Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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276 11 Newtonian Mechanics

is the only one satisfying the conditions m(0) = m

,˙m(0) = C and such

that for every t ∈ I the vector

˙m(t) is parallel to v(t) along m(·). To see

this, let for some t ∈ I the curve ¯m(·)inT

m(t)

M be obtained by the parallel

translation of ˙m to the point m(t). Then, by Theorem 2.32 we have

dτ

¯m(t + τ)

|τ=0

˙m(t).

It is clear that the vector



v(τ)dτ + C ∈ T

is parallel to ¯m(t) along m(·).

In other words, the vector

˙m(t) is parallel to v(t) along m(·). Let us set

v(t)=Γ ¯α



t, m(t)˙m(t)



.Then(11.19) means that the vector

˙m(t)canbe

obtained by transporting ¯α (t, m(t), ˙m(t)) ∈ T

m(t)

M along m(·)ﬁrsttothe

point m

= m(0) and then back to m(t). The theorem follows. 

Let m(t), t ∈ I, be a trajectory of the mechanical system, i.e., a solution

of (11.2).

Deﬁnition 11.52. The velocity hodograph of the trajectory m(t) is the curve

v : I → T

m(0)

M such that v(t) is parallel to ˙m(t) along m(·).

It is not hard to see that the velocity hodograph of a solution of (11.19)

satisﬁes the equation

v(t)=



Γ ¯α



τ,Sv(τ),

dτ

Sv(τ )



dτ + C, (11.20)

It is clear that if v is a solution of (11.20), then Sv is a solution of (11.19),

i.e., by Theorem 11.51, a trajectory of the mechanical system.

Remark 11.53. The notion of a velocity hodograph in the sense of Deﬁni-

tion 11.52 was introduced by Synge in [217] where analogs of the standard

properties of the hodograph were proved for some mechanical systems (see

also [218]). The hodograph equation (11.20) originally appeared in [88].

Remark 11.54. If we have a mechanical system on a group, then it is natural

to pick the initial condition m(0) = e.Thus,(11.20) becomes an equation in

the Lie algebra similar to the Euler equation in the body or space coordinates.

All three equations are equivalent to Newton’s equation on the conﬁguration

space. However, for an arbitrary conﬁguration space, (11.20) is the only one

among these three that makes sense.

Let us denote the operator which sends v ∈ C

(I,T

M)to



Γ ¯α



τ,Sv(τ),

dτ

Sv(τ )



dτ + C ∈ C

(I,T



Γ ◦ ¯α ◦S

11.8 Integral Equations of Geometric Mechanics. The Velocity Hodograph 277

Theorem 11.55 The operator



Γ ◦ ¯α ◦S

: C

(I,T

M) → C

(I,T

is completely continuous.

Proof. Since S,¯α and Γ are continuous, so is the operator. Let U

be the ball

in C

(I,T

M) of radius K centered at the origin. Because ¯α is continuous,

Theorem 3.46 and Lemma 3.53 imply that





Γ ◦ ¯α ◦S



) is compact.



11.8.2 An integral formalism of geometric mechanics

with constraints

Let the conﬁguration space M be a complete Riemannian manifold equipped

with a constraint β, which may be non-holonomic (Section 11.6). To develop

an adequate integral formalism, we use parallel translation of admissible vec-

tors along admissible curves. Such a parallel translation arises from the re-

duced connection

Let m(t), t ∈ I, be an admissible C

-curve and X(t, m) an admissible

vector ﬁeld on M. Denote by Γ



t, m(t)



the curve in β

m(0)

such that the

vector X (t, m(t)) at m(0) is parallel to Γ

X (t, m(t)) along m(·) under the

reduced connection. The properties of the operator Γ

are similar to those

of Γ studied in Section 3.2.2.

