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ISSN 1560-3547, Regular and Chaotic Dynamics, 2015, Vol. 20, No. 3, pp. 205­224. c Pleiades Publishing, Ltd., 2015.

The Dynamics of Systems with Servoconstraints. I
Valery V. Kozlov*
Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
Received February 10, 2015; accepted March 5, 2015

Abstract--The paper discusses the dynamics of systems with B´ eghin's servo constraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servo constraint -- the pro jection of the angular velo city onto some direction fixed in the bo dy is equal to zero (a generalization of the nonholonomic Suslov problem) -- and the motion of the Chaplygin sleigh with servo constraints of a certain type. The dynamics of systems with B´ eghin's servo constraints is richer and more varied than the more usual dynamics of nonholonomic systems. MSC2010 numbers: 34D20, 70F25, 70Q05 DOI: 10.1134/S1560354715030016 Keywords: servoconstraints, symmetries, Lie groups, left-invariant constraints, systems with quadratic right-hand sides.

1. INTRODUCTION The paper is concerned with the dynamics of systems with servoconstraints. Fundamentals of the theory of systems with constraints realized by controlled forces are discussed in B´ eghin's thesis [1], and features of a general theory were shaped by Appel in [2]. The relation between B´ eghin's theory of servoconstraints and the basic principles of dynamics is described in [3]. In the current literature on controlled systems, servoconstraints are usually called a motion program. Equations of motion with the servoconstraint (x, x, t) = 0 have the form T d T - = F + N . (1.2) dt x x Here, x = (x1 ,... ,xn ) are generalized coordinates, x = (x1 ,... , xn ) is the velocity of the system, T is the kinetic energy, F is an external force, N is a predetermined covector field (field of forces), and is a Lagrange multiplier. Equations (1.1) and (1.2) should, of course, be considered jointly. The condition for realization of the servoconstraint is as follows: = 0. (1.3) A-1 N, x Here, A=
*

=0 x

(1.1)

2T x2

E-mail: kozlov@pran.ru

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is a positive definite operator of inertia. The condition (1.3) means that the parameter can be found as a function of x, x and t (without solving the equations of motion). A discussion of the condition (1.3) can be found in [3]. Some clarifications are in order here. The general theory of controlled mechanical systems with constraints should not be confused with B´ eghin's theory of servoconstraints (as is done, for example, in [23] and [24]). In general, systems with constraints that depend on control parameters were considered before eghin [1]. Let us mention Ya. I. Grdina's papers on the dynamics of living the classical work of B´ organisms (see [26], where references to his earlier papers can be found). In the simplest version, equations with a control parameter have the form T d T - = F ( x, t, ), (x, x, t, ) = 0. x, dt x x The function t (t) (control) is found as a result of simultaneous solution of this differentialalgebraic system. The same equations were studied in [23] and [24]. If = 0, then the parameter is found from the constraint equation (at least locally) as a function of the state x, x and time t. Substituting this function into the expression for force yields a closed system of second-order differential equations, from which we find the motion t x(t) and, thus, the control t (t). If the constraint equation does not contain a control parameter, then differentiating it with respect to time and using the Lagrange equation, we obtain A-1 F, where is a known function of x, x and t. If A-
1

x

= ,

F , x

= 0,

then (again by the implicit function theorem) the parameter is found as a function of the state of the system and time. In particular, if at some instant of time = 0, then this equality holds for all values of t. Finally, when the external force is not dependent on the parameter either, the differential-algebraic system is usually inconsistent and an additional force (constraint reaction) needs to be introduced for realization of motion with a constraint. In [23], constraints that depend on control parameters are called "conventional constraints". The force F is considered to be an external force, and the reaction of the conventional constraint is not introduced. More precisely, it is considered to be zero and, therefore, does no work on the virtual displacements x defined by the Chetaev equation , x x = 0.

The same virtual displacements are also used in [24­26]. Moreover, equations of motions in these articles are unnecessarily complicated by introducing an additional Lagrange multiplier: T d T - = F ( x, t, )+ x, , (x, x, t, ) = 0. dt x x x It is argued that in order to close the system, one has to specify the control law (or to determine it from some additional conditions) = ( x, t). x, In our opinion, in these equations one has to set = 0 (as is done in [23]).
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In B´ eghin's theory, constraints are independent of controls and the force acting on the system is equal to the sum of the external force F (also independent of the parameter) and the reaction of the servoconstraint N . For the servoconstraint to be ideal, virtual displacements satisfy another equation (N, x) = 0. Although this property was clearly expressed by B´ eghin (and repeated in Appel's classical treatise [2]), it is not always understood in terms of the basic principles of analytical mechanics. In general, the definition of virtual displacements is a key axiom of the dynamics of systems with constraints. Changing this axiom (while retaining the other axioms: the principle of release and the condition for ideal constraints) leads to different dynamics of a system with the constraint (T, F, ) (for a discussion of this set of issues, see [3]). An obvious analogy is Euclid's famous fifth postulate. It is clear that servomotors realize the motion of a mechanical system with a constraint with some error. An analysis of these errors, as well as issues of their minimization, are discussed, for example, in [24] and [27]. For dynamic systems of the general form, some such issues are discussed in [28]. The realization of motion of systems with servoconstraints can be called active because it is done by means of controlling devices. If we set N= , x

then the realization condition (1.3) is satisfied and Eqs. (1.1)­(1.2) describe the classical nonholonomic motion. On the other hand, it is well known that nonholonomic motion can be also realized by the passive method using additional forces of anisotropic viscous friction. This idea is due to Caratheodory. An overview of results on the passive realization of constraints and their discussion can be found in [29]. The problem of exact integration of Eqs. (1.1)­(1.2) is considerably more complex than the corresponding problem for the Hamiltonian equations and nonholonomic equations (where N = / x). In a typical situation, the well-known theorem on the change of kinetic energy d T + t dt T , x - T = (F, x), x (1.4)

from which the energy integral is derived in the stationary case and when external forces are potential, no longer holds here. The relation (1.4) holds only in the case where for all the "real" velocities x (satisfying Eq. (1.1)) (N, x) = 0. In other words, real velocities must be within virtual displacements. This condition is satisfied only in particular cases, such as = (N, x), where is a nonzero function in the phase space. If the covector field N is independent of x, then we obtain a homogeneous nonholonomic constraint for which the relation (1.4) is satisfied. On the other hand, it is not difficult to formulate the generalized Noether theorem, which gives the first integrals linear in the velocity. Let w(x) be a vector field on the configuration space whose flow (a one-parameter family of transformations of the configuration space) conserves the kinetic energy T ( x) (after these transformations are naturally lifted to the phase space). If x, (N, w) = 0, then d dt
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The relation (1.5) means that the symmetry field at each point of the configuration space is a virtual displacement of our system. In particular, if the system moves by inertia F = 0, then the equations of motion admit the integral T ,w x = const.

