Cases of integrability corresponding to the motion of a pendulum on the two-dimensional plane

Transcription

1 Cases of integrability corresponding to the motion of a pendulum on the two-dimensional plane MAXIM V. SHAMOLIN Lomonoso Moscow State Uniersity Institute of Mechanics Michurinskii Ae.,, 99 Moscow RUSSIAN FEDERATION shamolin@rambler.ru, shamolin@imec.msu.ru Abstract: We systematize some results on the study of the equations of plane-parallel motion of symmetric fixed rigid bodies pendulums located in a nonconseratie force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a plane-parallel motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconseratie tracing force; under action of this force, either the magnitude of the elocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable sero constraint, or the center of mass of the body moes rectilinearly and uniformly; this means that there exists a nonconseratie couple of forces in the system. Key Words: Rigid body, Pendulum, Resisting Medium, Dynamical Systems With Variable Dissipation, Integrability Model assumptions Let consider the homogeneous flat plate AB symmetrical relatie to the plane which perpendicular to the plane of motion and passing through the holder OD. The plate is rigidly fixed perpendicular to the tool holder OD located on the cylindrical hinge O, and it flows about homogeneous fluid flow. In this case, the body is a physical pendulum, in which the plate AB and the piot axis perpendicular to the plane of motion. The medium flow moes from infinity with constant elocity = 0. Assume that the holder does not create a resistance. We suppose that the total force S of medium flow interaction is parallel to the holder, and point N of application of this force is determined by at least the angle of attack α, which is made by the elocity ector D of the point D with respect to the flow and the holder, and also the reduced angular elocity ω = lω D, D = D l is the length of the holder, Ω is the algebraic alue of a projection of the pendulum angular elocity to the axle hinge. Such conditions arise when one uses the model of streamline flow around plane bodies [, ]. Therefore, the force S is directed along the normal to the plate to its side, which is opposite to the direction of the elocity D, and passes through a certain point N of the plate, which is displaced from the point D forward with respect to the flow see also [, 3]. The ector e = OD l determines the orientation of the holder. Then S = sα De, sα = s αsign cos α, 3 and the resistance coefficient s 0 depends only on the angle of attack α. By the plate symmetry properties with respect to the point D, the function sα is een. Let Dx x = Dxy be the coordinate system rigidly attached to the body, herewith, the axis Dx = Dx has a direction ector e see, and the axis Dx = Dy has the same direction with the ector DA. The space of positions of this physical pendulum is the circle one-dimensional sphere S ξ R : ξ mod π, 4 and its phase space is the tangent bundle of a circle T S ξ; ξ R : ξ mod π, 5 E-ISSN: Volume, 07

2 i.e., two-dimensional cylinder. To the alue Ω, we put in correspondence the skew-symmetric matrix Ω = 0 Ω Ω 0, Ω so. 6 The distance from the center D of the plate to the center of pressure the point N, has the form r N = r N = DN α, lω, 7 D r N = 0, x N = 0, y N in system Dx x = Dxy. Immediately, we note that the model used to describe the effects of fluid flow on fixed pendulum is similar to the model constructed for free body and, in further, takes into account of the rotational deriatie of the moment of the forces of medium influence with respect to the pendulum angular elocity see also [3]. An analysis of the problem of the physical pendulum in a flow will allow to find the qualitatie analogies in the dynamics of partially fixed bodies and free ones. Set of dynamical equations in Lie algebra so If I is a central moment of inertia of a rigid body pendulum then the general equation of motion has the following form: I Ω = DN α, lω D sα D, 8 since the moment of the medium interaction force equals the determinant of the following auxiliary matrix: 0 x N sαd, 9 0 sα D, 0 is the decomposition of the medium interaction force S in the coordinate system Dx x. Since the dimension of the Lie algebra so is equal to, the single equation 8 is a group equations