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DOI 10.1007/s10958-015-2218-7 Journal of Mathematical Sciences, Vol. 204, No. 6, February, 2015

ON THE CONSTRUCTION OF THE GENERAL SOLUTION OF A CLASS OF COMPLEX NONAUTONOMOUS EQUATIONS Yu. M. Okunev and M. V. Shamolin UDC 517.925; 531.01; 531.552

Abstract. In this pap er, we give a survey of cases of integrability for some class of complex, linear, nonautonomous, ordinary, second-order differential equations. We p erform a qualitative analysis of these cases and construct general solutions in the form of absolutely and uniformly converging series with resp ect to small parameters.

1. Preliminaries. Earlier [9, 11, 12, 21, 28­32] in the study of nonstationary motion with resp ect to the center of mass of a dynamically symmetric rigid b ody of spatial aerodynamical form with high carrier prop erties under the quadratic resistance of the medium, a nonautonomous linear complex second-order equation was obtained. In the framework of a linearized model that does not take into account damping moments of aerodynamical forces, one can observe the damping action of the lifting force and find restrictions on the aerodynamical coefficients under which angular oscillations of the b ody attenuate [17, 18, 22, 23, 26, 37, 38, 42, 43, 45­47, 49, 51, 52, 59, 60]. 2. Complex nonautonomous linear second-order equation. Consider the following complex equation: y + (c2 - c1 )v (t) - 2 - ¨ J2 J1 s2 i y J2 J1
2 s2

+ [k - (c2 - c1 )c1 ]v 2 (t) - 1 - where k , c1 , c2 , J1 , J2 , and satisfies the equation
s2

- 1-

J2 J1

s2 (c2 - c1 )v (t)i y = 0, (2.1)

are p ositive constants and the real-valued function of time v = v (t) v = -c1 v 2 , (2.2)

which can b e easily integrated: v (t) = v0 , 1+ c1 v0 t v0 = v (0). (2.3)

If the equality J2 /J1 = 0 holds, then this equation has a general solution that can b e expressed through a finite combination of elementary functions (see [7, 32, 36, 39­41, 48, 53, 57]. We examine Eq. (2.1) under the sufficiently general natural condition J2 > 0. J1 (2.4)

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 88, Geometry and Mechanics, 2013. 1072­3374/15/2046­0787 c 2015 Springer Science+Business Media New York 787


3. Reduced equation and the corresp onding nonautonomous linear complex Hamiltonian system. Using the substitution y (t) = u0 (t)z (t), where u0 (t) = (1 + c1 v0 t)(c Eq. (2.1) can b e reduced to the form z + f (t)z = 0, ¨ where 1 1 v f (t) = k - (c2 - c1 )c1 - (c1 - c2 )2 v 2 (t) - (c2 - c1 ) (t) 4 2
2 J2 1 J2 2 1 + (c2 - c1 )s2 i v (t)+ 2 . 2 J1 4 J1 s2
1

(3.1) J2 2J1

-c2 )/2c

1

â exp

1-

s2 it ,

(3.2)

(3.3)

(3.4)

Using natural changes of variables, we transforn Eq. (3.3) to the following form: w+ where z (t( )) = w( ), c1 1 = ln(1 + c1 v0 t) = , d d d = v (t) = v (t)c1 = v (t)c1 ( ), dt d1 d d d2 d2 = -c1 v (t) + c2 v 2 (t) 2 , 1 dt2 d d 1 J2 b1 = s2 i. 2 J1 () = (3.6) (3.7) (3.8) (3.9) (3.10) 1 1 b2 c2 k - 2 + b1 (c2 - c1 ) e - 1 e2 w = 0, 2 4 v0 c2 v0 1 (3.5)