From a qualitative point of view, systems with servoconstraints have a more complicated behavior than the more usual nonholonomic systems. The discussion of this set of issues and of the dynamics of systems with servoconstraints on Lie groups forms the content of this paper. For modern methods and problems of nonholonomic systems (analogs of some of them in the presence of servoconstraints are considered in this paper) see also the following series of works [4­11]. 2. SYSTEMS ON LIE GROUPS WITH LEFT-INVARIANT KINETIC ENERGY AND LEFT-INVARIANT SERVOCONTRAINTS Let G be an n-dimensional Lie group and g be its Lie algebra. Let T be a left-invariant Riemannian metric on G. From the point of view of dynamics, this metric can be interpreted as the kinetic energy of some mechanical system. According to Poincar´ [12], the dynamical equations e (Lagrange's equations) decouple and form a closed system of differential equations on the algebra g (or on the coalgebra g -- the dual linear space): mk = ci mi j ; k = 1,... ,n. jk (2.1)

Here, 1 ,... ,n (the Cartesian coordinates on the algebra g ) are the quasi-velocities of the system and m1 ,... ,mn (the coordinates on the coalgebra g ) are the momenta. They are related by the linear relationships mk = Ikj j , 1 k n, (2.2)

where Ikj is a constant tensor of inertia. The constants ci = -ci are structural constants of jk kj the Lie algebra; they satisfy the well-known Jacobi identities. The kinetic energy 1 1 T = (I , ) = (I -1 m, m) 2 2 is the first integral of Eqs. (2.1). A full description of the motion of the system requires that kinematic equations be added to Eqs. (2.1). For more details, see [13]. Remark. The velocity of the system = (1 ,... ,n ) is a tangent vector to the group G at the identity element. Its components transform according to the contravariant rule. Therefore (from the sequential point of view of tensor analysis), the numbers of components of the velocity should be written above. However, this is inconvenient when dealing with specific examples. Let = (a, ) = ai i = 0 (ai = const) (2.3)

be a left-invariant constraint linear in the velocities. Let the constant nonzero covector a be an eghin) that this constraint is realized by element of the coalgebra g . Assume (according to B´ means of the controlled force b = (b1 ,...,bn ), where (t) is an unknown function of time (control) and b g \{0}. Then the dynamics of the system with a left-invariant servoconstraint take the following form: mk = ci mi j + bk (1 jk k n), ai i = 0. (2.4)

If a and b are collinear, then the system (2.4) defines the nonholonomic motion with the constraint (2.3). In this case, Eqs. (2.4) admit the energy integral T = const. If the covectors a and b are noncollinear, then the energy is generally not conserved.
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The condition for realization of the constraint (2.3) is reduced to the inequality (I
-1

a, b) = 0.

(2.5)

It means that the covectors a and b are not orthogonal with respect to the "inner" metric on g induced by the original left-invariant metric on the group G. The condition (2.5) is a general condition for realization of the servoconstraint (1.3) expressed in terms of group variables. After the exclusion of the multiplier from Eq. (2.4), the system on the algebra g (or on the coalgebra g ) can be reduced to a closed system of n - 1 differential equations with homogeneous quadratic right-hand sides. This can be achieved in different ways. Let us specify one of them. To do this, we represent Eqs. (2.4) as a system on the coalgebra g (in other words, we invert the linear system (2.2) and replace 1 ,...,n in (2.4) with the linear functions of m1 ,... ,mn ). Then the constraint equation takes the form Aj mj = 0, (2.6)

where A = (A1 ,...,An ) is a nonzero element of the algebra g . By a linear change of variables, this vector is transformed to the form (0,... , 0, 1); that is, the constraint equation (2.6) takes the form mn = 0. Thus, the last equation of the dynamical system (2.4) is reduced to an algebraic equation from which the multiplier can be found as a quadratic form of m1 ,... ,mn . Substituting this expression for in the first n - 1 equations, we obtain the desired result. We shall call a closed system of p = n - 1 differential equations with quadratic right-hand sides a reduced system. 3. GENERAL PROPERTIES OF REDUCED SYSTEMS Thus, we consider a closed system of differential equations in Rp = {x1 ,...,xp } xj = vj (x1 ,... ,xp ), 1 j p, (3.1)

with v1 ,...,vp being homogeneous quadratic polynomials of p independent variables. The case p = 1 is trivial. For p = 2, the system (3.1) is integrable by quadratures. This is a consequence of the more general result about the integrability of autonomous systems on a plane with homogeneous right-hand sides (due to Leibniz). Since the case p = 2 is of particular interest to us, we present formulas that allow us, on the one hand, to integrate the system (3.1) and, on the other hand, to provide a qualitative analysis of the behavior of its tra jectories. To do this, we set z = x2 /x1 . Then z= where f (z ) = v2 (1,z ) - zv1 (1,z ) is a third-degree polynomial in z . Prop osition 1. If t x(t) is a solution of the system (3.1), then t -x(-t) is also a solution of the same system. This is a simple consequence of the quadraticity of the right-hand sides and the theorem of uniqueness of solutions. This is not the case for ordinary linear systems with constant coefficients: here t -x(t) is also a solution. It is clear that -x(-t + c) is a solution of the system (3.1) for all real c. Note that nontrivial equilibrium points of the system (3.1) (i.e., the points x0 = 0, where v (x0 ) = 0) cannot be isolated. If v (x0 ) = 0, then v (x0 ) = 0 for all real . The qualitative properties of solutions of the system (3.1) depend essentially on the existence of nontrivial equilibrium points.
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v2 x1 - v1 x2 = x1 f (z ), x2 1

(3.2)

(3.3)


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Theorem 1. If v (x) = 0 for x = 0, then the system (3.1) admits an invariant straight line x = , where is a nonzero vector from Rp and the parameter changes with time according to the differential equation = c2 , c = 0. (3.4)

