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?2 mx1 M m2
? mx1 mx1
= ? 3 + ?0 (x0 ), =? 2 2?x
3. + c2 ,
2 x0 2 x0 2x0 0
x1 1 Mm
x1 ?
+ ?1 (x0 );
u= u= ;
x0 x0 x0
m2 x1 ? c1 x0
mx1 0
4. ?= 2 + m2 ) + ? (x0 ), ?= ,
km(x2 + m2 )
k(x0 0
x0 x1 c1 + x0 x1
+ ?1 (x0 );
u= 2 u= 2 ;
2 x0 + m2
x0 + m
4
x2 2?1 x1 ?4 x2
? 2c1 x1
1
= 12 +
+ ?0 (x0 ), 0
5. = 2+ 2 + ? (x0 ),
4 9x0 3x0 4 9x0 3x0
x1 x1 c2
+ ?1 (x0 );
u= u= + 4/3 ;
3x0 3x0 x0
c2 x1
? = ?0 (x0 ), u = x1 ?1 (x0 );
6. ?= , u= ;
x0 + c1 x0 + c1
x2
F (?)d? = x1 + 0 ? c1 x0 ,
F (?)d? = x1 + ?0 (x0 ),
7.
2
u = c2 ? x0 .
u = ?1 (x0 ),
Через ? позначено довiльну гладку функцiю; M , c, c1 , c2 , k, m — довiльнi сталi.
164 В.I. Фущич, М.I. Сєров, Л.О. Тулупова

У випадку довiльної кiлькостi змiнних в рiвняннях (1) дослiдження умовної
симетрiї пов’язане з громiздкими перетвореннями i в цiй статтi не наводиться.
Однак деякi з операторiв умовної симетрiї n-вимiрних рiвнянь (1) можуть бути
одержанi безпосереднiм узагальненням операторiв (13). Такi узагальнення наве-
денi нижче разом з анзацами, побудованими за цими операторами, i вiдповiдними
точними розв’язками системи Нав’є–Стокса (1).
?a
Оператор Qa = x0 ?a + ?? + ?ua , F = µ:
mx0
?x ?x + k M (ln x0 + 1) ?
?
+ ?0 (?),
?= ?= +,
mx2 m2 x2
2
m? x0
0 0
?(?x) ?(?x) M ? ln x0 k?
?
u= + ?(?), ? = x0 ; u= + .
? x0 mx0 x0
m?a
Оператор Qa = x0 ?a + ?? + ?ua , F = µ?:
?x2
0

?2 ?2 m(?x) + k M m2
m(?x) k
=? =?
+ ?0 (?), + + 2,
3 4
3
2 ? 2 x0 x0 x0
?(?x) ?(?x + k) M m?
?
u= + ?(?), ? = x0 ; u= .
x2
? x0 0
m?a
?? + x0 ?ua , F = ?k 2 ?:
Оператор Qa = (x2 + m2 )?a +
0
k
m(?x) km
?=? ? = [m2 (?x + c2 ) ? c1 x0 ] 2
+ ?0 (?), ,
k(? 2 + m2 ) x0 + m2
??(?x) x0 ?(?x + c2 ) + c1 ?
u= 2 + ?(?), ? = x0 ; u = .
x2 + m2
? + m2 0

2nua
?? + ?ua , F = ?3 :
Оператор Qa = (2n + 1)x0 ?a + 3
?

?4 n x2 ?4 n x2 n?2
2n x?
+ ?0 (?),
= + = + 2+
2n + 1 x2
2
4 2n + 1 ? 2n + 1 ? 4 x0
0
2n x? ?4n/(2n+1)
+ 2 + kx0 ,
2n + 1 x0
x x ?
u= + ?(?), ? = x0 ; u= +.
(2n + 1)? (2n + 1)x0 x0
xa ? x0 ua
?? + ?ua , F = ?3 :
Оператор Qa = (2n + 1)x0 ?a + 3
x0 ?

?4 n x2 ?4 x2 ?2
1 x? n
? ?
+ ?0 (?),
= = +
(2n + 1)2 x2
2
4 2n + 1 ? 2n + 1 ? 4 4n
0
1 x? ?4n/(2n+1)
? 2 + kx0 ,
2n + 1 x0
x x
u= + ?(?), ? = x0 ; u= + ?.
(2n + 1)? (2n + 1)x0
Умовна симетрiя рiвнянь Нав’є–Стокса 165

2?a
?u?? + ?ua , F = ?3 :
Оператор Qa = 3x0 ?a + 3
?
2 2
?4 ?x2 ?4 2k(?x) k 2
2(?x)(??) ?x ?4/3
? 0
= + ? (?), = + + 2 + ?x0 ,
4 3? 3? 4 3x0 3x0 x0
?(?x) ?(?x) k?x
u= + ?(?), ? = x0 ; u= + .
3? 3x0 x0
?a
(?x ? x0 ?u)?? + ?ua , F = ?3 :
Оператор Qa = 3x0 ?a +
x0 ?3
2 2
?4 ?x2 ?4 (??)2
(?x)(??) ?x
? ?
+ ?0 (?),
= = +
4 3? 3? 4 3x0 3x0
(?x)(??) ?4/3
? + kx0 ,
3x0
?(?x) ?(?x)
u= + ?(?), ? = x0 ; u= + ?.
3? 3x0
Оператор Qa = ?x?a + ?a ub ?ub , F = F (?):
k
? = ?0 (?), ?= ,
x0 + ?
?(?x)
u = ?x?(?), ? = x0 ; u= + ?.
x0 + ?
Оператор Qa = f ?a + ??? , F = F (?):

x2
? ??x0 + k,
0
f (?)d? = ?x + ?0 (?), f (?)d? = ?x +
2
u = ? ? ?x0 .
u = ?(?), ? = x0 ;