As in Section 11.6.2 consider the orthonormal frame bundle ¯π : O

(M) →

M of β (i.e., b ∈ O

is an orthonormal frame in β

Consider the map

E : O

×R

→

H, k =dimβ

, deﬁned by the formula

(X)=Tπ

−1

(bX) |

,whereb ∈ O

(M) is regarded as an orthogonal

operator from R

to β

(cf. Deﬁnition 2.68 of basic vector ﬁelds in the non-

holonomic case). It is easy to see that

E is smooth and ﬁber-wise linear.

Let v(t) be a continuous curve in β

.Fixb

∈ Oβ

(M) and consider the

time-dependent vector ﬁeld



−1

v(t)



on O

(M). By deﬁnition, this vector

ﬁeld is smooth in b for a ﬁxed t and continuous in t for a ﬁxed b ∈ O

(M).

Hence, for this vector ﬁeld, the Cauchy problem has a solution. As above, it

is easy to show that the integral curves of



−1

v(t)



extend to the entire

interval I =[0,l]. Consider the integral curve b

(t) beginning at b

and its

projection S

v(t)=¯πb

(t). It is clear that S

v(·) is an admissible curve and,

in addition, for every t ∈ I the vector

v(t) is parallel to v(t) along Sβ

(·)

with respect to the reduced connection

The following result can be proved in the same way as Theorem 3.56.

Theorem 11.56 Let X(t, m) be an admissible vector ﬁeld which is jointly

continuous in all its variables. An admissible curve m(t) is an integral curve

278 11 Newtonian Mechanics

of X(t, m) if and only if it satisﬁes the equation

m(t)=S

◦ Γ

X (t, m(t)) . (11.21)

Theorem 11.57 A continuous curve v(t) ⊂ β

is a solution of the equation

v(t)=Γ



t, S

v(t)



if and only if S

v(t) satisﬁes (11.21).

The operators S

, Γ

and their compositions have the same compactness

and continuity properties as the integral operators from Section 3.2.

Now consider the integral equation

m(t)=S





P ¯α (τ,m(τ ), ˙m(τ)) dτ + C



, (11.22)

where C is a vector in β

. Taking into account the relationship between

parallel translations and covariant derivatives, we get the following result.

Theorem 11.58 An admissible curve with the initial conditions m(0) = m

and ˙m(0) = C satisﬁes (11.13) (i.e., m(·) is a trajectory of the system with

the constraint β) if and only if it is a solution of (11.22).

It is clear that the equation of the velocity hodograph of a solution of

(11.22)is

v(t)=



P ¯α



τ,S

v(τ),

dτ

v(τ)



dτ + C (11.23)

on the space of continuous curves in β

Remark 11.59. We emphasize that even if M is the Euclidean space R

the integral operators considered in this subsection (e.g., S

, S

◦ Γ

,etc.)

cannot be reduced to their classical analogs (antiderivatives, the Urysohn-

Volterra operator, etc.) unless the distribution β is trivial, i.e., all the spaces

are parallel (in the Euclidean sense) to β

⊂ T

= R

11.9 Mechanical Interpretation of Parallel Translation

and Systems with Delayed Control Forces

In this section, following [91, 92, 93, 94], we study a certain type of diﬀerential

equation with delay on Riemannian manifolds. In these equations one of

the terms on the right-hand side is obtained by parallel translation to the

corresponding point along a solution. The equations are analogous to those

diﬀerential equations on Euclidean space with discrete delay or where the

right-hand side depends only on time. Our analysis of the equations in terms

of geometric mechanics is based on the mechanical interpretation of parallel

translation.

11.9 Mechanical interpretation of parallel translation 279

The mechanical interpretation of Riemannian parallel translation was dis-

covered by Johann Radon and described by Blaschke in [159]. A similar idea

was independently used by Synge [217] in order to deﬁne the hodograph for

a geometric mechanical system. (See Remark 11.53.)

Radon proved that for a pendulum moving in the conﬁguration space of a

mechanical system, the direction of oscillation is a parallel translation along

its trajectory. In other words, the coordinate system attached to a gyroscope

(e.g., the stationary system for a ﬂat conﬁguration space) is parallel along a

trajectory.