This assertion and its generalizations play an important role in the first Lyapunov method for strongly nonlinear systems [14]. We give here the highlights of its proof. We equip Rp with the "standard" Euclidean structure and consider the Gauss map x (x) = v ( x) , v ( x)

x S p-1 (a unit sphere in Rp with center at the origin). We have a smooth map of the unit sphere to itself. The index of the vector field v at the point x = 0 is equal to the degree of mapping . The degree is even because each element of has an even number of preimages. Thus, the degree is different from(-1)p-1 , and therefore (by the well-known theorem from differential topology) has a fixed point x = . But then the system of differential equations (3.1) admits the invariant straight line x = , R. Substituting x = into Eqs. (3.1) yields = v ( ) = 2 v ( ) = 2 v ( ) . Hence we obtain Eq. (3.4), where c = v ( ) = 0. Theorem 1 has a number of important corollaries. 1 . If x = 0 is the only singular point of the quadratic vector field v , then it is unstable. Moreover, there is a solution with an arbitrarily small initial condition, which goes to infinity in finite time. The latter follows from the form of the solution of Eq. (3.4): (t) = 0 , 0 = (0). 1 - c0 t

2 . The phase flow of this system is not defined on the entire axis of time t. However, if the equilibrium x = 0 of the quadratic vector field v is not isolated, then it can be stable and the phase flow is defined on the entire axis R = {t}. Here is a simple example: x1 = x1 x2 , x2 = -x2 . 1 Here, the entire axis x1 = 0 consists of equilibrium points. 3 . Under the conditions of Theorem 1, the system (3.1) does not admit a positive definite first integral (in particular, as a positive definite quadratic form in x1 ,... ,xp ). Otherwise, the equilibrium x = 0 would be stable. In particular, the energy is not conserved for solutions of the original dynamical system (2.4) on a Lie algebra. Looking ahead, we can say that there are examples of systems on Lie groups with left-invariant B´ eghin's constraints whose reduced systems satisfy the condition of Theorem 1. On the contrary, equilibria of reduced nonholonomic systems on Lie groups are never isolated. Theorem 2. If the covectors a and b are col linear, then the trivial equilibrium x = 0 of the reduced system is stable and there is a straight line running through the origin and consisting entirely of equilibrium points. Indeed, in this case the left-invariant constraint (2.3) is nonholonomic, and therefore the dynamical system admits an energy integral -- a positive definite quadratic form on the algebra g . Its restriction to the hyperplane (2.3) yields a positive definite quadratic integral of the reduced system. Thus, the trivial equilibrium x = 0 is stable in the sense of Lyapunov. The existence of a straight line of equilibria follows from Theorem 1. Remark. One should not think that if the covectors a and b are noncollinear, then the reduced system has only the trivial equilibrium x = 0. The answer to this question depends, among other things, on the structure of the Lie group G. For example, if the group G is Abelian (ck = 0), then ij for all a and b the reduced system degenerates into a trivial system: x = 0. All points on Rp are stable equilibrium points.
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For p = 2, Theorem 1 is immediately derived from the formulas (3.2) and (3.3) because a polynomial of the third degree over the field of real numbers always has a real zero. Some clarifications are in order here. If the axis x1 = 0 is not invariant, the polynomial v1 (1,z ) has degree 2. Then the polynomial f from (3.3) is, indeed, of degree 3. In addition, in the right-hand half-plane (where x1 > 0), the relation z = x2 /x1 can take on all real values. Even more can be said: the system (3.1) on a plane with a single trivial equilibrium can have either one or three invariant straight lines (counted with multiplicities). This observation can be extended to the multidimensional case. The following theorem holds. Theorem 3. Again, let v (x) = 0 for x = 0. If the number of invariant straight lines (counted with multiplicities) of the system (3.1) is finite, then it is odd and does not exceed 2p - 1. For p = 2, we find that the system (3.1) has either one or three (counted with multiplicities) invariant straight lines. Theorem 1 is obviously contained within Theorem 3 (although the former is used to prove the latter). Contrary to the topological proof of Theorem 1, the proof of Theorem 3 is algebraic. Let us write explicitly the system of algebraic equations for invariant straight lines: v1 (x1 ,... ,xp ) - x1 = 0,... ,vp (x1 ,... ,xp ) - xp = 0. (3.5) This is a system of p homogeneous equations of degree 2 with respect to p +1 variables x1 ,... ,xp and . According to Bezout's theorem, the number of different complex disproportionate nonzero solutions of the system (3.5) (counted with multiplicities) exactly equals 2p . One of the real solutions of the system (3.5) x1 = ... = xp = 0, = 1 ( 3. 6) gives no invariant straight lines. Hence, the number of invariant straight lines does not exceed 2p - 1. Since v (x) = 0 for x = 0, the other nonzero solutions of the algebraic system (3.5) are disproportionate to the solution (3.6). One of the real solutions of the algebraic system (3.5) exists by Theorem 1. Other solutions are either real or pairs of complex conjugate solutions. In any case, the total number of real disproportionate solutions (counted with multiplicities) is odd. This proves Theorem 3. In conclusion of this section we make some comments regarding the systems with quadratic right-hand sides on a plane that admit nontrivial equilibrium points. Such systems are, of course, not typical. It turns out that after rescaling time they reduce to linear systems. Theorem 4. Let p = 2 and let the system (3.1) have a whole straight line ax1 + bx2 = 0 (a2 + b2 = 0) consisting of equilibrium points. Then upon rescaling time by t , d = (ax1 + bx2 )dt the system (3.1) reduces to a linear system. Without loss of generality, we can assume, for example, the axis x2 to be a straight line of equilibria. It can be always achieved by a suitable linear change of variables that does not alter the structure and form of the system (3.1). With this choice of variables, Eqs. (3.1) take the following form: x1 = x1 (Ax1 + Bx2 ), x2 = x1 (Cx1 + Dx2 ). Rescaling time as d = x1 dt reduces the system to a linear system. The tra jectories of the linear system x1 = Ax1 + Bx2 , x2 = Cx1 + Dx2 , (3.8) obviously consist of the tra jectories of the original system. It is only that the direction of motion in the original system depends also on the sign of the variable x1 . As an example, consider the linear system (3.8) with the equilibrium in the form of a stable node (Fig. 1a). Figure 1b shows a phase portrait of the original nonlinear system (3.7). In contrast to the linear system, the trivial equilibrium of the nonlinear system is unstable. It is worth emphasizing that the change of stability does not always occur during the transition from a linear to a nonlinear system.
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(3.7)


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(a)
Fig. 1.

