On Integrability of Certain Classes of Variable Dissipation Systems

MAXIM V. SHAMOLIN

Lomonosov Moscow State University

Institute of Mechanics

Michurinskii Ave., 1, 119192 Moscow

RUSSIAN FEDERATION

Abstract: We study the non-conservative systems for which the methods for studying, for example, Hamiltonian

systems is not applicable in general. Therefore, for such systems, it is necessary, in some sense, to “directly”

integrate the main equation of dynamics. Herewith, we offer more universal interpretation of both obtained cases

and new ones of complete integrability in transcendental functions in dynamics in a non-conservative force ﬁeld.

Key–Words: Dynamical Systems With Variable Dissipation, Integrability

Received: April 17, 2024. Revised: September 9, 2024. Accepted: October 11, 2024. Published: November 6, 2024.

The results of the proposed work are a devel-

opment of the previous studies, including a certain

applied problem from rigid body dynamics [1, 2],

where complete lists of transcendental ﬁrst integrals

expressed through a ﬁnite combination of elemen-

tary functions were obtained. Later on, this circum-

stance allows us to perform a complete analysis of

all phase trajectories and show those their properties

which have a roughness and are preserved for systems

of a more general form. The complete integrability of

such systems are related to symmetries of latent type.

As is known, the concept of integrability is sufﬁ-

ciently fuzzy in general. In its construction, it is nec-

essary to take into account the meaning in which it is

understood(we mean a certain criterion with respect

to which one makes a conclusion that the structure of

trajectories of the dynamical system considered is es-

pecially simple and “attractive and simple”, in which

class of functions, we seek for ﬁrst integrals, etc. (see

also [2, 3]).

In this work, we accept the approach that as the

class of functions for ﬁrst integrals, takes transcenden-

tal functions, and, moreover, elementary ones. Here,

the transcendence is understood not in the sense of el-

ementary function theory (for example, trigonometric

functions), but in the sense of existence of essentially

singular points for them (according to the classiﬁca-

tion accepted in the theory of function of one complex

variable). In this case, we need to formally continue

the function considered to the complex domain (see

also [3, 4]).

1 Preliminary Arguments and Re-

sults

Of course, in the general case, it is sufﬁciently difﬁcult

to construct a certain theory of integrability for non-

conservative systems (even of low dimension). But in

a number of cases where the system studied have addi-

tional symmetries, we succeed in ﬁnding ﬁrst integrals

through ﬁnite combinations of elementary functions

[5, 6].

Proposed work appeared from the plane problem

of motion of a rigid body in a resisting medium whose

contact surface with the medium is a part of its ex-

terior surface. In this case, the force ﬁeld is con-

structed from the reasons of the medium action on the

body under streamline (or separation) ﬂow around un-

der the quasi-stationarity conditions. It turns out that

the study of motion of such classes of bodies reduces

to systems with energy scattering ((purely) dissipative

systems or systems in a dissipative force ﬁeld) or sys-

tems with energy pumping (the so-called systems with

anti-dissipation, or systems with dispersing forces).

Note that such problems were already appeared in ap-

plied aerodynamics (see also [6]).

The previously considered problems have stimu-

lated (and stimulate) the development of quality tools

for studying, which essentially complement the quali-

tative theory of nonconservative systems with dissipa-

tion of both signs (see also [7]).

Also, we have qualitatively studied nonlinear ef-

fects of motion in plane and spatial rigid body dynam-

ics and have justiﬁed on the qualitative level the neces-

sity of introducing the deﬁnitions of relative rough-

ness and relative non-roughness of various degrees

(see also [8]).

PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024

The following series of results allowed us to pre-

pare this work in a very deﬁnite style.

i) We have elaborated the methods for qualita-

tive studying dissipative systems and systems with

anti-dissipation that allo us to obtain the condition

for bifurcation of birth of stable and unstable auto-

oscillations and also the conditions for absence of any

singular trajectories. We succeeded in generalizing

the method for studying plane topographical Poincar`

systems to higher dimensions. We have obtained suf-

ﬁcient Poisson stability conditions(density near itself)

of certain classes of non-closed trajectories of dynam-

ical systems [8, 9].

ii) In two- and three-dimensional rigid body dy-

namics, we have discovered complete lists of ﬁrst in-

tegrals of dissipative systems and systems with anti-

dissipation that are transcendental (in the sense of

classiﬁcation of their singularities) functions that are

expressed through elementary functions in a number

of cases. We have introduced new deﬁnitions of rela-

tive roughness and relative non-roughness of various

degrees, which have the integrated systems [9, 10].

iii) We have obtained multiparameter families of

topologically non-equivalent phase portraits arising in

purely dissipative systems (i.e., systems with variable

dissipation and nonzero (positive) mean). Almost ev-

ery portrait of such families is (absolutely) rough [10,

11].

iv) We have discovered new qualitative analogs

between the properties of motion of free bodies in a

resisting medium that is ﬁxed at inﬁnity, and bodies in

the run-of medium ﬂow [12].