Consider the motion of a mechanical system with a force ﬁeld ¯α and a

“control” force Φ. The latter depends on time, the velocity and on the co-

ordinates of the point. Faithfully modeling the delays that occur in real life

systems, Φ acts after time h. The equation of motion of such a system is as

follows:

˙m(t)=¯α (t, m(t), ˙m(t)) +



Φ (t − h, m(t − h), ˙m(t −h)) , (11.24)

where



denotes the Riemannian parallel translation along the solution.

Similarly, one may consider the evolution of a system with a velocity ﬁeld

V and delayed control “velocity” W , i.e., a system given by the equation

˙m(t)=V (t, m(t)) +



W (t − h, m(t − h)) . (11.25)

If M is a Euclidean space, (11.24)and(11.25) are quite simple diﬀerential

equations with discrete constant delay [4]. However, if M is not ﬂat, (11.24)

and (11.25) have much more complex properties.

First, since the parallel translation is deﬁned along C

-curves and depends

on a curve and its derivative, (11.25) is an equation of neutral type [4].

Secondly, the ﬁrst order equation corresponding to (11.24) has distributed

delay. The reason is that if M is not ﬂat, the parallel translations of a vertical

vector in TM do not necessarily coincide with the lift of the parallel trans-

lation in M. (Here the manifold TM is equipped with the standard metric

arising from the metric on M.)

Finally, note that the ﬁrst order equation, which is equivalent to (11.24)

and thus, as we have just explained, has distributed delay, is again an equation

of neutral type on TM because the right hand side is neither continuous in

the C

-topology nor deﬁned on arbitrary curves.

Let φ:[−h, 0] → M be a C

-curve.

Deﬁnition 11.60. A continuous curve m(·): [−h, ε) → M, ε>0isaso-

lution of (11.24) (respectively, (11.25)) on the interval [−h, ε) with initial

condition φ if m(·)isC

-smooth on (0,ε), coincides with φ on [−h, 0], and

satisﬁes (11.24) (respectively, (11.25)) on [ 0,ε).

It is useful to ﬁrst analyze the particular cases of (11.24)and(11.25)where

the control force depends on time only. These mechanical systems are given

280 11 Newtonian Mechanics

by the equations

˙m(t)=¯α (t, m(t), ˙m(t)) +



Φ(t) (11.26)

˙m(t)=V (t, m(t)) +



W (t) (11.27)

In this case Φ and W take values in the tangent space to M at the initial

point m

It is essential that (11.24)and(11.25) can be reduced to (11.26)

and (11.27), respectively. To see this, specify an initial condition φ ∈

([−h, 0],M) and consider the following T

φ(0)

M-valued functions of θ ∈

[−h, 0]: Φ(θ)=Φ(θ + h, φ(θ + h),

φ(θ + h)) and W (θ)=W (θ + h, φ(θ + h)).

Let t = θ + h. It is clear that the solutions of (11.26)and(11.27) with control

forces Φ(t)andW (t) coincide with those of (11.24)and(11.25), respectively.

Note that the equations above make sense only if their solutions are C

smooth, for the parallel translation is not deﬁned otherwise.

Theorem 11.61 Let V (t, m) be a continuous vector ﬁeld on M and W (t) a

continuous curve in T

M.Then:

(i) equation (11.27) has a local C

-solution;

(ii) the solution is unique provided that for any t ∈ I the ﬁeld V (t, m) is

locally Lipschitz continuous in m.

Proof. Let O(M) be the orthonormal frame bundle over M and H the

Levi-Civit´a connection on O(M). The tangent map of the natural projec-

tion π : O(M) → M induces the isomorphism Tπ: H

→ T

πb

M at ev-

ery point b ∈ O(M). Hence, at every b ∈ O(M), we obtain the vector

Tπ

−1

V (t, πb) ∈ H

⊂ T

O(M). These vectors form a horizontal vector ﬁeld

on O(M) (i.e., belonging to H).