on so, and, simply speaking, the motion equation. We see, that in the right-hand side of Eq. 8, first of all, it includes the angle of attack, therefore, this equation is not closed. In order to obtain a complete system of equations of motion of the pendulum, it is necessary to attach seeral sets of kinematic equations to the dynamic equation on the Lie algebra so. 3 First set of kinematic equations In order to obtain a complete system of equations of motion, it needs the set of kinematic equations which relate the elocities of the point D i.e., the formal center of the plate AB and the oer-running medium flow: D = D i α = Ω l + 0 i ξ, 0 i α = cos α sin α. The equation 0 resses the theorem of addition of elocities in projections on the related coordinate system Dx x. Indeed, the left-hand side of Eq. 0 is the elocity of the point D of the pendulum with respect to the flow in the projections on the related with the pendulum coordinate system Dx x. Herewith, the ector i α is the unit ector along the axis of the ector D. The ector i α is the image of the unit ector along the axis Dx, rotated around the ertical the axis Dx 3 by the angle α and has the decomposition. The right-hand side of the Eq. 0 is the sum of the elocities of the point D when you rotate the pendulum the first term, and the motion of the flow the second term. In this case, in the first term, we hae the coordinates of the ector OD = l, 0 in the coordinate system Dx x. We lain the second term of the right-hand side of Eq. 0 in more detail. We hae in it the coordinates of the ector =, 0 in the immoable space. In order to describe it in the projections on the related coordinate system Dx x, we need to make a reerse rotation of the pendulum at the angle ξ that is algebraically equialent to multiplying the alue on the ector i ξ. Thus, the first set of kinematic equations 0 has the following form in our case: D cos α = cos ξ, D sin α = lω + sin ξ. 4 Second set of kinematic equations We also need a set of kinematic equations which relate the angular elocity tensor Ω and coordinates ξ, ξ of the phase space 5 of pendulum studied, i.e., the tangent bundle T S ξ; ξ. We draw the reasoning style allowing arbitrary dimension. The desired equations are obtained from the E-ISSN: Volume, 07

3 following two sets of relations. Since the motion of the body takes place in a Euclidean space E n, n = formally, at the beginning, we ress the tuple consisting of a phase ariable Ω, through new ariable z from the tuple z: Ω = z. 3 Then we substitute the following relationship instead of the ariable z: z = ξ. 4 Thus, two sets of Eqs. 3 and 4 gie the second set of kinematic equations: Ω = ξ. 5 We see that three sets of the relations 8,, and 5 form the closed system of equations. These three sets of equations include the following two functions: r N = DN α, lω D, sα. 6 In this case, the function s is considered to be dependent only on α, and the function r N = DN may depend on, along with the angle α, generally speaking, the reduced angular elocity ω = lω/ D. 5 Problem on free body motion under assumption of tracing force Parallel to the present problem of the motion of the fixed body, we study the plane-parallel motion of the free symmetric rigid body with the frontal plane butt-end one-dimensional plate AB in the resistance force fields under the quasi-stationarity conditions [4, 5] with the same model of medium interaction. If, α are the polar coordinates of the elocity ector of the certain characteristic point D of the rigid body D is the center of the plate AB, Ω is the alue of its angular elocity, I, m are characteristics of inertia and mass, then the dynamical part of the equations of motion in which the tangent forces of the interaction of the body with the medium are absent, has the form cos α α sin α Ω sin α + σω = F x m, sin α + α cos α + Ω cos α σ Ω = 0, 7 I Ω = y N α, Ω sα, F x = S, S = sα, σ = CD, 8 in this case 0, y N α, Ω 9 are the coordinates of the point N of application of the force S in the coordinate system Dx x = Dxy related to the body. The first two equations of the system 7 describe the motion of the center of a mass in the two-dimensional Euclidean plane E in the projections on the coordinate system Dx x. In this case, Dx = Dx is the perpendicular to the plate passing through the center of mass C of the symmetric body and Dx = Dy is an axis along the plate. The third equation of the system 7 is obtained from the theorem on the change of the angular moment of a rigid body in the projection on the axis perpendicular to the plane of motion. Thus, the direct product R S so 0 of the two-dimensional cylinder and the Lie algebra so is the phase space of third-order system 7 of the dynamical equations. Herewith, since the medium influence force dos not depend on the position of the body in a plane, the system 7 of the dynamical equations is separated from the system of kinematic equations and may be studied independently see also [, 6]. 