Many results of this work were regularly reported

at numerous workshops, including the workshop “Ac-

tual Problems of Geometry and Mechanics” named af-

ter professor V. V. Troﬁmov led by D. V. Georgievskii

and M. V. Shamolin.

2 Variable Dissipation Dynamical

Systems as a Class of Systems Ad-

mitting a Complete Integration

2.1 Visual characteristic of variable dissipa-

tion dynamical systems

Since, in the initial modelling of the medium action

on a rigid body, we have used the experimental infor-

mation about the properties of streamline ﬂow, it be-

comes necessary to study the class of dynamical sys-

tems that has the property of (relative) structural sta-

bility (relative roughness). Therefore, it is quite natu-

ral to introduce these deﬁnitions for such systems. In

this case, many of the systems considered are (abso-

lutely) rough in the Andronov–Pontryagin sense [13,

14].

After some simpliﬁcations (for example, in the

two-dimensional dynamics), the dynamical part of the

general system of equations of plane-parallel motion

can be reduced to a pendulum-like second-order sys-

tem in which there is a linear nonconservative (sign-

alternating) force with coefﬁcients that have different

signs for different values of the periodic phase coordi-

nate in the system.

Therefore, in this case, we will speak of systems

with the so-called variable dissipation, where the term

“variable” refers not to the value of the dissipation

coefﬁcient, but to the possible alternation of its sign

(therefore, it is more reasonable to use the term “sign-

alternating”).

On the average, for the period of the existing peri-

odic coordinate, the dissipation can be either positive

(“purely” dissipative systems), or negative (systems

with dispersing forces), and also it can be equal to zero

(but not identically in this case). In the latter case, we

speak of the zero mean variable dissipation systems

(such systems can be associated with “almost” con-

servative systems).

As was already mentioned previously, we have

detected important mechanical analogs arising in the

comparison of qualitative properties of the stationary

motion of a free body and the equilibrium of a pen-

dulum in a medium ﬂow. Such analogs have a deep

support meaning since they allow us to extend the

properties of nonlinear dynamical systems for a pen-

dulum to dynamical systems for a free body. Both

these systems belong to the class of the so-called zero

mean variable dissipation pendulum-like dynamical

systems.

Under additional conditions, the equivalence de-

scribed above is extended to the case of spatial mo-

tion, which allows us to speak of the common charac-

ter of symmetries in a zero mean variable dissipation

system under the plane-parallel and spatial motions

(for the plane and spatial variants of a pendulum in

the medium ﬂow, see also [1–3]).

In this connection, previously, we have intro-

duced the deﬁnitions of relative structural stability

(relative roughness), and relative structural instability

(relative nonroughness) of various degrees. The latter

properties were proved for systems that arise, e.g., in

[4, 6].

The purely dissipative systems (as well as

(purely) anti-dissipative ones), which in our case can

belong to nonzero zero mean variable dissipation sys-

tems, are, as a rule, structurally stable ((absolutely)

rough), whereas zero mean variable dissipation sys-

tems (which, as a rule, have additional symmetries)

are either structurally unstable (nonrough) or only rel-

atively structurally stable (relatively nonrough). In the

PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024

general case, it is difﬁcult to prove the latter assertion.

For example, a dynamical system of the form

˙α= Ω + βsin α,

Ω = −βsin αcos α(1)

is relatively structurally stable (relatively rough) and

is topologically equivalent to the system describing a

clamped pendulum in an over-run medium ﬂow [7, 8].

Below we present its ﬁrst integral, which is a tran-

scendental (in the sense of the theory of functions

of one complex variable having essentially singular

points after its continuation to the complex domain)

function of phase variables expressed through a ﬁ-

nite combination of elementary functions. The phase

cylinder R2{α, Ω}of quasi-velocities of the system

considered has an interesting topological structure of

partition into trajectories (for more detail, see [9]).

Although the dynamical system considered is not

conservative, in the rotation domain (and only in it) of

the phase plane R2{α, Ω}, it admits the preservation

of an invariant measure with variable density. This

property characterizes the system considered as a zero

mean variable dissipation system.