Let O be an orthonormal frame in T

M.TheframeO gives rise to the

isomorphism O: R

→ T

M and, therefore, we have a horizontal time-

dependent basic vector ﬁeld E(O

−1

W (t)) on O(M) (see Deﬁnition 2.68).

Consider the vector ﬁeld

V (t, b)=Tπ

−1

V (t, πb)+E(O

−1

W (t)) on O(M).

By the existence theorem for ordinary diﬀerential equations, this vector ﬁeld

has an integral curve γ

∗

, γ

∗

(0) = O, deﬁned on the interval [ 0,). Since

the vector ﬁeld

V is horizontal, the frame γ

∗

(t) is parallel to O along γ =

πγ

∗

for every t ∈ [0,ε). By deﬁnition, we have ˙γ

∗

(t)=Tπ

−1

[V (t, γ(t)) +

∗

(t)(O

−1

W (t))], where the last term is the vector with coordinates O

−1

W (t)

in the basis γ

∗

(t). Hence, ˙γ(t)=V (t, γ(t)) + γ

∗

(t)O

−1

(W (t)). Furthermore,

by the deﬁnition of parallel translation, the last term is parallel to W (t) along

γ(·). This means that γ(t) is a solution of (11.27).

Note that the ﬁeld E



−1

W (t)



is smooth in b for a ﬁxed t.IfV (t, m)is

locally Lipschitz in m (for every ﬁxed t), then

V is locally Lipschitz in b and

the resulting solution is unique. 

11.9 Mechanical interpretation of parallel translation 281

Corollary 11.62 Assume that the ﬁelds V (t, m) and W(t, m) are jointly

continuous in all variables. Then for any C

-initial condition φ:[−h, 0] →

M, there exists a solution of (11.25). The solution is unique if V is locally

Lipschitz in m for any ﬁxed t.

Theorem 11.63 Assume that the ﬁeld ¯α satisﬁes the Carath´eodory condi-

tion (see, e.g., [74])andΦ(t) is an integrable function with values in T

Then for any initial condition C ∈ T

(i) equation (11.26) has a local C

-smooth solution;

(ii) the solution is unique if ¯α is locally Lipschitz continuous for all t.

Proof. Consider the direct product O(M) × R

, n =dimM, equipped with

the right action of O(n) given by the formula (b, x)g =(bg, g

−1

x), where

b ∈ O(M), x ∈ R

and g ∈ O(n). The quotient space of O(M) × R

under

the right action can be naturally identiﬁed with TM. We denote the natural

projection O(M ) ×R

→ TM by λ (see Section 2.7 and Notation 1.36).

Let (b, x) ∈ O(M) ×R

. It is easy to see that Tλ induces an isomorphism

of H

∈ T

O(M) with the horizontal space in T

λ(b,x)

TM and an isomorphism

of V

= T

with the vertical space V

λ(b,x)

. (Recall that the latter is the

tangent space to T

πb

M,whereπ : O(M) → M is the natural projection.)

Pick an orthonormal basis O in T

M and deﬁne the function O

−1

Φ(t)

with values in R

to be the coordinates of Φ(t) with respect to the basis O.

As in the proof of Theorem 11.61, the function gives rise to the horizontal

vector ﬁeld E(O

−1

Φ(t)) on O(M). We see that any basis b ∈ O(M) gives

rise to a vector TπE

−1

Φ(t)) in the tangent space T

πb

Consider the vector ﬁelds A, B,andC on O(M ) × R

such that for any

(b, x) ∈ O(M) ×R

we have A

(b,x)

= Tλ

−1

λ(b,x)

∈ H

,whereZ is the spray

of the Levi-Civit´a connection on M and

(b,x)

(t)=Tλ

−1

(¯α (t, ¯πλ(b, x),λ(b, x))) ∈ V

(b,x)

(t)=Tλ

−1



TπE



−1

Φ(t)