(b)

Theorem 5. Let p = 2 and suppose that the system (3.1) admits a first integral F in nondegenerate quadratic form. Then the system has an entire straight line of equilibrium points. If the integral F is positive (negative) definite, then this assertion is that considered earlier in [15]. In this case, Theorem 5 is a simple corollary of Theorem 1. We give the proof of Theorem 5 for a quadratic integral with the signature + -. By a nondegenerate linear change of variables, the integral F can be reduced to x1 x2 . (3.9)

Let the system (3.1) have the following form in the new variables: x1 = a1 x2 + b1 x1 x2 + c1 x2 , x2 = a2 x2 + b2 x1 x2 + c2 x2 . 1 2 1 2 (3.10) The condition F = 0 is equivalent to the following relations for the coefficients of this system: a2 = c1 = 0, a1 + b2 = 0, b1 + c2 = 0. Thus, (3.10) takes the following explicit form: x1 = x1 (a1 x1 + b1 x2 ), x2 = -x2 (a1 x1 + b1 x2 ).
Fig. 2.

(3.11)

Hence, the straight line a1 x1 + b1 x2 = 0 consists entirely of equilibrium points. If a1 = b1 = 0, then the system degener ates into the trivial system x1 = x2 = 0, all solutions of which are equilibrium points.

Note that after rescaling time as d = (a1 x1 + b1 x2 )dt (3.12)

the system (3.11) becomes a linear Hamiltonian system with the Hamiltonian F . A phase portrait of the system (3.11) is shown in Fig. 2 (the directions of motion along the tra jectories may, of course, be opposite). The phase portrait for the case of a positive definite quadratic integral is well known (see [16], where the classical nonholonomic Suslov problem is considered).
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4. REMARKS ON PHASE PORTRAITS ON A PLANE Consider a reduced system on the plane (p = 2). It has the form (3.10). In a typical situation, the system (3.10) has a single equilibrium point x1 = x2 = 0. More precisely, this property holds if the following inequalities are satisfied: (a1 c2 - c1 a2 )2 + a1 c1 b2 + b2 a2 c2 - a1 b1 b2 c2 - b1 c1 a2 b2 = 0, 2 1 (a2 + a2 )(c2 + c2 ) = 0. 1 2 1 2 What do the phase portraits of such systems (with a single trivial equilibrium) look like? As we have seen in Section 3, the system (3.10) has either one or three (counted with multiplicities) invariant straight lines passing through the origin, which is an equilibrium point.
x
2

2

x21
0.5

1

x2

1

0.5

-2

-1

0

1

2

x

1

-1

-0.5

0

x
0.5

1

1

-1

-0.5

0 -0.5

0.5

x1
1

-1

-0.5

-2

-1

-1

(a) (k = 2)
1

(b) (k = 0.5)
x2
1

(c) (k = 0)
x
2

0.5

0.5

-1

-0.5 -0.5

0.5

x1
1
-1 -0.5 0 -0.5 0.5

x

1

1

-1

(d) (k = -0.5)
Fig. 3.

(e) (k = -2)

-1

Using the formulas (3.2)­(3.3) and their modifications, one can come up with a classification of possible types of phase portraits of the system (3.10). Moreover, using pro jective coordinates one can even obtain information about the drift of the phase tra jectories to infinity (as was done, for example, in [18] for systems on a plane with polynomial right-hand sides). However, the author does not know whether this work has been done. Usually more complex systems involving terms linear in x1 and x2 are studied (see [18]). Nevertheless, a comprehensive representation of the possible types of phase portraits is given in Figs. 3a­3e, which show the phase portraits of the system x1 = x2 + x2 , x2 = (k +1)x2 +(2 - k )x1 x2 - x2 , 1 2 1 2 (4.1) containing the real parameter k . The system (4.1) obviously satisfies the condition of Theorem 1. For all k it has an invariant straight line x1 = x2 . If k > 0, then there are no other invariant straight lines, while if k < 0, two invariant straight lines are added (4.2) x2 = (-1+ -k )x1 and x2 = (-1 - -k)x1 .
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When k = 0 (the "bifurcation" value of the parameter), these straight lines coincide. Figures 3a­3e give a clear picture of the transformation of the phase portrait when the parameter passes through the point of bifurcation. For positive values of k , the phase tra jectories are doubly asymptotic to the invariant straight line x1 = x2 . They have the form of a "wave" whose crest rises infinitely as k + 0. When k < 0, there appears a new pair of invariant straight lines (4.2) between which all the phase tra jectories in the left half-plane tend to the equilibrium x1 = x2 = 0 as t + . 5. ROTATION OF A RIGID BODY WITH A LEFT-INVARIANT SERVOCONSTRAINT This is a simple but important example of the theory under development; an outline for its consideration was provided already in [3]. The nonholonomic version of this problem is discussed, for example, in [16] and [17]. Here, the Lie group G is the three-dimensional rotation group SO(3) in three-dimensional Euclidean space. The presence of a Euclidean metric on the algebra SO(3) allows us to identify covectors and vectors. Equations (2.4) on the algebra SO(3) take the following form: I + â I = b, (a, ) = 0. (5.1) The symbol "â" denotes a vector product, I is the tensor of inertia of the rigid body, and the constant nonzero vectors a and b specify the directions of the body-fixed axes l and n (mentioned in [3]). We choose the axes of the moving orthogonal coincides with the axis l. In these axes the vector components of the tensor of inertia and by 1 , 2 , onto the chosen axes. In this notation 3 = 0 is following explicit form: trihedron in such a manner that the third axis a has components 0, 0, 1. We denote by Iij the 3 the pro jections of the angular velocity vector the constraint equation, and Eqs. (5.1) take the

I11 1 + I12 2 +(I13 1 + I23 2 )2 = b1 , I12 1 + I22 2 - (I13 1 + I23 1 )1 = b2 , I13 1 + I23 2 +(I12 1 + I22 2 )1 - (I11 1 + I12 2 )2 = b3 .

(5.2)

Here we have already taken into account the constraint equation 3 = 0. To obtain a reduced system on the plane, the multiplier needs to be eliminated from Eq. (5.2). If b1 = b2 = 0 (and b3 = 0), we get the equations of the classical Suslov problem of nonholonomic mechanics. Its phase portrait is well-known (see [16]). To feel the difference in the qualitative behavior of the system in the general case (when b2 + b2 = 0), we set I13 = I23 = 0. From the last 1 2 equation we find the Lagrange multiplier
2 2 = I12 1 +(I22 - I11 )1 2 - I12 2 .

(5.3)

In the general case, this quadratic form is not identically zero. A linear integral is easily derived from the first two equations of (5.2) (b2 I11 - b1 I12 )1 +(b2 I12 - b1 I22 )2 = const.
2 This integral does not degenerate under the natural condition b2 +b2 = 0 (because I11 I22 -I12 > 0). 1 2 Its level lines are straight lines. The phase portrait is shown in Fig. 4. The two heavy straight lines (consisting of equilibrium points) are the straight lines which the equation = 0 decomposes into. We emphasize that is a neutral quadratic form of 1 and 2 . The depicted direction of motion along the rectilinear segments of the phase tra jectories corresponds to a positive value of the product I12 (b1 I22 - b2 I12 ). If I12 = 0, the coordinate axes 1 = 0 and 2 = 0 are filled with the equilibrium points (where = 0).