We see that the parameter b1 b ecomes purely imaginary. Equation (3.5) is equivalent to the following complex, linear, nonautonomous, Hamiltonian system with one degree of freedom: w =- ~ with the Hamiltonian H (~ w, ) = w, where = 1 J2 s2 b1 = i. c1 v0 2 J1 c1 v0 (3.13) k c2 c2 - c1 w2 w2 ~ + 2 - 22 + e - 2 e2 , 2 c1 2 c1 4c1 (3.12) H (~ w, ) w, , w w= H (~ w, ) w, w ~ (3.11)

Thus, we consider a linear, complex Hamiltonian system with the purely imaginary parameter . 788


4. Complete integral and the Jacobi equation. For integration of the system (3.11), it suffices to find its complete integral S = S (, w, ) , ~ where is an arbitrary complex constant, as a solution of the complex Hamilton­Jacobi equation: ~ S (, w, ) ~ S (, w, ) ~ + H w, ~ , w (see [1­6, 8, 10, 13­16, 19, 20, 24, 25, 27, 33­35, 44, 50, 54­56, 58]) We search for a function S = S (, w, ) in the form ~ ~ S = S (, w, ) = S0 (, ) ~ w2 ; 2 (4.2) =0 (4.1)

moreover, the partial differential equation (4.1) is reduced to the following Riccati ordinary differential equation (see [10, 16, 34]): k c2 c2 - c1 ~ dS0 (, ) 2 + S0 (, )+ 2 - 22 + ~ e - 2 e2 = 0. d c1 c1 4c1 (4.3)

5. Some particular solutions of the Riccati equation. Equation (4.3) is a Riccati equation, and its general solution, in general, cannot b e expressed through a finite combination of elementary functions (see [10, 16, 34]). However, it is known that if one knows a particular solution of this equation, then the general solution can also b e obtained. First, for simplicity, we search for a particular solution in the form S0 = + e . (5.1)

Then the coefficients and are defined by the following (in general, incompatible) complex algebraic equations: 2 = 2 , 2 = - k , c2 1 (5.2) c2 - c1 = 0, c1

+2 +

where k = k - c2 /4. 2 Equations (5.2) are compatible under at least one of the following conditions for the coefficient k: k = c1 (c2 - c1 ), k = 0. (5.3) (5.4)

The cases (5.3) and (5.4), resp ectively, are characterized by the existence of the following particular solutions of Eq. (4.3): S S
01 02

c2 - 1 - e , 2c1 c2 =- + e . 2c1 =

(5.5) (5.6) 789


6. Calculation of a complete integral of the Hamiltonian system in the case (5.3). If condition (5.3) holds, then Eq. (4.3) takes the form c2 dS0 ( ) 2 + S0 ( )+ - 1 - d 2c1
2

+

c2 - c1 e - 2 e2 = 0 c1

(6.1)

(we omit the complex constant ). We will search for the general solution of Eq. (6.1) in the form ~ S0 = S
01

+ z.

(6.2)

Indeed, this form is convenient since in the case (6.2) Eq. (6.1) is transformed into the Bernoulli equation (6.3) z +2S01 z + z 2 = 0, which, in turn, can b e easily reduced to the following linear inhomogeneous equation: d d 1 z - 2S
01

1 z

- 1 = 0. c2 2- c1 +2 exp{ } d + , ~

(6.4)

The solution of Eq. (6.4) is expressed as follows: c2 1 = exp -2+ - 2 exp{ } · exp z c1
0

(6.5)

where is an arbitrary complex constant. ~ We write equality (6.5) in the form z= + ~ exp{(2 - c2 /c1 ) +2 exp{ }}
0

.