2.2 One of the deﬁnitions of a zero mean

variable dissipation system

We will study systems of ordinary differential equa-

tions having a periodic phase coordinate. The systems

under study have those symmetries under which, on

the average, for the period in the periodic coordinates,

their phase volume is preserved. So, for example, the

following pendulum-like system with smooth and pe-

riodic in αof period Tright-hand side V(α, ω)of the

form

˙α=−ω+f(α),

˙ω=g(α),

f(α+T) = f(α), g(α+T) = g(α),

(2)

preserves its phase area on the phase cylinder over the

period T:

divV(α, ω)dα =

=ZT

0∂

∂α(−ω+f(α)) + ∂

∂ω g(α)dα =(3)

=ZT

f′(α)dα = 0.

The system considered is equivalent to the pendu-

lum equation

¨α−f′(α) ˙α+g(α)=0,(4)

in which the integral of the coefﬁcient f′(α)standing

by the dissipative term ˙αis equal to zero on the aver-

age for the period.

It is seen that the system considered has those

symmetries under which it becomes the so-called zero

mean variable dissipation system in the sense of the

following deﬁnition.

Deﬁnition 1 Consider a smooth autonomous system

of the (n+ 1)th order of normal form given on the

cylinder Rn{x} × S1{αmod 2π}, where αis a pe-

riodic coordinate of period T > 0. The divergence

of the right-hand side V(x, α)(which, in general, is a

function of all phase variables and is not identically

equal to zero) of this system is denoted by divV(x, α).

This system is called a zero (nonzero) mean variable

dissipation system if the function

divV(x, α)dα (5)

is equal (not equal) to zero identically. Moreover, in

some cases (for example, when singularities arise at

separate points of the circle S1{αmod 2π}), this in-

tegral is understood in the sense of principal value.

It should be noted that giving a general deﬁnition

of a zero (nonzero) mean variable dissipation system

is not simple. The deﬁnition just presented uses the

concept of divergence (as is known, the divergence of

the right-hand side of a system in normal form charac-

terizes the variation of the phase volume in the phase

space of this system).

3 Systems with Symmetries and Zero

Mean Variable Dissipation

Let us consider systems of the form (the dot denotes

the derivative in time)

˙α=fα(ω, sin α, cos α),

˙ωk=fk(ω, sin α, cos α), k = 1, . . . , n, (6)

given on the set

S1{αmod 2π}\K×Rn{ω},(7)

ω= (ω1, . . . , ωn), where the smooth functions

fλ(u1, u2, u3),λ=α, 1, . . . , n, of three variables

u1, u2, u3are as follows:

fλ(−u1,−u2, u3) = −fλ(u1, u2, u3),

fα(u1, u2,−u3) = fα(u1, u2, u3),

fk(u1, u2,−u3) = −fk(u1, u2, u3),

(8)

PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024

herewith, the functions fk(u1, u2, u3)are deﬁned for

u3= 0 for any k= 1, . . . , n.

The set Kis either empty or consists of ﬁnitely

many points of the circle S1{αmod 2π}.

The latter two variables u2and u3in the functions

fλ(u1, u2, u3)depend on one parameter α, but they

are allocated into different groups because of the fol-

lowing reasons. First, not in the whole domain, they

are one-to-one expressed through each other, and, sec-

ond, the ﬁrst of them is an odd function and the second

is an even function of α, which inﬂuences the symme-

tries of system (6) in different ways.

To the system (6), let us put in correspondence the

following nonautonomous system

dωk

dα =fk(ω, sin α, cos α)

fα(ω, sin α, cos α), k = 1, . . . , n, (9)

which via the substitution τ= sin α, reduces to the

form

dωk

dτ =fk(ω, τ, φk(τ))

fα(ω, τ, φα(τ)), k = 1, . . . , n, (10)

φλ(−τ) = φλ(τ), λ =α, 1, . . . , n.

The latter system can have in particular an alge-

braic right-hand side (i.e., it can be the ratio of two

polynomials); sometimes this helps us to ﬁnd its ﬁrst

integrals in explicit form.

The following assertion embeds the class of sys-

tems (6) in the class of zero mean variable dissipation

systems. The inverse embedding does not hold in gen-

eral.

Theorem 2 Systems of the form (6) are zero mean

variable dissipation dynamical systems.

This theorem is proved by using the certain sym-

metries (8) of system (6) only, listed above, and uses

the periodicity of the right-hand side of the system on

α.