5. Nonintegrable constraint If we consider a more general problem on the motion of a body under the action of a certain tracing force T passing through the center of mass and proiding the fulfillment of the equality const, during the motion see also [7, 8], then F x in system 7 must be replaced by T sα. As a result of an appropriate choice of the magnitude T of the tracing force, we can achiee the fulfillment of Eq. during the motion [9]. Indeed, if we formally ress the alue T by irtue of system 7, we obtain for cos α 0: T = T α, Ω = mσω + E-ISSN: Volume, 07

4 [ +sα mσ I y N α, Ω ] sin α. 3 cos α This procedure can be iewed from two standpoints. First, a transformation of the system has occurred at the presence of the tracing control force in the system which proides the corresponding class of motions. Second, we can consider this procedure as a procedure that allows one to reduce the order of the system. Indeed, system 7 generates an independent second-order system of the following form: α cos α + Ω cos α σ Ω = 0, I Ω = y N α, Ω sα, 4 the parameter is supplemented by the constant parameters specified aboe. We can see from 4 that the system cannot be soled uniquely with respect to α on the manifold O = α, Ω R : α = π + πk, k Z 5 Thus, formally speaking, the uniqueness theorem is iolated on manifold 5. This implies that system 4 outside of the manifold 5 and only outside it is equialent to the following system: α = Ω + σ I Ω = I y N y Nα, Ω sα cos α, α, Ω sα. 6 The uniqueness theorem is iolated for system 4 on the manifold 5 in the following sense: regular phase trajectories of system 4 pass through almost all points of the manifold 5 and intersect the manifold 5 at a right angle, and also there exists a phase trajectory that completely coincides with the specified point at all time instants. Howeer, these trajectories are different since they correspond to different alues of the tracing force. Let us proe this. As was shown aboe, to fulfill constraint, one must choose the alue of T for cos α 0 in the form 3. Let y N α, Ω lim α π/ cos α sα Ω = L. 7 Note that L < + if and only if lim y N α, Ω sα < +. 8 α π/ α For α = π/, the necessary magnitude of the tracing force can be found from the equality π T = T, Ω = mσω mσl. 9 I Ω is arbitrary. On the other hand, if we support the rotation around a certain point W of the Euclidean plane E by means of the tracing force, then the tracing force has the form T = T π, Ω = m R 0, 30 R 0 is the distance CW. Generally speaking, Eqs. 9 and 30 define different alues of the tracing force T for almost all points of manifold 5, and the proof is complete. 5. Constant elocity of the center of mass If we consider a more general problem on the motion of a body under the action of a certain tracing force T passing through the center of mass and proiding the fulfillment of the equality see also [0] V C const 3 during the motion V C is the elocity of the center of mass, then F x in system 7 must be replaced by zero since the nonconseratie couple of the forces acts on the body: T sα 0. 3 Obiously, we must choose the alue of the tracing force T as follows: T = T α, Ω = sα, T S. 33 The choice 33 of the magnitude of the tracing force T is a particular case of the possibility of separation of an independent second-order subsystem after a certain transformation of the third-order system 7. Indeed, let the following condition hold for T : T = T α, Ω = τ α, Ω + +τ α, Ω Ω + τ 3 α, Ω Ω = 34 = T α, Ω. We can rewrite system 7 as follows: + σω cos α σ sin α = T [ I y N α, Ω sα m α, Ω ] sα = cos α, E-ISSN: Volume, 07