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It can be seen that the phase portrait in Fig. 4 is radically different from the portrait of the nonholonomic Suslov problem (which for I13 = I23 = 0 consists entirely of equilibrium points). In particular, if I11 = I22 , the equilibrium 1 = 2 = 0 is always unstable. We now move on to discussing the general case. From this point on, we assume that b1 = 0 . This can always be achieved by rotating the orthogonal coordinate system through an appropriate angle about the third axis (coinciding with the axis l). According to [3], the condition for realization of the constraint 3 = 0
Fig. 4.

( 5. 4)

is reduced to the condition that the body-fixed axes l and n are not orthogonal with respect to the metric defined by the kinetic energy of the body (more precisely, with respect to the metric on the dual algebra (so(3)) defined by the tensor I -1 ). The same condition can be obtained in another way, by eliminating the Lagrange multiplier from the last two equations of (5.2) (the first equation does not contain the multiplier because b1 = 0 by assumption). After that we obtain a system of two equations with respect to 1 and 2 , and the condition for realization of the constraint (5.4) reduces to the condition for solvability of this system for the derivatives 1 and 2 . This yields the inequality
2 b3 (I11 I22 - I12 ) - b2 (I11 I23 - I12 I13 ) = 0.

(5.5)

This is the sought-for condition for realization of the constraint (5.4). In Eq. (5.5), of course, b2 + b2 = 0. 2 3 If b2 = 0 and b3 = 0, the condition (5.5) is certainly satisfied. This observation is a special case of the more general result about the realization of conventional nonholonomic constraints. On the contrary, if b3 = 0 while b2 = 0 (the axes l and n are orthogonal with respect to the Euclidean metric SO(3) induced by the metric of the original three-dimensional Euclidean space), the condition for realization of (5.5) takes the form I11 I23 = I12 I13 . This condition, of course, can be violated in the case of some special mass distributions of the rigid body. Let us find all the equilibrium points of the reduced system. To do this, we write explicitly Eqs. (5.2) after eliminating the Lagrange multiplier: I11 1 + I12 2 +(I13 1 + I23 2 )2 = 0, (b3 I12 - b2 I13 ) 1 +(b3 I22 - b2 I23 ) 2 - [(b3 I13 + b2 I12 )1 +(b3 I23 + b2 I22 )2 ]1 + b2 (I11 1 + I12 2 )2 = 0. It follows from Eq. (5.6) that in a state of equilibrium we have either 2 = 0 or I13 1 + I23 2 = 0. Let us start by considering the first alternative. In this case, Eq. (5.7) yields
2 (b3 I13 + b2 I12 )1 = 0.

(5.6) (5.7)

(5.8)

Therefore, if (b3 I13 + b2 I12 ) = 0, then the reduced system does not admit nontrivial equilibrium points. On the contrary, if (b3 I13 + b2 I12 ) = 0,
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then the entire axis 2 = 0 consists of equilibrium points. In this case, the reduced system (5.6)­ (5.7) becomes linear after rescaling time as d = 2 dt. This result is predicted by Theorem 4. What does the phase flow of the corresponding linear system look like? We answer this question in a particular case where I12 = I13 = 0. The linear system has the following form: I11 1 + I23 2 = 0, (b3 I22 - b2 I23 )2 +(b2 I11 - b2 I22 - b3 I23 )1 = 0. It admits the quadratic integral
2 2 I11 [b2 (I11 - I22 ) - b3 I23 ]1 - I23 (b3 I22 - b2 I23 )2 = 0.

(5.10)

(5.11)

The phase tra jectories of the system (5.10), like those of the original nonlinear system, consist of level lines of this function. Let b3 I23 = 0. Then one can always find the intervals of variation of the relation b2 /b3 for which the quadratic form (5.11) is (positive or negative) definite, as well as whole intervals when the form (5.11) is neutral. In the former case, the phase flow is similar to that of the nonholonomic Suslov problem, while in the latter case it has the form shown in Fig. 2. We now consider the second alternative, when the equality (5.8) is satisfied. Then, assuming that b2 = 0, it follows from Eq. (5.7) that
2 2 I12 1 +(I22 - I11 )1 2 - I12 2 = 0.

(5.12)

The general solution of Eq. (5.8) has the form 1 = kI23 , 2 = -kI13 ; k R. Substituting it into Eq. (5.12) with
2 2 I12 (I23 - I13 ) - (I22 - I11 )I13 I23 = 0

(5.13)

yields only the trivial equilibrium (since k = 0). The above can be summarized as follows: · if b2 = 0 and inequalities (5.9) and (5.13) are satisfied, the reduced system does not admit nontrivial equilibrium points. In particular, under these conditions the trivial equilibrium 1 = 2 = 0 of the system (5.6)­ (5.7) is unstable. The general results of Sections 3 and 4 can be applied to the investigation of this system. In particular, we note that if we have an equality instead of inequality (5.13), then the system of nonlinear equations (5.6)­(5.7) admits a straight line of equilibria and, by Theorem 4, reduces to a linear system after rescaling time. In conclusion of this Section we mention briefly the conditions for existence of an integral invariant (an invariant measure) with a smooth positive density in the reduced system (5.6)­ (5.7). According to [19], such an integral invariant exists if and only if the phase flow of the system preserves the standard Lebesgue measure on the plane. In other words, if we represent the system (5.6)­(5.7) in the standard form 1 = v1 (1 ,2 ), 2 = v2 (1 ,2 ), this condition reduces to the zero-divergence condition: v2 v1 + = 0. 1 2 (5.14)