Indeed, let calculate the speciﬁed divergence of

the vector ﬁeld of system (6). It equals to

∂fα(ω, sin α, cos α)

∂u2

cos α−

−∂fα(ω, sin α, cos α)

∂u3

sin α+

k=1

∂fk(ω, sin α, cos α)

∂u1

.(11)

The following integral on two ﬁrst summands

(11) is equal to zero:

Z2π

0∂fα(ω, sin α, cos α)

∂u2

dsin α+

+∂fα(ω, sin α, cos α)

∂u3

dcos α=

=Z2π

∂fα(ω, sin α, cos α)

∂α dα =hα(ω)≡0,(12)

since the function fα(ω, sin α, cos α)is periodic one

on α.

So, by virtue of the third equation (8), the follow-

ing property holds for any k= 1, . . . , n:

∂fk(ω, sin α, cos α)

∂u1

= cos α·∂gk(ω, sin α)

∂u1

,(13)

herewith, the function gk(u1, u2)is rather smooth for

any number k= 1, . . . , n.

Then the integral on period 2πfrom the right-

hand side of the equation (13) gives

Z2π

∂gk(ω, sin α)

∂u1

dsin α=hk(ω)≡0(14)

for any k= 1, . . . , n. And we get the assertion of the

theorem 2 from the equalities (12), (14).

The converse assertion is not true in general since

we can present a set of dynamical systems on the two-

dimensional cylinder, being zero mean variable dissi-

pation systems that have none of the above-listed sym-

metries.

In this work, we are mainly concerned with

the case where the functions fλ(ω, τ, φk(τ)) (λ=

α, 1, . . . , n) are polynomials in ω, τ.

Example 1. We consider, in particular,

pendulum-like systems on the two-dimensional cylin-

der S1{αmod 2π} × R1{ω}with parameter b > 0

from rigid body dynamics:

˙α=−ω+bsin α,

˙ω= sin αcos α, (15)

and

˙α=−ω+bsin αcos2α+bω2sin α,

˙ω= sin αcos α−bω sin2αcos α+

+bω3cos α,

(16)

in the variables (ω, τ), to these systems we can put

in correspondence the following equations with alge-

braic right-hand sides:

dω

dτ =τ

−ω+bτ ,(17)

and dω

dτ =τ+bω[ω2−τ2]

−ω+bτ +bτ[ω2−τ2](18)

which have the form (4.2). Moreover, these systems

are zero mean variable dissipation dynamical systems,

which is easily directly veriﬁed.

PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024

Indeed, the divergences of its right-hand sides are

equal to

bcos α

and

bcos α[4ω2+ cos2α−3 sin2α],

respectively. It is easy to verify that they belong to the

class of systems (6).

Moreover, each of these systems has a ﬁrst inte-

gral that is a transcendental (in the sense of the the-

ory of functions of one complex variable) function

expressed through a ﬁnite combination of elementary

functions.

Let us present one more important example of a

higher-order system having the listed properties.

Example 2. To the following system

˙α=−z2+bsin α,

˙z2= sin αcos α−z2

1cos α

sin α,

˙z1=z1z2cos α

sin α,

(19)

with parameter b, considered in the three-dimensional

domain

S1{αmod 2π}\{α= 0, α =π} × R2{z1, z2}(20)

(such a system can also be reduced to an equiva-

lent system on the tangent bundle T∗S2of the two-

dimensional sphere S2) and describing the spatial mo-

tion of a rigid body in a resisting medium, we put in

correspondence the following nonautonomous system

with algebraic right-hand side: (τ= sin α):

dz2

dτ =τ−z2

1/τ

−z2+bτ ,dz1

dτ =z1z2/τ

−z2+bτ .(21)

It is seen that system (19) is a zero mean vari-

able dissipation system; in order to achieve complete

correspondence with the deﬁnition, it sufﬁces to intro-

duce a new phase variable

z∗

1= ln |z1|.(22)

If we calculate the divergence of the right-hand

side of the system (19) in Cartesian coordinates

α, z∗

1, z2, then we shall get that it is equal to bcos α.

Herewith, if we consider (20), we shall have in the

sense of principal value:

lim

ε→0Zπ−ε

bcos α+ lim

ε→0Z2π−ε

π+ε

bcos α= 0.(23)

Moreover, the system (19) has two ﬁrst integrals

(i.e., a complete list) that are transcendental functions

and we expressed through a ﬁnite combination of el-

ementary functions, which, as was mentioned above,

becomes possible after putting in correspondence to

it a (nonautonomous in general) system of equations

with algebraic (polynomial) right-hand (21).

