ñòð. 85 |

I shall say here briefly about one of such symmetries.

Theorem 7. (Fushchych 1971, 1974 [30, 31]. The Dirac equation (5.9) is invariant

with respect to the following representation of the Poincar? algebra

e

?

(2)

Pa = ?i

(2)

(5.14)

P0 = H = ?0 ?a pa + ?0 m, , a = 1, 2, 3,

?xa

i

(2)

Jab = xa pb ? xb pa + Sab , (?a ?b ? ?b ?a ), (5.15)

Sab =

4

1

(2)

J0a = x0 pa ? (xa H + Hxa ). (5.16)

2

Thus we have two different representations of the Poincar? algebra AP (1, 3) (5.10)

e

and (5.14)–(5.16).

The representation (5.15) and (5.16) generates nonlocal transformations

(2) (2)

xa > xa = exp{iJab vb }xa exp{iJ0c vc } = Lorentz transform, (5.17)

(2) (2)

t > t = exp{iJ0b vb }t exp{?iJ0c vc } = t. (5.18)

Thus, time does not change in relativistic physics. Time is absolute in relativistic

physics.

352 W.I. Fushchych

There are two nonequivalent possibilities (duality) for transformations of coordi-

nates and time: Lorentz transformation (5.11), (5.12) and non-Lorentz transformation

(5.17), (5.18).

The Maxwell and Klein–Gordon–Fock equations are also invariant under nonlocal

transformations (5.17), (5.18) when time does not change. However energy and

momentum are transformed by the Lorentz law [31,32]. We have new relativity prin-

ciple (5.17), (5.18).

What is the reason of such a paradoxical statement? The reason is that the

(2)

operators J0a are non-Lie symmetry operators and the standard relation (S. Lie’s

theorems) between Lie groups and Lie algebras is broken.

So, physics is not equivalent to geometry and geometry is not physics. Physics is

Nature. Theoretical Physics is only a Model of Nature!

6 On some new motion equations

Some new motion equations are adduced in this section. These equations are generali-

zations of known classical equations. Symmetry of these equations has not been

investigated.

6.1 High order parabolic equation in quantum mechanics

The Schr?dinger equation (5.1) is not the only equation compatible with the Galilei

o

relativity principle. A more general equation was suggested in [1, 2]

(?1 S + ?2 S 2 + · · · + ?n S n )u = ?u,

(6.1)

p2

S ? p0 ? a , S 2 = S · S, S n = S n?1 S,

2m

?, ?1 , ?2 , . . . , ?n are arbitrary parameters. Equation (6.1) as well as the classical

equation (5.1) is invariant with respect to the Galilei transformations but it is not

invariant with respect to scale and projective transformations.

A new equation for two particles (waves):

12

p0 u 1 = p u1 + V1 t, x1 , x2 , . . . , x6 , u1 , u2 ,

2m1 a

12

p0 u 2 = p u2 + V2 t, x1 , x2 , . . . , x6 , u1 , u2 ,

2m2 a+3

u1 = u1 (t, x1 , x2 , x3 ), u2 = u2 (t, x4 , x5 , x6 ), V1 and V2 are potentials.

6.2 Nonlinear generalization of Maxwell equations

If we assume that the light velocity is not constant [34], we can suggest some

generalizations of the Maxwell equations

?H

= rot {c(E 2 , H 2 , E H)H}, = ?rot {c(E 2 , H 2 , E H)E},

?E

(6.2)

?t ?t

div {a(E , H , E H)E} = 0, div {b(E 2 , H 2 , E H)H} = 0,

2 2

Ansatz ’95 353

where a, b and c are some functions of electromagnetic field;

?E ?D

= rot {c(B 2 , D2 , B D)B} + j, = rot {c(E 2 , H 2 , E H)E} + j,

?t ?t

or (6.3)

?H ?B

= ?rot {c(B 2 , D2 , B D)D}, = ?rot {c(E 2 , H 2 , E H)E},

?t ?t

?1 D + ?2 2D = F1 (E 2 , H 2 , E H)E + F2 (E 2 , H 2 , E H)H,

(6.4)

?3 B + ?4 2B = R1 (E 2 , H 2 , E H)E + R2 (E 2 , H 2 , E H)H,

(6.5)

div D = ?, div B = 0,

where F1 , F2 , R1 , R2 are functions of fields E and H, c in equations (6.2), (6.3) can

be a function of (t, x), c = c(t, x), or depend on the gravity potential V , c = C(V ).