Since the functions v1 and v2 are quadratic in 1 and 2 , the left-hand side of Eq. (5.14) is a linear homogeneous function of 1 and 2 . Consequently, Eq. (5.14) ultimately reduces to the condition that the coefficients of 1 and 2 are equal to zero.
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Omitting straightforward calculations, we give the final result. The reduced system (5.6)­(5.7) admits an integral invariant with a smooth positive density only if the following two conditions are satisfied: 2 2 2 (5.15) b2 (2I12 - I11 I12 + I11 + I13 )+ b3 (I12 I13 - I11 I23 ) = 0, b2 (I12 I22 + I13 I23 )+ b3 (I22 I13 - I23 I12 ) = 0. (5.16) Thus, systems with an integral invariant form a subset of codimension 2 in the space of all reduced systems. In the particular case when I12 = I13 = 0 (the plane spanned by the vectors a and b is an invariant plane of the tensor of inertia of the rigid body), the condition (5.16) is satisfied automatically, while the condition (5.15) simplifies to b2 (I22 - I11 )+ b3 I23 = 0. Under the conditions (5.15) and (5.16) the reduced system admits a first integral in the form of a nonzero homogeneous polynomial of degree 3. Indeed, according to Euler, the differential 1-form v2 d1 - v1 d2 (5.17) is a total differential of the first integral dH of the original system. After the successive integrations of (5.17) we obtain the homogeneous third-degree integral H . Since v1 = -H / 2 and v2 = H / 1 , the reduced system takes the Hamiltonian form H H , 2 = . 2 1 Remark. The dynamical systems with nontrivial equilibria whose phase portraits are shown in Figs. 1b, 2 and 4 do not admit an integral invariant with a smooth positive density because all their tra jectories are asymptotic (as t + or t - ). Moreover, they do not even admit an invariant measure that is absolutely continuous with respect to the usual Lebesgue measure on a plane. The reduced system in the nonholonomic Suslov problem (see the discussion in [16]) also has this property. However, one should not think that all systems with quadratic right-hand sides (which do not vanish identically) that admit nontrivial equilibrium points possess this property. Here is a simple counterexample: x1 = x2 , x2 = 0. We also give an example of a system with a single 2 trivial equilibrium whose phase flow preserves the usual Lebesgue measure: x1 = x2 , x2 = x2 . This 2 1 system is Hamiltonian with the Hamiltonian function 1 = - x3 - x3 1 2 . 3 The set {H = 0} is obviously invariant. It reduces to the single straight line x1 = x2 whose existence is guaranteed by Theorem 1. H= 6. THE CHAPLYGIN SLEIGH AS A SYSTEM WITH A SERVOCONSTRAINT The Chaplygin sleigh is a rigid body moving on a horizontal plane with a nonintegrable constraint: the velocity of a certain point of the body (denoted by O) is always orthogonal to the body-fixed horizontal axis l. The classical nonholonomic version of this problem is well-studied (for instance, see [15, 20, 21]). We consider the dynamics of the Chaplygin sleigh assuming that the above-mentioned nonintegrable constraint is realized by controlled forces (according to B´ eghin). For brevity, we call such a system a servosleigh. The position of a rigid body on a plane is defined by three parameters: the Cartesian coordinates x and y of the specified point O and the rotation angle . We introduce a moving reference frame with the origin at the point O and one of the axes coinciding with the specified axis l. Let u and v be the pro jections of the velocity of the point O on these axes, and let be the angular velocity. The equation of nonintegrable constraint has the form v = 0. (6.1) We consider the simplest case when the center of mass of the rigid body coincides with the point O. The nonholonomic dynamics of such a system is trivial: the point O moves in a circle or
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in a straight line (a circle of infinite radius) with a constant velocity. The dynamics of the sleigh with a servoconstraint are more interesting. The configuration space of the servosleigh is a group of motions of the plane E (2). The kinetic energy equals to I 2 m2 (u + v 2 )+ , (6.2) 2 2 where m is the mass of the body and I is the moment of inertia with respect to the vertical straight line passing through the point O. Both the kinetic energy (6.2) and the constraint (6.1) are obviously left-invariant. Therefore, we can apply the general approach described in Section 3. T= Remark. Because of the well-known servosleigh problem can be considered body with a constraint from Section 5 perform an independent analysis of this of the sleigh on a plane. retraction of the group SO(3) to the group E (2), the as the limiting case of the problem of rotation of a rigid (these matters are discussed, for instance, in [15]). But we problem, adding a geometrical consideration of the motion

Eqs. (2.4) with the constraint (6.1) have the following form: mu = ma, 0 = -mu + b, I = c. Here, a and b are the "guiding" forces directed along the moving axes, and c is the extra moment of forces relative to a vertical axis passing through the point O; a, b, c = const. If a = c = 0 and b = 0, we have the classical nonholonomic motion. The condition for realization of the servoconstraint in the general case also reduces to the inequality b = 0. The reduced system is described by very simple differential equations: mc a (6.3) u = u , = u ; = , = b Ib The coordinate axes u = 0 and = 0 consist entirely of equilibrium points. The system (6.3) has the linear integral u - = = const. (6.4) We discuss the general case when both constants and are nonzero. The phase portrait of the system (6.3) is shown in Fig. 5. The location of the invariant straight lines (6.4) and the direction of motion along the tra jectories in Fig. 5 correspond to the case where > 0 and > 0. The phase portrait in Fig. 5 is a special case of the more general situation depicted in Fig. 4. Equations (6.3) can easily be explicitly integrated using the linear first integral (6.4). Eliminating the variable from the first equation, we obtain the equation for finding the linear velocity of the point O: u = u(u - ). Assuming that the constants and are positive, we find a solution with a tra jectory in the form of the interval 0 < u < / : . (6.5) u(t) = + et

Fig. 5.

This solution is doubly asymptotic: as t - and as t + , it tends, respectively, to / and 0. Solutions of the form (6.5) correspond to the tra jectories of (6.3) located in the fourth quadrant in Fig. 5 (where u > 0 and < 0). The angular velocity also decreases monotonically from 0 to -/. This simple analysis of the reduced system allows us to get an explicit representation of the sliding of the servosleigh on a horizontal plane. We continue to assume that all the constants , and are positive. The Cartesian coordinates x and y of the specified point O, as well as the angle of rotation of the sleigh, , can be found from the following kinematic relations x = u cos , y = u sin , = .
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We assume that the angle is measured from the fixed axis x in the positive direction. This is the angle between the axis x and the axis of the moving frame that does not coincide with the specified axis l. Let us find the curvature of the tra jectory of the point O: et ( + et ) 1 u - 1 |xy - xy | ¨ ¨ = . = u u ( 2 + y 2 )3/2 x It is important to note that the curvature does not depend on the angle of rotation of the sleigh. The curvature is always positive and exponentially rapidly tends to zero as t - ; conversely, as t + , it increases exponentially. The tra jectory of the point O is shown in Fig. 6. Its length is infinite from any point to the left and, conversely, finite from any point to the right. This is a simple consequence of the arc length formula
t2 t2