(6.6)

exp{(2 - c2 /c1 ) +2 exp{ }}d

Thus, the general solution of Eq. (6.1) has the form ~ S0 (, ) = where 1 ( ) = exp 2- c2 c1 +2 exp{ } . (6.8) c2 1 ( ) , - 1 - e + 2c1 + 0 1 ( )d ~ (6.7)

So, in the case (5.3) we have obtained a complete integral of the Hamiltonian system (3.11), (3.12) corresp onding to Eq. (3.5). It is known (see [34, 54, 55]) that if the complete integral (4.2) is known, then integrals of the Hamiltonian system (3.11) , (3.12) can b e found from the following equalities: S (, w, ) ~ = w, ~ w S (, w, ) ~ ~ = - = const. ~ From the second of Eqs. (6.9) we have ~ S0 (, ) w2 =- ~ 2 1 ( )
2

(6.9)

+ ~

0

1 ( )d

~ 1 w2 = - = const. 2 2

(6.10)

790


Relation (6.10) allows one to find the dep endence of the solution of the reduced ~ indep endent variable and on two arbitrary complex constants and in the ~ c2 ~ - 1 - exp{ } · + 1 ( )d ~ w( ) = exp 2c1
0

equation (3.5) on the case (5.3): , (6.11)

where, as ab ove, 1 ( ) = exp 2- c2 c1 +2 exp{ } .

7. Calculation of a complete integral of the Hamiltonian system in the case (5.4). Under condition (5.4), Eq. (4.3) takes the following form (we omit the complex constant ): ~ c2 c2 - c1 dS0 ( ) 2 + S0 ( )+ - 22 + e - 2 e2 = 0. d c1 4c1 We will search for the general solution of Eq. (7.1) in the form S0 = S
02

(7.1)

+ z.

(7.2)

Indeed, this is conveniaent since in the case (7.2), Eq. (6.1) is transformed into the Bernoulli equation z +2S02 z + z 2 = 0, which, in turn, can b e easily reduced to the linear inhomogeneous equation d d 1 z - 2S
02

(7.3)

1 z

- 1 = 0.

(7.4)

The solution of Eq. (7.4) is represented in the following form: c2 c2 1 = exp - +2 exp{ } · exp - 2 exp{ } d + , ~ z c1 c1
0

(7.5)

where is an arbitrary complex constant. ~ We write Eq. (7.5) in the form z= + ~ exp{c2 /c1 - 2 exp{ }}
0

.

(7.6)

exp{c2 /c1 - 2 exp{ }}d

Thus, the general solution of Eq. (7.1) has the form ~ S0 (, ) = - c2 + e + 2c1 2 ( )


,

(7.7)

+ ~

0

2 ( )d

where 2 ( ) = exp c2 - 2 exp{ } . c1 (7.8)

Thus, in the case (5.4) we have obtained the complete integral of the Hamiltonian system (3.11), (3.12) corresp onding to Eq. (3.5). 791


It is known (see [34, 54, 55]) that if the complete integral (4.2) is known, then integrals of the Hamiltonian system (3.11) , (3.12) can b e found from the following equalities: S (, w, ) ~ = w, ~ w S (, w, ) ~ ~ = - = const. ~ From the second equation (7.9) we have ~ S0 (, ) w2 =- ~ 2 2 ( )
2

(7.9)

+ ~

0

2 ( )d

~ w2 2 = - = const. 2 2

(7.10)

Relation (7.10) allows one to find the dep endence of the solution of the reduced equation (3.5) on the ~ indep endent variable and on two arbitrary complex constants and in the case (5.4): ~ c2 ~ + exp{ } · + 2 ( )d , ~ (7.11) w( ) = exp - 2c1
0

where, as ab ove, c2 - 2 exp{ } . c1 Thus, the solution of the initial equation (2.1) in the cases (5.3) and (5.4) can b e found from formulas (7.9) and (6.9) after reduction of the auxiliary variables w and to the initial variables y and t. 2 ( ) = exp 8. Reduction to initial variables. We return to the initial variables and rewrite the solutions of the complex differential equations found ab ove. The required solution of Eq. (2.1) is expressed by the formula y (t) = w(t)u1 (t)u2 (t), where u1 (t) = (1 + c1 v0 t) u2 (t) = exp
c 1 -c 2 2c1

(8.1) J2 2J1

· exp

1-

s2 it ,

(8.2) (8.3)

= (1 + c1 v0 t)1/2 . 2 We obtain the following intermediate result: y (t) = w(t) (1 + c1 v0 t) Recalling the representation e = 1 + c1 v0 t of the indep endent variable for the cases (5.3) and (5.4), resp ectively, we obtain t ~ w(t) = (1 + c1 v0 t) where 1 (t) = (1 + c1 v0 t) 792
c2 -2c1 2c1 2c1 -c2 2c1

· exp

1-

J2 2J1

s2 it

.