The above-presented systems (15), (16), and (19),

along with the property that they belong to the class

of systems (6) and are zero mean variable dissipation

systems, also have a complete list of transcendental

ﬁrst integrals expressed through a ﬁnite combination

of elementary functions.

Therefore, to search for the ﬁrst integrals of the

system considered, it is better to reduce systems of

the form (6) to systems (10) with polynomial right-

hand sides, on whose form the possibility of integra-

tion in elementary functions of the initial system de-

pends. Therefore, we proceed as follows: we seek suf-

ﬁcient conditions for integrability in elementary func-

tions of systems of equations with polynomial right-

hand sides studying systems of the most general form

in this process.

4 Systems on Tangent Bundle of

Two-Dimensional Sphere

Let consider the following dynamic system

θ+b˙

θcos θ+ sin θcos θ−˙

ψ2sin θ

cos θ= 0,

ψ+b˙

ψcos θ+˙

θ˙

ψh1+cos2θ

sin θcos θi= 0 (24)

on tangent bundle T∗S2of two-dimensional sphere

S2{θ, ψ}. This system describes the spherical pendu-

lum, placed in the accumulating medium ﬂow. Here-

with, both conservative moment is present in the sys-

tem

sin θcos θ, (25)

and the force moment depending on the velocity as

linear one with the variable coefﬁcient:

b ˙

ψ!cos θ. (26)

The coefﬁcients remaining in the equations are

the coefﬁcients of connectedness, i.e.,

Γθ

ψψ =−sin θ

cos θ,Γψ

θψ =1 + cos2θ

sin θcos θ.(27)

The system (24) has an order 3 practically, since

the variable ψis a cyclic, herewith, the derivative ˙

ψis

present in the system only.

Proposition 3 The equation

ψ= 0 (28)

deﬁne the family of integral planes for the system (24).

PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024

Furthermore, the equation (28) reduces the sys-

tem (24) to an equation describing the cylindric pen-

dulum which placed in the accumulating medium

ﬂow.

Proposition 4 The system (24) is equivalent to the

following system:

θ=−z2+bsin θ,

˙z2= sin θcos θ−z2

1cos θ

sin θ,

˙z1=z1z2cos θ

sin θ,

ψ=z1cos θ

sin θ

(29)

on the tangent bundle T∗S2{z1, z2, θ, ψ}of two-

dimensional sphere S2{θ, ψ}.

Moreover, the ﬁrst three equations of the system

(29) form the closed three order system and coincide

with the equations of the system (19) (if we denote

α=θ). The separation of fourth equation of the sys-

tem (29) has also occurred by the reason of cyclicity

of the variable ψ.

On the construction of phase pattern of the system

(24), expressed in Pic. ??, see [9, 10].

Example 3. Let us study a system of the form

(19), which reduces to (21), and also the system

˙α=−z2+b(z2

1+z2

2) sin α+

+bsin αcos2α,

˙z2= sin αcos α+bz2(z2

1+z2

2) cos α−

−bz2sin2αcos α−z2

1cos α

sin α,

˙z1=bz1(z2

1+z2

2) cos α−

−bz1sin2αcos α+z1z2cos α

sin α,

(30)

which also arises in the three-dimensional dynamics

of a rigid body interacting with a medium and which

corresponds to the following system with algebraic

right-hand side:

dz2

dτ =τ+bz2(z2

1+z2

2)−bz2τ2−z2

1/τ

−z2+bτ(z2

1+z2

2)+bτ(1−τ2),

dz1

dτ =bz1(z2

1+z2

2)−bz1τ2+z1z2/τ

−z2+bτ(z2

1+z2

2)+bτ(1−τ2).(31)

Therefore, as before, we consider a pair of sys-

tems: the initial system (30) and the algebraic system

(31) corresponding to it.

In a similar way, we pass to homogeneous coor-

dinates uk,k= 1,2,by the formulas

zk=ukτ. (32)

Using the latter change, we reduce system (21) to

the form

τdu2

dτ +u2=τ−u2

1τ

−u2τ+bτ ,

τdu1

dτ +u1=u1u2τ

−u2τ+bτ ,(33)

corresponding to the equation

du2

du1

=1−bu2+u2

2−u2

2u1u2−bu1

.(34)

This equation is integrated in elementary functions

since the identity

d 1−βu2+u2

u1!+du1= 0,(35)

is integrated, and in the coordinates (τ, z1, z2), it has

a ﬁrst integral of the form (compare with [10])

1+z2

2−βz2τ+τ2

z1τ=const.