5 α + Ω σ cos α [ I y N α, Ω ] sα i N = i π 40 = sα T m σω sin α = α, Ω Ω = I y N α, Ω sα. sin α, 35 If we introduce the new dimensionless phase ariable and the differentiation by the formulas Ω = n ω, < >= n < >, n > 0, 36 n = const, then system 35 is reduced to the following form: = Ψα, ω, 37 α = ω + σn ω [ sin ] α+ + σ In y N α, n ω sα cos α T α,n ω sα mn sin α, ω = y In N α, n ω sα [ ] ω σ In y N α, n ω sα sin α+ +σn ω 3 cos α ω T α,n ω sα mn cos α, Ψα, ω = σn ω cos α+ [ ] σ + y N α, n ω sα sin α+ In 38 + T α, n ω sα cos α. mn We see that the independent second-order subsystem 38 can be substituted into the third-order system 37, 38 and can be considered separately on its own two-dimensional phase cylinder. In particular, if condition 33 holds, then the method of separation of an independent second-order subsystem is also applicable. 6 Case the moment of nonconseratie forces is independent of the angular elocity We take the function r N as follows the plate AB is gien by the equation x N 0: r N = 0 x N = Rαi N, 39 see. In our case Thus, the equality i N = 0. 4 x N = Rα 4 holds and shows that for the considered system, the moment of the nonconseratie forces is independent of the angular elocity it depends only on the angle α. And so, for the construction of the force field, we use the pair of dynamical functions Rα, sα; the information about them is of a qualitatie nature. Similarly to the choice of the Chaplygin analytical functions see [], we take the dynamical functions s and R as follows: s Rα = A sin α, sα = B cos α, A, B > Reduced systems Theorem The simultaneous equations 8,, 5 under conditions 39, 43 can be reduced to the dynamical system on the tangent bundle 5 of the onedimensional sphere 4. Indeed, if we introduce the dimensionless parameter and the differentiation by the formulas b = ln 0, n 0 = AB I, < >= n 0 < >, 44 then the obtained equation has the following form: ξ + b ξ cos ξ + sin ξ cos ξ = After the transition from the ariables z about the ariables z see 4 to the ariables w w = n 0 z b sin ξ, 46 Eq. 45 is equialent to the system on the tangent bundle ξ = w b sin ξ, w = sin ξ cos ξ, 47 T S w ; ξ R : ξ mod π 48 of the one-dimensional sphere S ξ R : ξ mod π. E-ISSN: Volume, 07

6 6. General remarks on integrability of system In order to integrate the second-order system 47, we hae to obtain, generally speaking, one independent first integral. 6.. The system under the absence of a force field Let study the system 47 on the tangent bundle T S w ; ξ of the one-dimensional sphere S ξ. At the same time, we get out of this system the conseratie one. Furthermore, we assume that the function 7 is identically equal to zero in particular, b = 0, and also the coefficient sin ξ cos ξ in the second equation of system 47 is absent. The system studied has the form ξ = w, 49 w = The system 49, 50 describes the motion of a rigid body in the absence of an external force field. Theorem System 49, 50 has one analytical first integral as follows: Φ w ; ξ = w = C 5 This first integral 5 states that as the external force field is not present, it is presered in general, nonzero the component of the angular elocity of a two-dimensional rigid body, precisely Ω Ω 0 5 In particular, the existence of the first integral 5 is lained by the equation w = n Ω C The system under the presence of a conseratie force field Now let us study the system 47 under assumption b = 0. In this case, we obtain the conseratie system. Precisely, the coefficient sin ξ cos ξ in the second equation of system 47 unlike the system 49, 50 characterizes the presence of the force field. The system studied has the form α = w, 54 w = sin ξ cos ξ. 55 Thus, the system 54, 55 describes the motion of a rigid body in a conseratie field of external forces. Theorem 3 System 54, 55 has one analytical first integral as follows: Φ w ; ξ = w + sin ξ = C = const, 56 The first integral 56 is an integral of the total energy. 6.3 Transcendental first integral We turn now to the integration of the desired secondorder system 47 without any simplifications, i.e., in the presence of all coefficients. We put in correspondence to system 47 the following nonautonomous first-order equation: dw dξ = sin ξ cos ξ w b sin ξ. 57 Using the substitution τ = sin ξ, we rewrite Eq. 57 in the algebraic form dw dτ = τ w b τ. 58 Further, introducing the homogeneous ariable by the formula w = uτ, we reduce Eq. 58 to the following quadrature: b udu + b u + u = dτ τ. 59 Integration of quadrature 59 leads to the following three cases. Simple calculations yield the following first integrals. I. b 4 < 0. ln + b u + u + + b 4 b arctg u + b 4 b + ln τ 60 II. b 4 > 0. ln + b u + u b b 4 ln u + b + b 4 u + b b III. b 4 = 0. + ln τ ln u + + ln τ 6 u In other words, in the ariables ξ, w the found first integrals hae the following forms: E-ISSN: Volume, 07