∈ V

By deﬁnition, A is a smooth ﬁeld. Since ¯α satisﬁes the Carath´eodory con-

dition,sodoesB(t). By the hypothesis of the Theorem, the ﬁeld C(t)is

smooth on O(M) × R

for every ﬁxed t and is measurable in t for any ﬁxed

(b, x). Therefore, the ﬁeld A + B(t)+C(t) satisﬁes the hypothesis of the clas-

sical theorem which guarantees the existence of a local solution of the Cauchy

problem. Furthermore, if ¯α is locally Lipschitz in t, then the hypothesis of the

uniqueness theorem is also satisﬁed (see [62], Theorems 1 and 2 of Section 1).

Note that local solutions are, by construction, absolutely continuous curves.

Let (b(t),x(t)) be the local solution with initial condition (O, O

−1

C). The

curve λ(b(t),x(t)) is absolutely continuous in TM and, hence, the tangent

vector

Y (t)=Tλ(A + B(t)+C(t)) = Z

λ(b(t),x(t))

+ Tλ(B(t)+C(t))

λ(b(t),x(t))

282 11 Newtonian Mechanics

exists for almost all t. The vector Z

λ(b(t),x(t))

belongs to the connection and

both vectors Tλ(B(t))

λ(b(t),x(t))

and Tλ(C(t))

λ(b(t),x(t))

are in the vertical

subspace. Hence, TπY(t)=TπZ. On the other hand, since Z is the spray,

T ¯πZ

λ(b(t),x(t))

= λ(b(t),x(t)). As one can easily see, this means that the curve

λ(b(t),x(t)) in TM has the form (γ(t), ˙γ(t)), where γ(t)=πλ(b(t),x(t)) is a

-curve. In particular, the parallel translation is deﬁned along γ.

By deﬁnition, the projection of (b(t),x(t)) to O(m) is horizontal, and so

b(t) is a parallel frame ﬁeld along γ. Hence, for every t the vector Tλ(C(t)) ∈

γ(t)

M is parallel to Φ(t) along γ. Taking into account the deﬁnition of the

covariant derivative, we see that γ satisﬁes (11.26). It is clear that γ(0) = m

and ˙γ(0) = C. 

Corollary 11.64 Assume that ¯α satisﬁes the Carath´eodory condition and

Φ(t, m, X) is jointly measurable in all variables. Then for any C

-curve

φ:[−h, 0] → M, there exists a local solution of (11.24) with initial condi-

tion φ provided that Φ(t, φ(t),

φ(t)) is integrable on [−h, 0]. If for every

ﬁxed t the vector ﬁeld ¯α is locally Lipschitz in (m, X), then the solution is

unique.

Theorem 11.65 Let the Riemannian metric ·, · be complete. Assume also

that for some point m

∈ M the inequalities ¯α(t, m, X)≤Ψ (t)L(ρ(m

,m))

and, respectively, V (t, m)≤Ψ(t)L(ρ(m

,m)) hold where the function L is

deﬁned in Section 3.1.4 and satisﬁes (3.16), ρ is the Riemannian distance

on M,andΨ is a positive function integrable on ﬁnite intervals. Then the

solutions of (11.24) and (11.25), respectively, are deﬁned on [−h, ∞).

Proof. Without loss of generality we may assume that inf L = C>0. (Oth-

erwise, we simply replace L by L + C.) Let us rewrite (11.24)and(11.25)

in their equivalent forms (11.26)and(11.27), respectively, and consider the

complete Riemannian metric ·, ·

∗

introduced in Section 3.1.4. In this metric,

V (t, m)

∗

≤ Ψ(t). The norm of W (t) with respect to ·, ·,beingacontinu-

ous function, is bounded on [−h, 0] by a constant K>0. It is easy to show

that any local solution m(t)of(11.27) satisﬁes the inequality

 ˙m(t) <Ψ(t)+

where the norm ·is taken with respect to , .Thus,m(t) extends to

[−h, h]. Covering any given interval I by intervals of length h,onecanprove

that the solution extends to I.For(11.24) the proof is similar. 