System (30) after its reduction corresponds to the

system

τdu2

dτ +u2=τ+bu2τ3(u2

1+u2

2)−bu2τ3−u2

1τ

−u2τ+bτ3(u2

1+u2

2)+bτ(1−τ2),

τdu1

dτ +u1=bu1τ3(u2

1+u2

2)−bu1τ3+u1u2τ

−u2τ+bτ3(u2

1+u2

2)+bτ(1−τ2),(36)

which also reduces to the form (34).

5 Certain Generalizations

Let us pose the following question: What are the pos-

sibilities of integrating in elementary functions the

system

dx =ax+by+cz+c1z2/x+c2zy/x+c3y2/x

d1x+ey+fz ,

dx =gx+hy+iz+i1z2/x+i2zy/x+i3y2/x

d1x+ey+fz ,(37)

of a more general form, which includes the systems

(21) and (31) considered above in three-dimensional

phase domains and which has a singularity of the form

1/x?

Previously, a number of results concerning this

question were already obtained (see also [5, 6]). Let

us present these results and complement them by orig-

inal arguments.

As previously, introducing the substitutions

y=ux, z =vx, (38)

we obtain that system (37) reduces to the following

system

xdv

dx +v=ax+bux+cvx+c1v2x+c2vux+c3u2x

d1x+eux+fvx ,(39)

xdu

dx +u=gx+hux+ivx+i1v2x+i2vux+i3u2x

d1x+eux+fvx ,(40)

PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024

which is equivalent to

xdv

dx =

=ax+bux+(c−d1)vx+(c1−f)v2x+(c2−e)vux+c3u2x

d1x+eux+fvx ,

(41)

xdu

dx =

=gx+(h−d1)ux+ivx+i1v2x+(i2−f)vux+(i3−e)u2x

d1x+eux+fvx ,

(42)

we put in correspondence the following equation with

algebraic right-hand side:

du =a+bu+cv+c1v2+c2vu+c3u2−v[d1+eu+fv]

g+hu+iv+i1v2+i2vu+i3u2−u[d1+eu+fv].(43)

The integration of the latter equation reduces to

that of the following equation in total differentials:

[g+hu+iv+i1v2+i2vu+i3u2−d1u−eu2−fuv]dv =

= [a+bu +cv +c1v2+

+c2vu +c3u2−d1v−euv −fv2]du. (44)

We have (in general) a 15-parameter family of

equations of the form (44). To integrate the latter iden-

tity in elementary functions as a homogeneous equa-

tion, it sufﬁces to impose seven relations

g= 0, i = 0, i1= 0,

e=c2, h =c, i2= 2c1−f. (45)

Introduce nine parameters β1, . . . , β9and con-

sider them as independent parameters:

β1=a, β2=b, β3=c, β4=c1, β5=c2,

β6=c3, β7=d1, β8=f, β9=i3.(46)

Therefore, under the group of conditions (45),

and (46), Eq. (44) reduces to the form

du =β1+β2u+(β3−β7)v+(β4−β8)v2+β6u2

(β3−β7)u+2(β4−β8)vu+(β9−β5)u2,(47)

and the system (41), (42), respectively, to the form

xdv

dx =β1+β2u+(β3−β7)v+(β4−β8)v2+β6u2

β7+β5u+β8v,(48)

xdu

dx =(β3−β7)u+2(β4−β8)vu+(β9−β5)u2

β7+β5u+β8v,(49)

after that, the equation (47) is integrated in ﬁnite com-

bination of elementary functions.

Indeed, integrating the identity (44), we obtain the

relation

d(β3−β7)v

u+

+d"(β4−β8)v2

u#+d[(β9−β5)v] + dβ1

u−

−d[β2ln |u|]−d[β6u] = 0,(50)

which for the beginning allows us to obtain the fol-

lowing invariant relation:

(β3−β7)v

u+(β4−β8)v2

u+ (β9−β5)v+β1

u−

−β2ln |u| − β6u=C1=const,(51)

and in the coordinates (x, y, z), it allows us to obtain

the ﬁrst integral in the form

yx −β2ln 

x

=const,(52)

A= (β4−β8)z2−

−β6y2+ (β3−β7)zx + (β9−β5)zy +β1x2.