7 I. b 4 < 0. [sin ξ + b w sin ξ + w ] b arctg w + b sin ξ 4 b 4 b sin ξ 63 II. b 4 > 0. [sin ξ + b w sin ξ + w ] w + b sin ξ + b 4 sin ξ w + b sin ξ b 4 sin ξ III. b 4 = 0. b / b 4 = 64 sin ξ w sin ξ 65 w sin ξ Therefore, in the considered case the system of dynamical equations 47 has the first integral ressed by relations or 60 6, which is a transcendental function of its phase ariables in the sense of complex analysis and is ressed as a finite combination of elementary functions. Theorem 4 Three sets of relations 8,, 5 under conditions 39, 43 possess the first integral the complete set, which is a transcendental function in the sense of complex analysis and is ressed as a finite combination of elementary functions. 6.4 Topological analogies Now we present two groups of analogies related to the system 7, which describes the motion of a free body in the presence of a tracking force. The first group of analogies deals with the case of the presence the nonintegrable constraint in the system. In this case the dynamical part of the motion equations under certain conditions is reduced to a system 6. Under onditions 39, 43 the system 6 has the form α = ω + b sin α, ω 66 = sin α cos α, if we introduce the dimensionless parameter, the ariable, and the differentiation analogously to 44: b = σn 0, n 0 = AB I, Ω = n 0ω, 67 < >= n 0 < >. Theorem 5 System 66 for the case of a free body is equialent to the system 47 for the case of a fixed pendulum. Indeed, it is sufficient to substitute ξ = α, w = ω, b = b. 68 Corollary 6. The angle of attack α for a free body is equialent to the angle of body deiation ξ of a fixed pendulum.. The distance σ = CD for a free body corresponds to the length of a holder l = OD of a fixed pendulum. 3. The first integral of a system 66 can be automatically obtained through the Eqs or after substitutions 68 see also []: I. b 4 < 0. II. b 4 > 0. [sin α bω sin α + ω ] b 4 b ω b sin α arctg = 4 b sin α 69 [sin α bω sin α + ω ] ω b sin α + b 4 sin α ω b sin α b 4 sin α III. b 4 = 0. ω sin α b/ b 4 = 70 sin α 7 ω sin α The second group of analogies deals with the case of a motion with the constant elocity of the center of mass of a body, i.e., when the property 3 holds. In this case the dynamical part of the motion equations under certain conditions is reduced to a system 38. Then, under conditions 3, 39, 43, and 67, the reduced dynamical part of the motion equations system 38 has the form of analytical system α = ω + b sin α cos α + bω sin α, ω = sin α cos α bω sin α cos α + bω 3 cos α, 7 E-ISSN: Volume, 07

8 in this case, we choose the constant n as follows: n = n If the problem on the first integral of the system 66 is soled using Corollary 6, the same problem for the system 7 can be soled by the following theorem 7. For this we introduce the following notations and new ariables comp. with [3]: C = b, C = b > 0, C 3 = b < 0, u = ω sin α, = ω + sin α, 74 u = t, = q, then the problem on licit form of the desired first integral reduces to soling of the linear inhomogeneous equation: dq dt = a t q + a t, 75 a t = C 3t + C C 3 t C, 76 a t = 4C t C 3 t C. The general solution of Eq. 75 has the following form see [, 3]: I. b <. q t = kt C 3 t + C b arctg + b 4 b b t + const. 77 II. b >. q t = kt C 3 t + C C + C 3 t C / C C 3 C + const. 78 C 3 t III. b =. in this case, I. b <. q t = kt t kt = b 8 t + const, 79 [ ] b b sin ζ cos ζ + 4 b 4 b 80 II. b >. +const, b tgζ = + b t. kt = ± ζ b/ b 4 b b + b b 4 ζ b/ 4+ + const, 8 b ζ t =. b + + ζ III. b =. kt = t + t. 8 t Thus, Eqs allow us to find the desired first integral of system 7 using the notations and substitutions 74. Theorem 7 The first integral of the system 7 is a transcendental function of its own phase ariables and is ressed as a finite combination of elementary functions. Because of cumbersome character of form of the first integral obtained, we represent this form in the case III only: sin α + ω 4ω sin α + 4ω sin α ω ω sin α = C = 83 Theorem 8 The first integral of system 66 is constant on the phase trajectories of the system 7. Let us perform the proof for the case b =. Indeed, we rewrite the first integral 83 of system 7 as follows: n0 sin α + Ω n 0 sin α Ω n 0 bn 0 Ω sin α + b Ω Ω n 0 sin α 84 We see that the numerator of the second factor is proportional to the square of the elocity of the center of mass V C of the body with constant coefficient n 0. But, by irtue of 3, this alue is constant on trajectories of the system 7. Therefore, the function n0 sin α + Ω VC n 0 sin α Ω Ω n 0 sin α = const 85 E-ISSN: Volume, 07