В цих формулах ? — довiльний вектор, для якого виконується умова (?)2 = 1;
? — довiльний вектор; M , k, m, c1 , c2 — довiльнi сталi; n — розмiрнiсть простору.
Зауважимо, що n-вимiрне узагальнення оператора Q1 знайти не вдалося, а
оператор Q5 узагальнено чотирма рiзними способами.
Таким чином, наведенi результати вказують на те, що рiвняння Нав’є–Стокса
мають прихованi симетрiї, якi не можна одержати за допомогою алгоритму Лi. Цi
симетрiї можна використати для знаходження точних розв’язкiв даних рiвнянь.

1. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения урав-
нений нелинейной математической физики, Киев, Наук. думка, 1989, 336 с.
2. Фущич В.И., Условная симметрия уравнений нелинейной математической физики, Укр.
мат. журн., 1991, 43, № 11, 1456–1470.
3. Серов Н.И., Условная инвариантность и точные решения нелинейного уравнения тепло-
проводности, Укр. мат. журн., 1990, 42, № 10, 1370–1376.
4. Fushchych W.I., Serov N.I., Tulupova L.A., The conditional invariance and exact solutions of
the nonlinear di?usion equation, Dopovidi Ukr. Acad. Nauk, 1993, № 4, 37–40.
W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 166–170.

High-order equations of motion in quantum
mechanics and Galilean relativity
W.I. FUSHCHYCH, Z.I. SYMENOH
Linear partial di?erential equations of arbitrary order invariant under the Galilei
transformations are described. Symmetry classi?cation of potentials for these equa-
tions in two-dimensional space is carried out. High-order nonlinear partial differential
equations invariant under the Galilei, extended Galilei and full Galilei algebras are
studied.

Non-relativistic quantum mechanics is based on the equation
L? ? (S + V )? = 0, (1)
where S = p0 ? p2 /2m, p0 = i?/?x0 = i?/?t, pa = ?i?/?xa , V = V (x, ?? ?). In the
a
case where V is a function only of x, equation (1) coincides with the standard linear
Schr?dinger equation.
o
The fundamental property of (1) (in the case V = 0) is the fact that this equa-
tion is compatible with the Galilean relativity principle. In other words, equation (1)
(V = 0) is invariant under the Galilei group G(1, 3). The Lie algebra AG(1, 3) =
P0 , Pa , Jab , Ga of the Galilei group is generated (see, e.g., [1, 2]) by the operators
Jab = xa pb ? xb pa ,
P 0 = p0 , Pa = pa , a = b, a, b = 1, 2, 3,
(2)
Ga = tpa ? mxa .

The operators Ga generate the standard Galilei transformations
t > t = t, xa > xa = xa + va t.
De?nition 1. We say that the equation of type (1) is compatible with the Galilei
principle of relativity if it is invariant under the operators P0 , Pa , Jab , Ga .
Let X be one of the operators P0 , Pa , Jab , Ga .
De?nition 2. Equation (1) is invariant under the operator X if the following condi-
tion is true:
(3)
X L? = 0,
L?=0
(2)

where X is the second Lie prolongation of the operator X [1–4].
(2)

The equation of type (3) is a Lie condition of invariance of the equation under the
Lie algebra. In our case, it is the condition of invariance under the algebra AG(1, 3).
Theorem 1 [1, 2, 5]. Among linear equations of the ?rst order in t and of the second
order in the space variables x there exists the unique equation (1) (V = ? = const)
invariant under the algebra AG(1, 3) with the basic elements (2).
J. Phys. A: Math. Gen., 1997, 30, № 6, L131–L135.
High-order equations of motion in quantum mechanics and Galilean relativity 167

Conclusion. We can regard the theorem formulated above as a method of deriving
the Schr?dinger equation from the Galilei principle of relativity [5, 6].
o
In the present paper, we give the answer on the following question: Do there exist
equations not equivalent to the Schr?dinger equation for which the Galilei principle
o
of relativity is true?
In [6, 7], the following generalization of the Sch?dinger equation was proposed
o
(?1 S + ?2 S 2 + · · · + ?n S n + V )? = 0, (4)
S 2 = SS, . . . , S n = S n?1 S, ?1 , ?2 , . . . , ?n are arbitrary parameters.
If V = 0, equation (4), as well as equation (1), is invariant under the algebra
AG(1, 3), i.e. this equation is compatible with the Galilei principle of relativity. Is
this equation unique among high-order linear equations? In what follows, we get the
positive answer for this question.
More precisely, we solve the following problems:
(i) We describe all linear equations of arbitrary order invariant under the algebra
AG(1, 3).

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