Therefore, we can make a conclusion about the

integrability in elementary functions of the follow-

ing, in general, nonconservative third-order system

depending on nine parameters:

dx =β1x+β2y+β3z+β4z2/x+β5zy/x+β6y2/x

β7x+β5y+β8z,

dx =β3y+(2β4−β8)zy/x+β9y2/x

β7x+β5y+β8z.(53)

Corollary 5 The third-order system

˙α=β7sin α+β5z1+β8z2,

˙z2=β1sin αcos α+β2z1cos α+

+β3z2cos α+

+β4z2

2cos α

sin α+β5z1z2cos α

sin α+β6z2

1cos α

sin α,

˙z1=β3z1cos α+ (2β4−β8)z1z2cos α

sin α+

+β9z2

1cos α

sin α,

(54)

on the set

S1{αmod 2π}\{α= 0, α =π}×R2{z1, z2},(55)

which depends on nine parameters, has a ﬁrst inte-

gral, which is in general transcendental and expressed

through elementary functions:

z1sin α−β2ln 

sin α

=const,(56)

B= (β4−β8)z2

2−β6z2

+(β3−β7)z2sin α+ (β9−β5)z2z1+β1sin2α2.

In particular, for β1= 1, β2=β3=β4=β5=

β9= 0,β6=β8=−1,β7=bsystem (54) reduces

to system (19).

To ﬁnd the additional ﬁrst integral of the nonau-

tonomous system (37), we use the found ﬁrst integral

(52), which is expressed through a ﬁnite combination

of elementary functions.

PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024

For the beginning let transform the relation (51)

as follows:

(β4−β8)v2+

+ [(β9−β5)u+ (β3−β7)] v+f1(u) = 0,(57)

where

f1(u) = β1−β6u2−β2uln |u| − C1u.

Herewith, the value vformally can be found from

the equality

v1,2(u) = 1

2(β4−β8)×

×(β5−β9)u+ (β7−β3)±qf2(u),(58)

where

f2(u) = A1+A2u+A3u2+A4uln |u|,

A1= (β3−β7)2−4β1(β4−β8),

A2= 2(β9−β5)(β3−β7)+4C1(β4−β8),

A3= (β9−β5)2+ 4β6(β4−β8),

A4= 4β2(β4−β8).

Then the quadrature studied for the search of ad-

ditional (in general) transcendental ﬁrst integral (for

example, of the system (48), (49) or (41), (42), here-

with, the equation (49) is used) has the following form

Zdx

x=Z[β7+β5u+β8v1,2(u)]du

=Z[B1+B2u+B3pf2(u)]du

B4upf2(u),(59)

Bk=const, k = 1,...,4,

C= (β3−β7)u+ (β9−β5)u2+ 2(β4−β8)uv1,2(u).

And the quadrature studied for the search of ad-

ditional (in general) transcendental ﬁrst integral (for

example, of the system (48), (49) or (41), (42), here-

with, the equation (48) is used) has the following form

Zdx

x=Z[β7+β5u(v) + β8v]dv

D,(60)

D=β1+β2u(v)+(β3−β7)v+(β4−β8)v2+β6u2(v).

herewith, the function u(v)should be obtain as a re-

sult of resolving of implicit equation (51) respectively

to u(that, in general case, is not always evident).

The following lemma gives the necessary condi-

tions of the expression of integrals in (60) through the

ﬁnite combination of elementary functions.

Lemma 6 The indeﬁnite integral in (60) is expressed

through the ﬁnite combination of elementary functions

for A4= 0, i.e., either

β2= 0 (61)

β4=β8.(62)

Theorem 7 The system (54) under assumption of

necessary conditions of lemma 6 (the property (61)

holds in given case), has the compete set of ﬁrst inte-

grals, which expressing through the ﬁnite combination

of elementary functions.

Therefore, the dynamical systems considered in

this work refer to zero mean variable dissipation sys-

tems in the existing periodic coordinate. Moreover,

such systems often have a complete list of ﬁrst inte-

grals expressed through elementary functions.

So, it was shown above the certain cases of com-

plete integrability in spatial dynamics of a rigid body

motion in a nonconservative ﬁeld. Herewith, we dealt

with with three properties, which are independent for

the ﬁrst glance:

i) the distinguished class of systems (6) with the

symmetries above;

ii) the fact that this class of systems consists of

systems with zero mean variable dissipation (in the

variable α), which allows us to consider them as “al-

most” conservative systems;

iii) in certain (although lower-dimensional) cases,

these systems have the complete tuple of ﬁrst inte-

grals, which are transcendental in general (from the

viewpoint of complex analysis).

The method for reducing the initial systems of

equations with right-hand sides containing polynomi-

als in trigonometric functions to systems with polyno-

mial right-hand sides allows us to search for the ﬁrst

integrals (or to prove their absence) for systems of a

more general form, but not only for systems that have

the above symmetries (see also [10, 11]).