We conclude this section by noting that the shift operators along solutions

of (11.24), (11.25) and some other neutral type equations were studied in

[28, 91, 92, 93]. The existence of ﬁxed points of these operators (i.e., periodic

solutions for a periodic right-hand side) was proved by the methods of [28, 33].

Chapter 12

Accessible Points and Sub-Manifolds

of Mechanical Systems. Controllability

12.1 Discussion of the Problem

In this chapter we study the question of whether or not two points m

and

in the conﬁguration space of a mechanical system can be connected by

a trajectory. It is known (see, e.g., [144]) that for a second order diﬀerential

equation (i.e., in particular, for Newton’s law) on Euclidean space such a

trajectory exists provided that the right-hand side of the diﬀerential equation

is bounded and continuous. More precisely, for any two points m

and m

and any interval [a, b], there exists a solution m(t) such that m(a)=m

and m(b)=m

. When the right-hand side is linearly bounded, some similar

results are known for small time intervals.

Note that if the right-hand side is quadratic in velocity, even in R

there

may be pairs of points that cannot be connected by a solution. In Section 12.2

we give a simple example of this phenomenon.

The situation becomes much more complicated for a non-linear conﬁgura-

tion space. In Section 12.2, we illustrate this with four examples of mechanical

systems on the two-dimensional sphere. In the ﬁrst example, the force ﬁeld

is smooth and independent of time and velocity (and so it is bounded). How-

ever, none of the trajectories beginning at the South Pole reach the North

Pole. In the second example, the force ﬁeld is still bounded, autonomous, and

smooth but now depends on the velocity. In this case there is no trajectory

connecting any two antipodal points on the sphere. In the third example, we

consider a gyroscopic force on S

(hence, the force ﬁeld is linear in velocity).

The behavior of trajectories in this system turns out to be quite similar to

the second example. The same behavior occurs in the fourth example where

the force ﬁeld is quadratic in velocity.

There is a deep geometric reason for the diﬀerence in the behavior of

trajectories on ﬂat and “curved” conﬁguration spaces. The points on a sphere

in the above-mentioned examples are conjugate along all geodesics joining

them. Since in the ﬂat case conjugate points are absent, this is not true in R

Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9

283

284 12 Accessible Points and Sub-Manifolds of Mechanical Systems. Controllability

Below we show that if the force ﬁeld on a complete Riemannian manifold

has less than quadratic growth in velocity, then for any two points m

and

, there exists a trajectory joining m

and m

provided that the points

are not conjugate along some geodesic. This is true only for a small enough

time interval, even if the force ﬁeld is uniformly bounded (however, we show

that from our construction it follows that for uniformly bounded forces on ﬂat

conﬁguration spaces the required solution exists on every ﬁnite time interval).

For force ﬁelds quadratic in velocity we ﬁnd a geometric condition on the

distance between points (not conjugate along at least one geodesic), and

conditions on the geometry of the manifold and on the right-hand side of the

equation, under which the solution, joining the points, exists at least on a

small enough time interval. In fact the existence of the required trajectory in

the case of less than quadratic growth follows from the fact that the latter

case always satisﬁes the above-mentioned condition. We also ﬁnd a subclass

of systems with quadratic growth for which, under the above condition, the

solution exists on every ﬁnite time interval.

Besides its mechanical interpretation, the problem where the force is quad-

ratic in velocity is important since it is a generalization of the well-known

classical question which asks whether it is possible to join two given points in

a manifold by a geodesic curve of a certain connection (see, e.g., [161]). Recall

(see Theorem 2.36)thatif∇ and

∇ are covariant derivatives of two diﬀerent

connections on a manifold M,thereexistsa(1, 2)-tensor ﬁeld S(·, ·)onM

such that for any two vector ﬁelds X and Y on M the equality

∇

Y =

∇

Y + S(X, Y ) holds. From this it follows that in terms of the covariant

derivative ∇ the geodesics of the connection

∇ are always described by an

equation of the form

˙m(t)=α(m(t), ˙m(t)), (12.1)

where α(m, X)=S

(X, X) is a vector ﬁeld on M that is quadratic in X ∈

M at any point m ∈ M.