9 is also constant on these trajectories. Further, let us raise the left-hand side of Eq. 85 to the power / and conclude that the following function is constant on the trajectories of the system 7: Ω + n0 sin α Ω n 0 sin α Ω n 0 sin α 86 And now, we diide Eq. 86 by e and obtain the function n0 sin α Ω n 0 sin α = const, Ω n 0 sin α 87 which is constant on the phase trajectories of the system 7. But the first integral 87 is completely similar to the first integral 7, as required. Thus, we hae the following topological and mechanical analogies in the sense lained aboe. A motion of a fixed physical pendulum on a cylindrical hinge in a flowing medium nonconseratie force fields. A plane-parallel free motion of a rigid body in a nonconseratie force field under a tracing force in the presence of a nonintegrable constraint. 3 A plane-parallel composite motion of a rigid body rotating about its center of mass, which moes rectilinearly and uniformly, in a nonconseratie force field. 7 Case the moment of nonconseratie forces depends on the angular elocity 7. Dependence on the angular elocity This section is deoted to dynamics of the twodimensional rigid body on the plane. This subsection is deoted to the study of the case of the motion the moment of forces depends on the angular elocity. We introduce this dependence in the general case; this will allow us to generalize this dependence both to three-, and multi-dimensional bodies. Let x = x N, x N be the coordinates of the point N of application of a nonconseratie force interaction with a medium on the one-dimensional plate and Q = Q, Q be the components independent of the angular elocity. We introduce only the linear dependence of the functions x N, x N = x N, y N on the angular elocity since the introduction of this dependence itself is not a priori obious see [, 3]. Thus, we accept the following dependence: x = Q + R, 88 R = R, R is a ector-alued function containing the angular elocity. Here, the dependence of the function R on the angular elocity is gyroscopic: R = R R = 0 Ω D Ω 0 = h h, 89 h, h are certain positie parameters comp. with [, 4]. Now, for our problem, since x N = x N 0, we hae Ω x N = y N = Q h. 90 D Thus, the function r N is selected in the following form the plate AB is defined by the equation x N 0: r N = 0 see 6,. In our case x N i N = i π = Rαi N D Ωh, 9 h, h = Ω = 0 Ω Ω 0 i N = 0 Thus, the following relation h, x N = Rα h Ω D 94 holds, which shows that an additional dependence of the damping or accelerating in some domains of the phase space moment of the nonconseratie forces is also present in the system considered i.e., the moment depends on the angular elocity. And so, for the construction of the force field, we use the pair of dynamical functions Rα, sα; the information about them is of a qualitatie nature. Similarly to the choice of the Chaplygin analytical functions see [, 5], we take the dynamical functions s and R as follows: Rα = A sin α, sα = B cos α, A, B > E-ISSN: Volume, 07

10 7. Reduced systems Theorem 9 The simultaneous equations 8,, 5 under conditions 9, 95 can be reduced to the dynamical system on the tangent bundle 5 of the onedimensional sphere 4. Indeed, if we introduce the dimensionless parameters and the differentiation by the formulas b = ln 0, n 0 = AB I, H = h B In 0, 96 < >= n 0 < >, then the obtained equation has the following form: ξ + b H ξ cos ξ + sin ξ cos ξ = After the transition from the ariables z about the ariables z see 4 to the ariables w w = + b H Eq. 97 is equialent to the system z + b sin ξ, 98 n 0 ξ = + b H w b sin ξ, w = sin ξ cos ξ + H w cos ξ. 7.3 Transcendental first integral 99 We put in correspondence to system 99 the following nonautonomous first-order equation: dw dξ = sin ξ cos ξ + H w cos ξ + b H