(8.4)

(8.5)

· exp{-(1 + c1 v0 t)}· + ~
0
2c1 -c2 c1

1 ( )d ,

(8.6)

exp{2(1 + c1 v0 t)},


and
- ~ w(t) = (1 + c1 v0 t)
c2 2c1

· exp{(1 + c1 v0 t)}· + ~
0
c2

t

2 ( )d , (8.7)

where

2 (t) = (1 + c1 v0 t) c1 exp{-2(1 + c1 v0 t)}. Thus, returning to the initial notation and taking into account the relation b1 1 J2 s2 = = i, c1 v0 2 J1 c1 v0 we finally obtain for the cases (5.3) and (5.4), resp ectively, t J2 ~ y (t) = exp 1- s2 it · + 1 ( )d , ~ J1 0 t ~ y (t) = (1 + c1 v0 t)
c 1 -c 2 c1

(8.8)

(8.9)

· exp{s2 it}· + ~
0

2 ( )d .

(8.10)

9. Construction of the general solution of the equation in the form of a series with resp ect to a small parameter. We introduce the following notation: k1 = Then Eq. (3.5) b ecomes 4k - c2 2 , 4c2 1 c0 = c2 - c1 . c1 (9.1)

w + k1 + c0 e - 2 e2 w = 0. We search for a solution of this equation in the form of a series in :


(9.2)

w(, ) =
n=0

wn ( )n .

(9.3)

Substituting solution (9.3) into Eq. (9.2), we obtain the following infinite system of linear differential equations: w0 + k1 w0 = 0, w1 + k1 w1 + c0 e w0 = 0, wn + k1 wn + c0 e w
n -1

(9.4) (9.5) n 2. (9.6)

- e wn

2

-2

= 0,

System (9.4)­(9.6) defines an infinite system of recurrent linear inhomogeneous equation. We search for its solution in the form (9.7) wn ( ) = wnO ( )+ wnH ( ), n 0, where wnO ( ) is a solution of the corresp onding homogeneous equation wnO + k1 w and w
nH nO

= 0,

n 0, = 0, n 0.

(9.8)

( ) is a solution of the corresp onding inhomogeneous equation w
nH

+ k1 wnH + c0 e w w
-2

n -1

- e2 wn

-2

(9.9)

Formally ( ) w-1 ( ) 0 w
0H

and ( ) 0 793


by Eq. (9.4). Obviously, for n 0, the general solution of the homogeneous equation (9.8) has the form
n wnO = C1 e
+ 0



n + C2 e

- 0



,

(9.10) (9.11)

where
± 0 = 0 ±

k1 i. s N; k1 i.
- 1

Moreover, we will henceforth need the values
± s = s ±

k1 i,

(9.12) (9.13)


they satisfy the relation The equation for w
1H ±2 s

+ k1 = s2 ± 2s
+ 1

b ecomes w
1H 0 + k1 w1H + c0 C1 e 0 + C2 e

= 0.

(9.14)

We will search for its particular solution in the form w
1H

( ) = A1 e 1

+ 1



+ A1 e 2

- 1



.

(9.15)

The constants A1 , k = 1, 2, are defined by the equalities k A1 C 1 A1 2 C
0 1 0 2

=-

c0 C 0 = - -2 2 . 1 + k1


0 c0 C1 , +2 1 + k1

(9.16)

Then the general solution of Eq. (9.5) has the form
1 w1 ( ) = C1 e
+ 0



1 + C2 e 2H

- 0

+ A1 C 1

0 1

e

+ 1



+ A1 C 2

0 2

e

- 1



.