x2
t1

+ y dt =
t1

2

u(t) dt

and the formula (6.5). Thus, in the "distant past" the body moves in a straight line practically without rotating, then the body begins to rotate with an increasing angular velocity, and the tra jectory of the point O begins to curl in a spiral. At infinitely large positive values of time the point O becomes fixed, and the body eventually goes into rotation with a constant velocity about a fixed vertical axis passing through the point O. Of course, the point Fig. 6. O can move in the "curl" shown in Fig. 6 in the opposite direction. In this case, the sleigh undergoes an asymptotic transition from the rotation about a vertical axis to the translational motion in a straight line. Such motions correspond to the initial data from the second quadrant of the phase plane of the reduced system. In general, the curls are mapped to each other by similarity transformations and mirror symmetry. This is a simple consequence of the homogeneity of the system (6.3). In the case of a mirror reflection transformation, the sleigh rotates in the opposite direction. Motions going to infinity (or coming from infinity) in a finite time are of less interest, and their analysis is not presented here. 7. SERVOSLEIGH ON AN INCLINED PLANE Let us complicate the problem and consider the dynamics of a servosleigh on an inclined plane. In contrast to Section 6, the rigid body is now acted upon by the force of gravity. The equations of motion become more complex and the dynamical equations no longer decouple from the kinematic relations (as they did in Section 3). In the notation of Section 6, the equations of motion of the servosleigh have the following form: mu = a - p sin , 0 = -mu + - p cos , I = c. (7.1) Here we have already taken into account the constraint equations v = 0 and, in addition, set b = 1 (this can always be done assuming that the constraint v = 0 is realizable). The angle of rotation is measured from the horizontal straight line on the inclined plane; p > 0 is the weight of the body (more precisely, the product of the body weight by the sine of the angle between the inclined and horizontal planes). The second equation of (7.1) is used to find the multiplier . Substituting the formula for into the first and third equations of (7.1) yields mu = mau + ap cos - p sin , I = mcu + cp cos . Adding the kinematic relation = ,
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we obtain a closed system of autonomous differential equations in three-dimensional space, which is 2 -periodic in the angle variable . If a = c = 0, we have an integrable (and well-studied) nonholonomic system; for almost all initial conditions the nonholonomic sleigh does not descend along the inclined plane, but moves, on average, horizontally. For c = 0, the system (7.2)­(7.3) is also easily integrable by quadratures. However, for c = 0 the system (7.2)­(7.3) is hardly an integrable one. This motivates qualitative analysis of the system. The first step in this direction is to find particular solutions and study their stability. If c = 0, a family of particular solutions exists when = 0 = const: the sleigh descends along an inclined straight line with a constant acceleration (if a cos 0 = sin 0 , the sleigh moves with a constant velocity). However, all these solutions are unstable. In the general case, with c = 0, there is only one such motion when = ±/2: the sleigh moves along the straight line of quickest descent. We investigate the stability of this motion, assuming for definiteness = /2 (the case = -/2 is no different from this one). Theorem 6. If c > 0, the motion of the servosleigh along a vertical line (when = /2) is stable with respect to = /2 - and . When c 0, the motion is unstable. Let us discuss the physical meaning of the condition of the theorem. If = , then u = -p/m. Consequently, at large values of time the linear velocity of the 2 sleigh is negative. Let the sleigh move by inertia; then the second term on the right-hand side of the second equation of (7.2) needs to be dropped. If u(t) < 0, the positive control torque (c > 0) stabilizes the straight-line motion of the sleigh. To prove Theorem 6, we need Lemma 1. Let t u(t) be a solution of the equation u = (t)u + (t) with the initial condition u(0) 0. Then u(t) < 0 in the time interval (0, ) where (t) < 0. Indeed, the function u(·) (as a solution of (7.4)) is expressed by the formula t s t (s) exp - ( ) d ds . u(0) + exp (s)ds
0 0 0

(7.4)

These values are negative as long as the values of (·) are negative. Theorem 6 is proved by continuous induction on time. First, note that the first equation of (7.2) has the form (7.4) where = aw, = g (a cos - sin ), where g = p/m is the g on and , which in the second equation of R = { } in a potential (7.5) ravitational acceleration; and are complex functions of time (they depend turn are functions of time as solutions of the system). On the other hand, (7.2) can be expressed as an equation of particle motion in the straight line field and a resisting medium: V = 0, V = cp(1 - cos ). I - (mcu) + Since c > 0, the potential V has a strict local minimum at the point = 0. If from some instant of time u(t) < 0, then the equilibrium = 0, = 0 is stable by virtue of the well-known inequality d I 2 + V ( ) = (mcu) 2 dt 2 0. (7.6)

If the sleigh moves along the vertical straight line ( = - = 0), then it is obvious that at 2 some instant the velocity u becomes negative (the sleigh moves down). By continuity, this property holds for small initial perturbations of and .
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According to (7.5), the function is negative for small values of . Thus, if (t) is small, then (by the lemma) u(t) < 0 from some instant t. But if u(t) is negative, the values of (and ) remain small according to inequality (7.6). Using induction on time, we deduce the result of the theorem. When c > 0, the asymptotic stability apparently holds with respect to and . At least this is true of the linearized equation that has the form pc ¨ > 0. (7.7) + t + = 0, = I To prove the asymptotic stability of the equilibrium = = 0 of the linear system, one can use the series expansion of solutions of Eq. (7.7). We represent (7.7) as the system x(At + B )x, x = (, )T , 00 01 , B = . A= 0 - - 0

(7.8)

Since the eigenvalues of the matrix A (the numbers 0 and -) are different, solutions of the system (7.8) can be sought (as shown in [22]) in the form of the following series: x1 x2 + 2 + ... exp(t2 + t). (7.9) x(t) = t x0 + t t Here x0 , x1 , ... are the vectors from R2 , 2 is the eigenvalue of the matrix A, and are to be found. Since > 0, the eigenvalue of 2 = - corresponds to the solution exponentially rapidly tends to zero. The zero eigenvalue of the matrix corresponds to a solution of (7.8) in the form of an inverse power series of t ( and the constants (7.9), which superA (the case = 0) = -1, = 0):

x 1 (x0 + 1 + ...), xj R2 . t t It also tends to zero as t +. Thus, the equilibrium of the linear system (7.7) is, indeed, asymptotically stable. 8. ROTATION OF A RIGID BODY WITH A LEFT-INVARIANT SERVOCONSTRAINT IN A GRAVITY FIELD The rotation of a rigid body about a fixed point with a left-invariant linear servoconstraint in an axisymmetric force field with potential V ( ) is described by the following system of differential equations generalizing the system (5.1): I + â I = â V + b, + â = 0, (a, ) = 0. (8.1)