Nonlinear wave equations for E and H have form

?2E ?2H

? c2 ?E = 0, ? c2 ?H = 0, (6.6)

2 2

?t ?t

or

?2E ?2H

? ?(c2 E) = 0, ? ?(c2 H) = 0; (6.7)

2 2

?t ?t

or

?2 ?2

1 1

? ?E = 0, ? ?H = 0, (6.8)

E H

?t2 c2 ?t2 c2

with one of the conditions

?E

+ ?H

1 ?t ?t

2

(6.9)

c=

2 (rot H)2 + (rot E)2

or

?c2 ?c2

(6.10)

= 0.

?xµ ?xµ

or

?c2

= ?(E 2 H 2 , E H)F?? c? , (6.11)

cµ

?xµ

c? is the four-velocity of the light (electromagnetic field), c2 = c? c? .

Equations of hydrodynamical type for electromagnetic field have form

?E

= a1 ? ? (c ? H) + a2 ? ? (c ? E) ,

?t

?H (6.12)

= b1 ? ? (c ? E) + b2 ? ? (c ? H) ,

?t

?c

+ (c?)c = R1 E + R2 H,

?t

354 W.I. Fushchych

c is the three-velocity of the light, where a1 , a2 , b1 , b2 , R1 , R2 are functions of E 2 ,

H 2 , E H.

Maxwell’s equations in a moving frame with the velocity can be generalized in

such forms

?E ?E ?H ?H

+ ? 1 vk + ?2 rot H = 0, + ? 3 vk + ?4 rot E = 0,

?t ?xk ?t ?xk

or

?E ?H ?H ?E

+ ? 1 vk + ?2 rot H = 0, + ? 3 vk + ?4 rot E = 0,

?t ?xk ?t ?xk

?vk

+ vl ?vk = 0.

with the conditions ?t ?xl

6.3 Equations for fields with the spin 1/2

Fields with the spin 1/2 are described, as a rule, by first-order equations, by the Dirac

equation. However, such fields can be also described by second-order equations. Some

of such equations are adduced below:

? ? ?

pµ pµ ? = F1 (??)?, ??µ pµ ? = F2 (??); (6.13)

? ? ?

pµ pµ ? = R1 (??)?, (??µ ?)pµ ? = F2 (??)?; (6.14)

? ? ? ?

pµ pµ ? = F1 (??)?, (??µ ?)(?pµ ?) = F3 (??); (6.15)

?

pµ pµ ? + ??µ pµ ? = F (??)?; (6.16)

? ? ? ?

p0 ? = {(??0 ?)(??k ?)pk + m??0 ?}?.

pµ pµ ? = F1 (??)?,

6.4 How to extend symmetry of an equation

with arbitrary coefficients?

Let us consider the a second-order equation

?2u ?u

(6.17)

aµ? (x) µ ? + bµ (x) µ + F (u) = 0.

?x ?x ?x

Equation (6.17) with arbitrary fixed coefficients has only a trivial symmetry (x >

x = x, u > u = u). However, if we do not fix coefficient functions aµ? (x), bµ (x),

such an equation can have wide symmetry. E.g., if aµ? , bµ satisfy the equations

?u ?u

2aµ? = (6.18)

F1 (u)

?xµ ?x?

or

?2u

?u

2bµ = F2 (u) 2aµ? (6.19)

, = F3 (u),

?xµ ?xµ ?x?

then the nonlinear system (6.17), (6.18), (6.19) is invariant with respect to the Poin-

car? group P(1,3). Let us emphasize that here even if we put F1 = 0, F2 = 0,

e

Ansatz ’95 355

equations (6.17), (6.18), (6.19) are a nonlinear system of equations. With some parti-

cular functions F1 and F2 , it is possible to construct ansatzes which reduce system

(6.17), (6.18), (6.19) to the system of ordinary differential equations.