(9.17)

Further, we have the equation for w w
2H 1 + k1 w2H + c0 C1 e
+ 1

:
- 1



1 + C2 e

0 1 0 - C1 e
+ 2

+ c0 A1 C 1 ( ) = A1 C 1
1 1



+ c0 A1 C 2


0 2

0 - C2 e

- 2



= 0.

(9.18)

We will search for its particular solution in the form w
2H

e

+ 1



+ A1 C 2

1 2

e

- 1

+ A2 e 1 -C -C
0 1 0 2

+ 2



+ A2 e 2

- 2



.

(9.19)

The constants A2 , k = 1, 2, are defined by the equalities k A2 C 1 A2 2 C
0 1 0 2

=- =-

c0 A1 C 1 c0 A1 C 2
-2 2 +2 2

0 1 0 2

+ k1 + k1

, (9.20) .

Then the general solution of Eq. (9.6) for n = 2 has the form
2 w2 ( ) = C1 e
+ 0



2 + C2 e

- 0



+ A1 C 1

1 1

e

+ 1

1 2

+ A1 C 2 794

e

- 1



+ A2 C 1

0 1

e

+ 2



+ A2 C 2

0 2

e

- 2



.

(9.21)


Further, we have the equation for w w
3H 2 + k1 w3H + c0 C1 e
+ 1

3H

:
- 1



2 + C2 e 1 2



+ c0 A1 C 1
- 2

1 1

1 - C1 e 0 1

+ 2

0 1

+ c0 A1 C 2

1 - C2 e



+ c0 A2 C 1

- A1 C 1
0 2

e

+ 3



+ c0 A2 C 2 We will search for its particular solution in the form w
3H

- A1 C 2

0 2

e

- 3



= 0.

(9.22)

( ) = A1 C 1

2 1

e

+ 1



+ A1 C 2

2 2

e

- 1



+ A2 C 1

1 1

e

+ 2

1 2

+ A2 C 2 The constants A3 , k = 1, 2, are defined by the equalities k A3 C 1
0 1

e

- 2



+ A3 e 1
0 2 -2 3

+ 3



+ A3 e 2
0 2

- 3



.

(9.23)

=-

c0 A2 C 1

0 1 +2 3

- A1 C 1 + k1

0 1

,

A3 C 2

0 2

=-

c0 A2 C 2

- A1 C 2 + k1

.

(9.24)

Then the general solution of Eq. (9.6) for n = 3 has the form
3 w3 ( ) = C1 e
+ 0



3 + C2 e

- 0



+ A1 C 1
2 2

2 1

e
- 1

+ 1



+ A1 C 2

e



+ A2 C 1

1 1

e

+ 2



+ A2 C 2

1 2

e
0 1

- 2


+ 3

+ A3 C 1
n

e



+ A3 C 2

0 2

e

- 3



. (9.25)

It is easy to prove that the general solution of the equation for wn has the form
n wn ( ) = C1 e
+ 0



n + C2 e

- 0



+
s=1

As C 1

n -s 1

e

+ s



+ As C 2

n -s 2

e

- s



,

(9.26)

where, by (9.12), As (p) = - 1 moreover, A0 (p) p, k A-1 (p) 0, k k = 1, 2. (9.28) c0 As-1 (p) - As-2 (p) 1 1 , s2 +2s k1 i As (p) = - 2 c0 As-1 (p) - As-2 (p) 2 2 , s2 - 2s k1 i s 1; (9.27)

10. Convergence theorem. Now we examine the variables and with the general term (9.26)­(9.28) it on an arbitrarily large segment of length L for Indeed, since the initial conditions are chosen from p ositive constants Mk , k = 1,... , 4, such that for any |wn ( )| |C | + |C | +
n s=1 n 1 n 2 n

convergence of the functional series (9.3) of two dep ending on . For this purp ose, we estimate [0,L]. a b ounded set of the complex plane, there exist [0,L] we have the estimate es
nL n s=1