w b sin ξ. 00 Using the substitution τ = sin ξ, we rewrite Eq. 00 in the algebraic form dw dτ = τ + H w + b H w b τ. 0 Further, introducing the homogeneous ariable by the formula w = uτ, we reduce Eq. 0 to the following quadrature: b + b H udu + b + H u + + b H u = dτ τ. 0 Integration of quadrature 0 leads to the following three cases. Simple calculations yield the following first integrals. I. b H <. ln + b + H u + + b H u + b + 4 b H arctg + b H u + b + H 4 b H + ln τ = II. b H > b H ln + b + H u + + b H u + + ln τ b + b H b H 4 A + b H ln 4 A b H 4 = 04 = const, A = + b H 3/ u + b + H + b H. III. b H =. ln u + b + H + b H b H + ln τ = 05 + b H u + b + H In the ariables ξ, w the found first integrals hae the cumbersome character of their form. Neertheless, we represent this form in the case III in the licit form: w + b + H + b H sin ξ b + H sin ξ = + b H w + b + H sin ξ 06 Therefore, in the considered case the system of dynamical equations 99 has the first integral ressed by relations or, in particular, in the case III 06, which is a transcendental function of its phase ariables in the sense of complex analysis and is ressed as a finite combination of elementary functions. Theorem 0 Three sets of relations 8,, 5 under conditions 9, 95 possess the first integral the complete set, which is a transcendental function in the sense of complex analysis and is ressed as a finite combination of elementary functions. E-ISSN: Volume, 07

11 7.4 Topological analogies Now we present two groups of analogies related to the system 7, which describes the motion of a free body in the presence of a tracking force. The first group of analogies deals with the case of the presence the nonintegrable constraint in the system. In this case the dynamical part of the motion equations under certain conditions is reduced to a system 6. Under conditions 9, 95 the system 6 has the form α = + bh ω + b sin α, ω = sin α cos α H ω cos α, 07 if we introduce the dimensionless parameters, the ariable, and the differentiation analogously to 44: b = σn 0, n 0 = AB I, H = h B In 0, Ω = n 0 ω, < >= n 0 < >. 08 Theorem System 07 for the case of a free body is equialent to the system 99 for the case of a fixed pendulum. Indeed, it is sufficient to substitute ξ = α, w = ω, b = b, H = H. 09 Corollary. The angle of attack α for a free body is equialent to the angle of body deiation ξ of a fixed pendulum.. The distance σ = CD for a free body corresponds to the length of a holder l = OD of a fixed pendulum. 3. The first integral of a system 07 can be automatically obtained through the Eqs or after substitutions 09 see also [6, 7]. In the ariables α, ω the found first integrals hae the cumbersome character of their form. Neertheless, we represent this form in the case III in the licit form: ω b + H + bh sin α b H sin α + bh ω b + H sin α = 0 The second group of analogies deals with the case of a motion with the constant elocity of the center of mass of a body, i.e., when the property 3 holds. In this case the dynamical part of the motion equations under certain conditions is reduced to a system 38. Then, under conditions 3, 9, 95, 08 the reduced dynamical part of the motion equations system 38 has the form of analytical system α = ω + b sin α cos α+ +bω sin α bh ω cos α, ω = sin α cos α bω sin α cos α+ +bω 3 cos α+ +bh ω sin α cos α H ω cos α, in this case, we choose the constant n as follows: n = n 0. If the problem on the first integral of the system 07 is soled using Corollary, the same problem for the system can be soled by the following Theorem 3. For this we introduce the following notations and new ariables comp. with [8, 9]: A = b bh H, A = + b + bh + H > 0, A 3 = b + bh H, u = ω sin α, = ω + sin α, u = t, = q, 3 then the problem on licit form of the desired first integral reduces to soling of the linear inhomogeneous equation: dq dt = a t q + a t, 4 a t = A t A A t + bh t + A 3, a t = b t + H t /4 A t + bh t + A 3. 5 The general solution of Eq. 4 has the following form [0]: I. b H <. q t = kt A t + bh t + A 3 b bh H 4 b H B, 6 B = arctg + b + bh + H 4 b H t + E-ISSN: Volume, 07