So, considering (6.17) as a nonlinear equation with additional conditions for aµ? ,

b? , we can construct the exact solution for eqation (6.17). The adduced idea about

extension of the symmetry of (6.17) can be used for construction of exact solutions

for motion equations in gravity theory.

The second example of equations which have wide symmetry is

? 2 F??

(6.20)

vµ v? µ ? = 0,

?x ?x

?v?

(6.21)

vµ = 0.

?xµ

If in (6.20) vµ are fixed functions the equation, as a rule, has trivial symmetry.

1. Fushchych W., Symmetry in problems of mathematical physics, in Algebraic-Theortic Studies in

Mathematical Physics, Kiev, Inst. of Math. Ukrainian Acad. Sci., 1981, 6–44.

2. Fushchych W., On symmetry and exact solutions of some multidimensional equations of mathematical

physics problems, in Algebraic-Theoretical Methods in Mathematical Physics, Kiev, Inst. of Math.

Ukrainian Acad. Sci., 1983, 4–23.

3. Fushchych W.,Serov M., The symmetry and exact solutions of the nonlinear multi-dimensional Li-

ouville, d’Alembert and eikonal equations, J. Phys. A: Math. Gen., 1983, 16, ¹ 15, 3645–3658.

4. Fushchych W., Zhdanov R., On some new exact solutions of the nonlinear d’Alembert–Hamilton

system, Phys. Lett. A, 1989, 141, ¹ 3–4, 113–115.

5. Fushchych W., Zhdanov R., Revenko I., General solutions of the nonlinear wave and eikonal equations,

Ukrainian Math. J., 1991, 43, ¹ 11, 1471–1486.

6. Fushchych W., Zhdanov R., Yehorchenko I., On the reduction of the nonlinear multi-dimensional wave

eqautions and compatibility of the d’Alembert–Hamilton system, J. Math. Anal. Appl., 1991, 161, ¹ 2,

352–360.

7. Patera J., Winternitz P., Zassenhaus H., Continuous subgroups of the fundamental groups of physics,

J. Math. Phys., 1975, 16, ¹ 8, 1597–1624.

8. Patera J., Winternitz P., Zassenhaus H., Maximal Abelian subalgebras of real and complex symplectic

Lie algebras, J. Math. Phys., 1983, 24, ¹ 8, 1973–1985.

9. Fushchych W., Barannyk A., Barannyk L., Fedorchuk V., Continuous subgroups of the Poincare group

P (1, 4), J. Phys. A: Math. Gen., 1985, 18, ¹ 5, 2893–2899.

10. Barannyk L., Fushchych W., On continuous subgroups of the generalized Schr?dinger groups, J. Math.

o

Phys., 1989, 30, ¹ 2, 280–290.

11. Fushchych W., Barannyk L., Barannyk A., Subgroup analysis of the Galilei and Poincar? groups and

e

reductions of nonlinear equations, Kiev, Naukova Dumka,1991, 300 p.

12. Clarkson P., Kruskal M., New similarity reductions of the Boussinesq equation, J. Math. Phys., 1989,

30, ¹ 10, 2201–2213.

13. Fushchych W., Nikitin A., Symmetries of Maxwell’s equation, Reidel Dordrecht, 1987, 230 p. (Russian

version 1983, Naukova Dumka 1983).

14. Fushchych W., How to extend symmetry of differential equations?, in Symmetry and Solutions of

Nonlinear Mathematical Physics, Kiev, Inst. of Math. Ukrainian Acad. Sci., 1987, 4–16.

15. Fushchych W., Tsyfra I., On reduction and solutions of nonlinear wave equations with broken

symmetry, J.Phys. A: Math. Gen., 1987, 20, ¹ 2, 45–47.

16. Fushchych W., Serov M., Chopyk V., Conditional invariance and nonlinear heat equations, Dopovidi

ñòð. 85 |