As C 1 +
n

n -s 1 n -s 2 n

+ As C 2 e
sL

n -s 2

M1 +
s=1

As 1

C

n -s 1

As 2

C

M1 + ne
n s=1

As C 1

n -s 1

+ As C 2

n -s 2

M1 + nenL

M2
s=1 s=1

1 + M3 s2

1 s2

M1 + M4

n2 enL = M1 + M4 an , (10.1) (n!)2 795


where the real-valued numerical series
+ +

an =
n=1 n=1

n2 enL (n!)2

(10.2)

converges. Indeed, eL an+1 = lim = 0. n an n (n +1)2 Further, we calculate the convergence radius R of the complex p ower series lim
n=0

(10.3)

an n .

(10.4)

We have the equalities 1 = lim n an = lim n R n Thus, the convergence radius is equal to . Theorem (convergence theorem). The power series (9.3) with the general term (9.26)­(9.28) converges absolutely and uniformly for any C on the segment [0,L] for any L > 0. Acknowledgment. This work was partially supp orted by the Russian Foundation for Basic Research (Pro ject No. 12-01-00020-a). REFERENCES 1. A. A. Andronov, Col lected Papers [in Russian], Izd. Akad. Nauk SSSR, Moscow (1956). 2. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Mayer, Qualitative Theory of SecondOrder Dynamical Systems, Wiley, New York (1966). 3. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Mayer, Theory of Bifurcations of Dynamical Systems on the Plane [in Russian], Nauka, Moscow (1967). 4. A. A. Andronov and L. S. Pontryagin, "Rough systems," Dokl. Akad. Nauk SSSR, 14, No. 5, 247­250 (1937). 5. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscil lations, Pergamon Press (1966). 6. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag (1989) 7. D. K. Arrowsmith and C. M. Place, Ordinary Differential Equations. A Qualitative Approach with Applications, Chapman and Hall, London­New York (1982). 8. I. Bendixson, "Sur les courb es d´finies par les ´ ations diff´ entielles," Acta Math., 24, 1­30 e equ er (1901). 9. I. T. Borisenok, B. Ya. Lokshin, and V. A. Privalov, "On the dynamics of flights of axisymmetric rotating b odies in air," Izv. Akad. Nauk SSSR. Mekh. Tverdogo Tela, 2, 35­42 (1984). 10. A. D. Bryuno, Local Method of Nonlinear Analysis of Differential Equations [in Russian], Nauka, Moscow (1979). 11. G. S. Byushgens and R. V. Studnev, Dynamics of Longitudinal and Lateral Motion [in Russian], Mashinostroenie, Moscow (1969). 12. G. S. Byushgens and R. V. Studnev, Dynamics of Aircrafts. Spatial Motion [in Russian], Mashinostroenie, Moscow (1983). 13. S. A. Chaplygin, Selected Works [in Russian], Nauka, Moscow (1976). 14. C. Godbillon, G´ m´ eo etrie Diff´ entiel le et M´ anique Analytique, Hermann, Paris (1969). er ec 796
n

n2 enL n = eL lim 2 n n! (n!)

2/n

= 0.

(10.5)