12 II. b H >. bh +. 4 b H q t = kt A t + bh t + A 3 4 b H + C b bh H / 4 b H 4 b H, C 7 C = + b + bh + H t + bh. III. b H =. q t = kt t + bh A b H + b + bh + H t + bh. 8 To find a solution of the nonhomogeneous equations 4, 5, we must ress the alue of k as a function of t, which is ressed as a finite combination of elementary functions. Thus, Eqs. 6 8 allow to obtain the desired first integral of the system using the notations and substitutions 3. Theorem 3 The first integral of the system is a transcendental function of its own phase ariables and is ressed as a finite combination of elementary functions. Because of cumbersome character of form of the first integral obtained, we represent this form in the case III only: b H sin α + bh ω b + H sin α 4ω sin α + 4ω ω sin α/b + H = C 9 Theorem 4 The first integral of system 07 is constant on the phase trajectories of the system. Let us perform the proof for the case b H =. Indeed, we rewrite the first integral 9 as follows: n 0 b H sin α + bh Ω n 0 b + H sin α n 0 4n 0 Ω sin α + 4Ω 0 Ω n 0 sin α/b + H We see that the numerator of the second factor is proportional to the square of the elocity of the center of mass V C of the body with constant coefficient n 0. But, by irtue of 3, this alue is constant on trajectories of the system. Therefore, the function n 0 b H sin α + bh Ω n 0 b + H sin α V C Ω n 0 sin α/b + H Further, let us raise the left-hand side of Eq. to the power / and conclude that the following function is constant on the trajectories of the system : n 0 b H sin α + bh Ω n 0 b + H sin α Ω n 0 sin α/b + H And now, it is clear that the function is equialent to the function 0, since in the case III the following relation b + H = 4 + bh 3 holds. Thus, we hae the following topological and mechanical analogies in the sense lained aboe. A motion of a fixed physical pendulum on a cylindrical hinge in a flowing medium nonconseratie force fields under assumption of additional dependence of the moment of the forces on the angular elocity. A plane-parallel free motion of a rigid body in a nonconseratie force field under a tracing force in the presence of a nonintegrable constraint under assumption of additional dependence of the moment of the forces on the angular elocity. 3 A plane-parallel composite motion of a rigid body rotating about its center of mass, which moes rectilinearly and uniformly, in a nonconseratie force field under assumption of additional dependence of the moment of the forces on the angular elocity. Acknowledgements: This work was supported by the Russian Foundation for Basic Research, project no a. References: [] Shamolin M. V. Some questions of the qualitatie theory of ordinary differential equations and dynamics of a rigid body interacting with a medium, J. Math. Sci., 00, 0, no., pp E-ISSN: Volume, 07

13 [] Shamolin M. V. New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium, J. Math. Sci., 003, 4, no., pp [3] Shamolin M. V. Foundations of differential and topological diagnostics, J. Math. Sci., 003, 4, no., pp [4] Shamolin M. V. Classes of ariable dissipation systems with nonzero mean in the dynamics of a rigid body, J. Math. Sci., 004,, no., pp [5] Shamolin M. V. On integrability in elementary functions of certain classes of nonconseratie dynamical systems, J. Math. Sci., 009, 6, no. 5, pp [6] Shamolin M. V. Dynamical systems with ariable dissipation: approaches, methods, and applications, J. Math. Sci., 009, 6, no. 6, pp [7] Shamolin M. V. Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconseratie field, J. Math. Sci., 00, 65, no. 6, pp [8] Trofimo V. V., and Shamolin M. V. Geometric and dynamical inariants of integrable Hamiltonian and dissipatie systems, J. Math. Sci., 0, 80, no. 4, pp [9] Shamolin M. V. Comparison of complete integrability cases in Dynamics of a two-, three-, and four-dimensional rigid body in a nonconseratie field, J. Math. Sci., 0, 87, no. 3, pp [0] Shamolin M. V. Some questions of qualitatie theory in dynamics of systems with the ariable dissipation, J. Math. Sci., 03, 89, no., pp [] Shamolin M. V. Variety of Integrable Cases in Dynamics of Low- and Multi-Dimensional Rigid Bodies in Nonconseratie Force Fields, J. Math. Sci., 05, 04, no. 4, pp [] Shamolin M. V. Classification of Integrable Cases in the Dynamics of a Four-Dimensional Rigid Body in a Nonconseratie Field in the Presence of a Tracking Force, J. Math. Sci., 05, 04, no. 6, pp [3] Shamolin M. V. Some Classes of Integrable Problems in Spatial Dynamics of a Rigid Body in a Nonconseratie Force Field, J. Math. Sci., 05, 0, no. 3, pp E-ISSN: Volume, 07