (Lx(k+i) , x ) = 0 ??? i ? {1, . . . , n ? 1};
(13)
k=1
n
k(Lx(k) , x(k) ) = L.
(14)
k=1
???? ?? ?????? (12) ??????????? ?????????
L[x(t), x (t), . . . , x(n) (t)](t (s))? = L[x(s), x (s), . . . , x(n) (s)],
(15)
?? ???????????? ???????????????? ????? ????????, ??? ?????? (14) ????? ???????????
?????????
n
k(Lx(k) , x(k) ) = ?L,
k=1
? ????????? (13) ????????? ? ????.
6. ??????????????? ???????????? ? ??????? ????????????? ?????
???? ?????????? L ????????? ??????? ? ???????????? ????????? x(t) ?? c(t) (??????????? ????????? (11)) ? ????????? ??????? ? ???????????? ?????? t = t(s) (???????????
????????? (15)), ?? ???? ????? (?, ?) ????? ???????? ???????? ???????????? ??????????? L.
???????? ?????????????? ????????? ??? ??????????? ????????????, ???????????
???????????? ?????? ? ? ?? ?? ????? ?????????? ? ???????????? ???????????? ?????? ?????????, ???????? ???????????? ????? ??????? ???????? x, x , . . . , x(k)
)
(k) ) (x, x)
(x,
x
.
.
.
(x,
x
(x , x)
(x , x ) . . .
(x , x(k) ) ?k = .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(x(k) , x) (x(k) , x ) . . . (x(k) , x(k) )
?????????? ???????????
43
?? ???????? ?????? ????????. ??? ??? ????? ? ?????? ??????? ?????????:
?k [x(t)](t (s))k(k+1) = ?k [x(s)].
?k [c(t)x(t)] = (c(t))2k+2 ?k [x(t)];
??? ????????, ??? ?k ???????? ???????????? ? ???????? ???????????? (2k+2, k(k+1)). ???
???? ????? ?????????? ???????? (3) ??? ?????????? ? ?? ??????? ?? ??????????????,
?????????, ????? ?????????? L ???? ?????? ???????????? (0, 1). ?? ???? ????????????
L1 ? L2 ? ????????? ???????????? (k1 , l1 ) ? (k2 , l2 ) ????? ??????????????? ?????????? ?
???????? ???????????? (0, 1), ? ??????
L = (L1 )p и (L2 )q ,
(16)
??? (p, q) ? ??????? ??????? ?????????
k1 p + k2 q = 0,
l1 p + l2 q = 1.
?????????????,
L(cx) = (L1 (cx))p и (L2 (cx))q = (ck1 L1 (x))p и (ck2 L2 (x))q = ck1 p+k2 q (L1 (x))p и (L2 (x))q = L(x),
?????????????, ?????????? (16) ???????? ?????????? ??????? ??????? ???????????? ????????? ?? ??????? c(t). ????? ????,
L[x(t)]t (s) = (L1 [x(t)])p и (L2 [x(t)])q t (s) = (L1 [x(s)](s (t))l1 )p и (L2 [x(s)](s (t))l2 )q t (s) =
= (s (t))l1 p+l2 q (L1 [x(s)])p и (L2 [x(s)])q t (s) = s (t)L[x(s)]t (s) = L[x(s)],
??????? ?????????? (16) ???????? ?????????? ?????? ??????? ???????????? ?????? ?????????. ???????, ??? ?k = 0 ??? k > m, ??? m ? ???? ???????????? ?????.
?????????? ???????????? ???? (16), ???????????? ?? ?k , ?????
L = (?1 ) 2 и (?0 )?1 .
1
??? ????? ??????????? ???????? ????????? ???????? ????????
(x, x ) (x, x )2 ? ?1
?L
?L
x
x
=
?
+
2
x +
1
1 x,
?(x, x )
?(x, x)
(?1 ) 2 ?0
(?0 )2 (?1 ) 2
?L
?L
(x, x )
x
x
+
2
x
=
?
x
+
Lx =
1
1 ,
?(x, x )
?(x , x )
?0 (?1 ) 2
(?1 ) 2
Lx =
M = [x, Lx ] =
[x, x ]
1 ,
(?1 ) 2
?1
E = L ? (x , Lx ) = L ?
1 = 0,
?0 (?1 ) 2
(x, x ) (x, x)(x, x )
= 0,
Q = S1 = (x, Lx ) =
1 ?
1
(?1 ) 2
?0 (?1 ) 2
(x, x )2
(x, x )2 ? ?1
+
+ L = 0.
S0 = (x, Lx ) + (x , Lx ) = ?
1
1
?0 (?1 ) 2
?0 (?1 ) 2
????????? E = 0, ????? (x , Lx ) = L. ??? ?????????????? ?????????? (14) ??? n = 1.
????????? Q = 0 ??????????? ? ???????????? (3): ???? ?????????? L ????????? ???????????? ????????? ?? ???????, ?? ???????? Q ???????????? ????? ????. ??, ??? S0 = 0,
?????????????? ?????????? (10) ??? p = 0.
44
?.?. ????????
??????????
[1] ?????????? ?.?. ???????? ?????????? ?????? ????. ???????????. ???. ???. ??-?. ? 1992. ? 59 ?. ???.
? ?????? ? 1005-B92.
[2] ???????? ?. ??????? ???????????????? ??????? ? ???????????? ?????????? (???. ? ????.). ? ?.:
???, 1986. ? 360 ?.
[3] ????????? ?.?. ???????????? ??????????. ? ?.: ?????, 1958. ? 162 ?.
?.?. ????????
c?????? ?????????????, ??????? ??????????????????? ? ???????????????????? ?????????,
?????????? ?????? ?????????????? ???????????????? ???????????? ????????????,
606520, ????????????? ???????, ?. ????????, ??. ???????????, ?. 1?,
e-mail: oxyzt@ya.ru
V.A. Luk?yanov
Senior Lecturer, Chair of General Educational and Professional Disciplines,
Zavolzh?ye Branch of Nizhni Novgorod Technical State University,
1a Pavlovskii str., Zavolzh?ye, Nizhegorodski region, 606520 Russia,
e-mail: oxyzt@ya.ru
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