15. V. V. Golub ev, Lectures on Analytic Theory of Differential Equations [in Russian], Gostekhizdat, Moscow­Leningrad (1950). 16. V. V. Golub ev, Lectures on Integration of Equations of Motion of a Heavy Rigid Body About a Fixed Point [in Russian], Gostekhizdat, Moscow­Leningrad (1953). 17. M. I. Gurevich, Theory of Jets of an Ideal Liquid [in Russian], Nauka, Moscow (1979). 18. V. A. Eroshin, V. A. Samsonov, and M. V. Shamolin, "A model problem on the deceleration of a b ody in a resisting medium in jet flow," Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, 3, 23­27 (1995). 19. V. V. Kozlov, "Integrability and nonintegrability in Hamiltonian mechanics," Usp. Mat. Nauk, 38, No. 1, 3­67 (1983). 20. S. Lefschetz, Differential Equations: Geometric Theory, Interscience Publ., New York­London (1957). 21. B. Ya. Lokshin, Yu. M. Okunev, V. A. Samsonov, and M. V. Shamolin, "Some integrable cases of spatial oscillations of a rigid b ody in a resisting medium," in: Proc. XXI Scientific Readings on Astronautics, Moscow, January 28­31, 1997, Institute for History of Natural Science and Technology of the Russian Academy of Sciences, Moscow (1997), pp. 82­83. 22. B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov, Introduction to the Problem of Motion of a Body in a Resisting Medium [in Russian], Moscow State Univ., Moscow (1986). 23. B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov, Introduction to the Problem of Motion of a Point and a Body in a Resisting Medium [in Russian], Moscow State Univ., Moscow (1992). 24. Yu. I. Manin, "Algebraic asp ects of theory of nonlinear differential equations," In: Progress in Science and Technology, Series on Contemporary Problems in Mathematics [in Russian], Vol. 11, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1978), pp. 5­112. 25. W. Miller, Symmetry and Separation of Variables, Addison-Wesley, Reading, Massachusetts (1977). 26. Yu. A. Mitrop ol'sky, Problems of the Asymptotic Theory of Nonstationary Oscil lations [in Russian], Nauka, Moscow (1964). 27. V. V. Nemytsky and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Moscow (1949). 28. Yu. M. Okunev and V. I. Borzov, Structural and Algorithmic Aspects of Model ling for Control Problems [in Russian], Moscow State Univ., Moscow (1983). 29. Yu. M. Okunev, V. A. Privalov, and V. A. Samsonov, "Some problems on the motion of a b ody in a resisting medium," in: Proc. Al l-Union Conf. "Nonlinear phenomena," Nauka, Moscow (1991), pp. 140­144. 30. Yu. M. Okunev and V. A. Sadovnichy, "Model dynamical systems for one problem of external ballistics and their analytic solutions," in: Problems of Modern Mechanics [in Russian], Moscow State Univ., Moscow (1998), pp. 28­46. 31. Yu. M. Okunev, V. A. Sadovnichy, V. A. Samsonov, and G. G. Cherny, "A complex for modelling problems of flight dynamics," Vestn. Mosk. Univ., Ser. 1, Mat., Mekh., 6, 66­75 (1996). 32. ddd Yu. M. Okunev and M. V. Shamolin, "On the integrability in elementary functions of some classes of complex nonautonomous equations," J. Math. Sci., 165, No. 6, 732­742 (2010). 33. F. W. J. Olver, Asymptotics and Special Functions, Academic Press (1974). 34. L. A. Pars, A Treatise on Analytical Dynamics, Heinemann, London (1965). 35. A. Poincar´ On Curves Defined by Differential Equations [Russian translation], OGIZ, Moscow­ e, Leningrad (1947).