For the Levi-Civit´a connection on a complete Riemannian manifold the

existence of a geodesic joining any two points m

and m

follows from the

Hopf-Rinow theorem (Theorem 3.68). However this is not the case in general,

even for a Riemannian connection with non-zero torsion: in [20, 85, 140]ex-

amples of Riemannian connections are presented for which this problem is not

solvable (in particular, on a compact manifold, the two-dimensional torus).

We consider the general case of systems with discontinuous forces or forces

with control, i.e., whose Newton law is described by diﬀerential inclusion

(11.17) (see Section 11.7). The results for continuous single-valued forces

(i.e., for equations of the form (11.2)) follow as simple corollaries. The inter-

pretation of set-valued forces as forces with control allows us to investigate

the so-called controllability problem, i.e., the problem of whether there exists

a time-dependent control under which a trajectory of a mechanical system

starting at a given point m

can reach another point m

of the conﬁguration

space.

12.2 Examples of Points that Cannot be Connected by a Trajectory 285

We also consider the problem for constrained systems. In this case we look

for a solution that connects a given point and a certain submanifold.

The method of investigation is based on the use of integral operators with

parallel translation and the velocity hodograph equation (see Section 11.8).

Below, in Section 13.2, we apply the machinery developed in this Chapter

to the investigation of geodesics of some connections on Lorentz manifolds.

The two-point boundary value problem for (11.17)and(11.2) with non-

conjugate points has been investigated under various conditions more re-

strictive than ours. For equation (11.2) (i.e., for single-valued force ﬁelds)

its solvability was shown by the author for continuous force ﬁelds in [88]

(bounded case) and in [101] (for linear growth in X), by E. Yakovlev, e.g., in

[232], for smooth force ﬁelds under some complicated conditions and by V.

Ginzburg in [85] for smooth force ﬁelds with less than quadratic growth in X.

The solvability of this problem for inclusion (11.17) has been demonstrated

for set-valued force ﬁelds of several types (B. Gel’man and Y. Gliklikh [80],

Y. Gliklikh and A. Obukhovski˘ı[124, 125], M. Kisielewicz [158], etc.) but

only in the uniformly bounded case.

Here we follow our joint work with P. Zykov [131, 133]. Note that in [241]

P. Zykov found some conditions for the solvability of the problem in cases

where the right-hand sides have greater than quadratic growth in velocity

(see Remarks 12.22 and 12.26).

12.2 Examples of Points that Cannot be Connected by a

Trajectory

Example 12.1. Let X =(x, y) ∈ R

and let a>0 be a real number; denote

the norm in R

by ·.InR

consider the following system of type (2):



¨x(t)=−a

X˙y

¨y(t)=a

X˙x

with initial conditions X(0) = 0,

X(0) = X

. Since here the vectors

X and

X are orthogonal to each other along the solution, 

X is constant. Let

X

 = C and represent the vector X

in the form X

= C(−sin ϕ

, cos ϕ

Then the solution of the above-mentioned Cauchy problem takes the form

x(t)=

cos(Cat + ϕ

) −

cos ϕ

, y(t)=

sin(Cat + ϕ

) −

sin ϕ

. Hence

any solution is a circle of radius

and it does not leave the disc of radius

centered at the initial point. We should emphasize that the radius decreases

as a increases.

Example 12.2. [90, 94]. Consider the mechanical system on the unit sphere

in R

with the force ﬁeld ¯α(¯r)=(−y, x,0), where ¯r =(x, y, z) ∈ S

.The

motion of the system is given by the following equations in R

¯r =¯α(¯r) − 2K·¯r

or, equivalently,