Here, is the vertical unit vector in the moving reference system attached to the rigid body. If the body rotates in a gravity field, then V ( ) = (c, ), (8.2) where the constant vector c is the product of the weight of the body by the radius vector of its center of mass in the moving reference frame. Without loss of generality, we can assume that the left-invariant constraint has the form 3 = 0. Prop osition 2. If the conditions (5.15) and (5.16) are satisfied, then the phase flow of the system (8.1) preserves the standard measure in R6 = R3 { }â R3 { }. The proof is by an side of the system o In the absence of a potential force field immediate verification of the following fact: the divergence of the right-hand f differential equations (8.1) is zero if the conditions (5.15)­(5.16) are satisfied. force field, this is established in Section 5. It turns out that the addition of a does not affect the magnitude of divergence.
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In the nonholonomic model (when b2 = 0, while b3 = 0) the conditions (5.15) and (5.16) are equivalent to I13 = I23 = 0. This means precisely that the vector a is an eigenvector of the inertia operator. The measure preservation result in this case is noted in [16]. Let us now consider some aspects of the problem of rotation of a rigid body in a uniform force field with potential (8.2). Theorem 7. Let Ia = a ( R) and suppose that the vector c is orthogonal to the vectors a and b. Then Eqs. (8.1) admit the integral linear in the velocity F = (I , c). (8.3) In the nonholonomic model the vectors a and b are collinear. In this case we have one condition (a, c) = 0; the integral (8.3) was obtained by E. I. Kharlamova in [30]. We emphasize that if a is an eigenvector of the inertia operator, then in the general case (when b2 = 0) the equations of motion do not admit an invariant measure with smooth positive density. Under the conditions of Theorem 7, the nonholonomic equations are integrable by quadratures, because in addition to the invariant measure and the integral (8.3) the equations of motion admit three first integrals 1 (I , )+(c, ), (a, ), (, ). (8.4) 2 The integrability property follows from the classical Euler ­ Jacobi theorem on the integrating factor. For servoconstraints of the general type the integrals are (8.3) and the last two functions of (8.4), which is clearly not sufficient for the integrability of the equations of rotation of a rigid body. The linear integral (8.3) has the form of the Noether integral: F= T ,c , 1 T = (I , ). 2

We show that it really can be derived from a generalized (in comparison to the definition provided in the introduction) Noether theorem. To do this, we introduce the vector field w=c (8.5) and show that it is a generalized symmetry field. It is a left-invariant vector field on the group SO(3). We recall that the flow of any right-invariant vector field conserves the left-invariant kinetic energy. In the general case, the flows of left-invariant fields do not possess this property. Since (b, c) = 0, the field (8.5) is a field of virtual displacements; it satisfies the equation (b, x) = 0. Further, the flow of the vector field (8.5) conserves the potential energy. Indeed, the field w = c generates the following system of differential equations in R3 = { }: = â c. Then, by virtue of this system, the total derivative of the potential (8.2) is (V , ) = (c, â c) = 0. It remains to show that the flow of the vector field w lifted to the phase space conserves the kinetic energy (on the constraint hyperplane). The total derivative of the kinetic energy (the function on the algebra so(3)) along the lifted field w is T , = T ,c â = (c, â I ). (8.6)

This quadratic form in is, of course, not identically zero (if the inertia operator is not proportional to the unit operator). However, it equals zero on the plane (a, ) = 0 if the conditions of Theorem 7 are satisfied (this is just where the slight generalization of the Noether theorem is). Indeed, the vector is orthogonal to the vector a (according to the constraint equation). Further, (I , a) = (, I a) = (, a) = 0. Therefore, the vector â I is collinear with the vector a. But then the right-hand side of Eq. (8.6) equals zero, since (c, a) = 0 by the condition of the theorem.
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9. SOME PROBLEMS In conclusion, we present a list of problems that arise in connection with the theory of motion of systems with B´ eghin's servoconstraints. 1 . Find the conditions for conservation of the phase volume of the reduced system (2.4) in the case when the group G is unimodular (i.e., when ck = 0 for all 1 i n; in particular, ik all compact Lie groups are unimodular). To do so requires finding the divergence of the uniform vector field. For the classical Euler ­ Poincar´ equations, this divergence vanishes if the group G is e unimodular [16]. 2 . It would be interesting to find multidimensional generalizations of Theorems 4 and 5. Here is one of the cases: if the system (3.1) has an entire plane of equilibria ai xi = 0, then upon rescaling time as d = ( ai xi )dt

it reduces to a linear system. Multidimensional analogues of Theorem 5 are of greater interest. 3 . Provide a classification of the phase portraits of dynamical systems on a plane with homogeneous quadratic right-hand sides (taking into account an analysis of the drift of tra jectories to infinity). 4 . In the framework of this classification, specify all types of phase portraits in the problem of rotation of a top with a left-invariant servoconstraint. The first step in this direction is to find conditions under which the corresponding reduced system on a plane has exactly one or three invariant straight lines. 5 . Study the motion of a servosleigh on a horizontal plane in the general unbalanced case when the center of mass of a rigid body does not lie on a vertical straight line passing through the point of contact. 6 . Consider the motion of a servosleigh along an inclined plane and find out the general conditions under which the sleigh does not descend, but moves, on average, horizontally. 7 . Investigate the integrability of Eqs. (8.1) with potential (8.2), which describe the rotation of a heavy rigid body with a Suslov servoconstraint. ACKNOWLEDGMENTS The author thanks A. V. Borisov and I. S. Mamaev for fruitful discussions. The study was financed by the grant from the Russian Science Foundation (Pro ject No. 14-5000005). REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. ´ B´ eghin, M. H., Etude th´ eorique des compas gyrostatiques Anschutz et Sperry, Paris: Impr. nationale, ¨ 1921; see also: Annales hydrographiques, 1921. Appel, P., Trait´ de M´ anique rationnel le: Vol. 2. Dynamique des syst` e ec emes. M´ anique analytique, 6th ec ed., Paris: Gauthier-Villars, 1953. Kozlov, V. V., Principles of Dynamics and Servo constraints, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1989, no. 5, pp. 59­66 (Russian). Borisov, A. V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443­490. Borisov, A. V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Physics­Uspekhi, 2003, vol. 46, no. 4, pp. 393­403; see also: Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 407­418. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in the Neighborho od of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 443­464. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 465­483. Borisov, A. V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26­36; see also: Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33­45. Borisov, A. V., Kazakov, A. O., and Sataev, I. R., The Reversal and Chaotic Attractor in the Nonholonomic Mo del of Chaplygin's Top, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 718­733.
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REGULAR AND CHAOTIC DYNAMICS

Vol. 20

No. 3

2015