6 Conclusion

The results of the presented work were appeared ow-

ing to the study the applied problem of the rigid body

motion in a resisting medium, where we have ob-

tained complete lists of transcendental ﬁrst integrals

expressed through a ﬁnite combination of elemen-

tary functions. This circumstance allows the author

to carry out the analysis of all phase trajectories and

show those their properties which have the roughness

and are preserved for systems of a more general form.

The complete integrability of such system is related

PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024

to their symmetries of latent type. Therefore, it is of

interest to study a sufﬁciently wide class of dynamical

systems having analogous latent symmetries [12, 13].

So, for example, the instability of the simplest

body motion, the rectilinear translational drag, is used

for methodological purposes, precisely, for ﬁnding the

unknown parameters of the medium action on a rigid

body under the quasi-stationarity conditions.

The experiment on the motion of homogeneous

circular cylinders in the water carried out in Institute

of Mechanics of M. V. Lomonosov State University

justiﬁed that in modelling the medium action on the

rigid body, it is also necessary to take into account an

additional parameter that brings a dissipation to the

system.

In studying the class of body drags with ﬁnite an-

gle of attack, the principal problem is ﬁnding those

conditions under which there exist auto-oscillations in

a ﬁnite neighborhood of the rectilinear translational

drag. Therefore, there arises the necessity of a com-

plete nonlinear study [14, 15].

Generally speaking, the dynamics of a rigid body

interacting with a medium is just the ﬁeld where there

arise either nonzero mean variable dissipation systems

or systems in which the energy loss in the mean dur-

ing a period can vanish. In the work, we have obtained

such a methodology owing to which it becomes possi-

ble to ﬁnally and analytically study a number of plane

and spatial model problems.

In qualitative describing the body interaction with

a medium, because of using the experimental infor-

mation about the properties of the streamline ﬂow

around, there arises a deﬁnite dispersion in modelling

the force-model characteristics. This makes it natural

to introduce the deﬁnitions of relative roughness (rela-

tive structural stability) and to prove such a roughness

for the system studied. Moreover, many systems con-

sidered are merely (absolutely) Andronov–Pontryagin

rough [15, 16].

References:

[1] Shamolin M. V. New Cases of Homogeneous

Integrable Systems with Dissipation on Tan-

gent Bundles of Three-Dimensional Mani-

folds, Doklady Mathematics, 2020, 102, no. 3,

pp. 518–523.

[2] Okunev Yu. M., Shamolin M. V. On the Con-

struction of the General Solution of a Class of

Complex Nonautonomous Equations, J. Math.

Sci.,204, no. 6, 2015, pp. 787–799.

[3] Shamolin M. V. Classes of variable dissipation

systems with nonzero mean in the dynamics of

a rigid body, J. Math. Sci., 2004, 122, no. 1,

pp. 2841–2915.

[4] Shamolin M. V. On integrability in elementary

functions of certain classes of nonconservative

dynamical systems, J. Math. Sci., 2009, 161,

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able dissipation: approaches, methods, and ap-

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[6] Prandtl L., Betz A. Ergebmisse der Aerodi-

namischen Versuchsastalt zu G¨

ottingen, Berlin,

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[8] Shamolin M. V. Comparison of complete inte-

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[10] Shamolin M. V. Variety of Integrable Cases

in Dynamics of Low- and Multi-Dimensional

Rigid Bodies in Nonconservative Force Fields,

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[11] Shamolin M. V. Classiﬁcation of Integrable

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[12] Shamolin M. V. Some Classes of Integrable

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[13] Shamolin M. V. New Cases of Integrable

Odd-Order Systems with Dissipation, Doklady

Mathematics, 2020, 101, no. 2, pp. 158–164.

[14] Shamolin M. V. New Cases of Homoge-

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Tangent Bundles of Two-Dimensional Mani-

folds, Doklady Mathematics, 2020, 102, no. 2,

pp. 443–448.

[15] Peixoto M. Structural stability on two-

dimensional manifolds, Topology, 1962, 1,

no. 2, pp. 101–120.

[16] Peixoto M. On an approximation theorem of

Kupka and Smale, J. Diff. Eq., 1966, 3,

pp. 214–227.

PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024

Contribution of Individual Authors to the

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Policy)

The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to declare that

is relevant to the content of this article.

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PROOF

DOI: 10.37394/232020.2024.4.6

Maxim V. Shamolin

E-ISSN: 2732-9941

Volume 4, 2024