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36. V. A. Samsonov and M. V. Shamolin, "On the problem of b ody motion in a resisting medium," Vestn. MGU, Mat., Mekh., 3, 51­54 (1989). 37. M. V. Shamolin, "Classification of phase p ortraits in problem of b ody motion in a resisting medium in the presence of a linear damping moment," Prikl. Mat. Mekh., 57, No. 4, 40­49 (1993). 38. M. V. Shamolin, "Introduction to the problem of b ody drag in a resisting medium and a new two-parameter family of phase p ortraits, " Vestn. MGU, Ser. 1, Mat., Mekh., 4, 57­69 (1996). 39. M. V. Shamolin, "Spatial Poincar´ top ographical systems and comparison systems," Usp. Mat. e Nauk, 52, No. 3, 177­178 (1997). 40. M. V. Shamolin, "On integrability in transcendental functions," Usp. Mat. Nauk, 53, No. 3, 209­210 (1998). 41. M. V. Shamolin, "On limit sets of differential equations near singular p oints," Usp. Mat. Nauk, 55, No. 3, 187­188 (2000). 42. M. V. Shamolin, "New integrable cases and families of p ortraits in the plane and spatial dynamics of a rigid b ody interacting with a medium," J. Math. Sci., 114, No. 1, 919­975 (2003). 43. M. V. Shamolin, "Model problem of b ody motion in a resisting medium taking into account the dep endence of resistance force on angular velocity," In: Scientifuc Report of Institute of Mechanics, Moscow State University [in Russian], No. 4818, Institute of Mechanics, Moscow State University, Moscow (2006), p. 44. 44. M. V. Shamolin, "Integrability in transcendental elementary functions," In: Abstracts of Sessions of Workshop "Current Problems of Geometry and Mechanics," Contemporary Mathematics, Fundamental Directions [in Russian], Vol. 23 (2007), p. 40. 45. M. V. Shamolin, Methods for Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007). 46. M. V. Shamolin, "Complete integrability of equations of motion for a spatial p endulum in the flow of a medium taking into account rotational derivatives of the moment of its action force," Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela, 3, 187­192 (2007). 47. M. V. Shamolin, "Dynamical systems with variable dissipation: approaches, methods, and applications," Fundam. Prikl. Mat., 14, No. 3, 3­237 (2008). 48. M. V. Shamolin, "On integrability in elementary functions of certain classes of nonconservative dynamical systems," J. Math. Sci., 161, No. 5, 734­778 (2009). 49. M. V. Shamolin, "On the stability of rectilinear translation motion," Prikl. Mekh., 45, No. 6, 125­140 (2009). 50. M. V. Shamolin, "New cases of integrability in spatial rigid-b ody dynamics," Dokl. Ross. Akad. Nauk, 431, No. 3, 339­343 (2010). 51. M. V. Shamolin, "Motion of a rigid b ody in a resisting medium," Mat. Model., 23, No. 12, 79­104 (2011). 52. M. V. Shamolin, "Multi-parametric family of phase p ortraits in the dynamics of a rigid-b ody interacting with a medium," Vestn. Mosk. Univ., Ser. 1, Mat. Mekh., 3, 24­30 (2011). 53. M. V. Shamolin, "A new case of integrability in the spatial dynamics of a rigid b ody interacting with a medium taking into account a linear damping," Dokl. Ross. Akad. Nauk, 442, No. 4, 479­481 (2012). 54. G. K. Suslov, Theoretical Mechanics [in Russian], Gostekhizdat, Moscow (1946). 55. Ya. V. Tatarinov, Lectures on Classical Dynamics [in Russian], Moscow State Univ., Moscow (1984). 56. A. N. Tikhonov, "Systems of differential equations containing small parameters as coefficients of derivatives," Mat. Sb., 31, No. 3, 575­586 (1952).

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57. V. V. Trofimov and M. V. Shamolin, "Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems," Fundam. Prikl. Mat., 16, No. 4, 3­229 (2010). 58. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. With an Introduction to the Problem of Three Bodies, At the University Press, Cambridge (1960). 59. N. E. Zhukovsky, "On the soaring of birds," in: Complete Works [in Russian], Vol. 5, Fizmatlit, Moscow (1937), pp. 49­59. 60. N. E. Zhukovsky, "On the fall of a light oblong b ody rotating ab out its longitudinal axis," in: Complete Works [in Russian], Vol. 5, Fizmatlit, Moscow (1937), pp. 72­80, 100­115. Yu. M. Okunev Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia E-mail: common@imec.msu.ru M. V. Shamolin Institute of Mechanics of the M. V. Lomonosov Moscow State University, Moscow, Russia E-mail: shamolin@imec.msu.ru

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