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266 11 Newtonian Mechanics

that the Euclidean distance in R

is a function of the diﬀerences of the same

coordinates.

Suppose in addition that the force ﬁeld ¯α(t, x

− y

−grad U(x

−y

)whereU is a smooth real-valued function

on R

Consider on the conﬁguration space R

of this system the one-parameter

diﬀeomorphism group of the form

)=(x

+ s, x

+ s, y

Diﬀeomorphisms of this group do not change the diﬀerences x

−y

, x

−y

− y

. For this reason, and since U and K do not depend on the location

of a point in conﬁguration space, this group preserves the Lagrangian. The

generator takes the form

∂

∂x

∂

∂y

, and by Noether’s theorem the system has

a conservation law of the form 

∂

∂x

∂

∂y

, ˙m =m

˙x

˙y

. Notice that

˙x

˙y

is the ﬁrst coordinate of the complete momentum of the system

that by deﬁnition (see [3]) equals m

˙x+m

˙y where m

and m

are the masses

of the particles. By analogy, introducing the one-parameter diﬀeomorphism

groups by adding t to the other coordinates of particles, we obtain that the

other coordinates of the complete momentum are also integrals of motion.

This means that the complete momentum of the system is preserved.

There is another example where a one-parameter diﬀeomorphism group

preserves the Lagrangian. This is a mechanical system on a group such that

the Riemannian metric deﬁning the kinetic energy is left-invariant and the

force equals zero. For the system of a rigid body with stationary point (see

Section 11.2) the conservation law that arises here, by Noether’s theorem, is

known as the conservation law of angular momentum. In Section 16.3 below

we present the proof of an analogous fact on an inﬁnite-dimensional group

of diﬀeomorphisms that has a hydrodynamical interpretation, as well as a

certain inﬁnite-dimensional version of Noether’s theorem.

Remark 11.20. Taking into account Noether’s theorem, it is possible to

derive the energy conservation law from the fact that both the potential and

kinetic energies of a conservative mechanical system do not depend on time,

i.e., they are preserved under shifts in t of the form t + s.

11.6 Geometric Mechanics with Linear Constraints

This section is an introduction to the modern geometric approach to me-

chanics with constraints, which goes back to the paper [71] by Faddeev and

Vershik. (See also [224, 225, 226] and the bibliography therein.) Here we focus

on systems with quadratic kinetic energy, as in previous sections, and linear

constraints only. Note that the papers mentioned above are devoted to La-

grangian mechanics with more general Lagrangians and possibly non-linear

constraints.

11.6 Geometric Mechanics with Linear Constraints 267

11.6.1 The notion of a linear mechanical constraint

Consider a mechanical system, in the sense of Section 11.1, on a conﬁguration

space M.

Deﬁnition 11.21. A linear constraint in the system is a smooth distribution

(i.e., a sub-bundle of the tangent bundle) β on M in the sense of Deﬁni-

tion 1.41.

In what follows we call a linear constraint simply a constraint.

Deﬁnition 11.22. A tangent vector is called admissible if it lies in the distri-

bution β. A curve in M is admissible if all its tangent vectors are admissible.

A constraint β imposes a restriction on the motion of the system: all its

trajectories must be admissible.

Let P : TM → β be the operator of orthogonal projection (with respect to

the Riemannian metric on M that determines the kinetic energy) of tangent

spaces onto their subspaces β, i.e., we have P

: T

M → β

for every m ∈

M. Let us deﬁne the reduced covariant derivative

∇a on admissible vectors by

the formula

∇

Y = P∇

Y ,where∇ is the covariant derivative of the Levi-

Civit´a connection. Denote by

= P

the reduced covariant derivative along

acurve.Let¯α(t, m, X) be the vector force ﬁeld of the mechanical system.

The equation of motion of the mechanical system with the constraint β is

the following analog of Newton’s equation:

˙m(t)=P ¯α(t, m, ˙m). (11.13)

In the same way as for (11.2)and(3.17), one may show that a curve m(t)

is a solution of (11.13) if and only if its derivative ˙m(t), regarded as a curve

in the total space of the bundle β, is an integral curve of the vector ﬁeld

Y = TP(Z)+



P ¯α(t, m, ˙m)



. (11.14)

It is not hard to see that TπY(m, X)=X ∈ β

and Y ∈ T

(m,X)

β.

If the distribution β is involutory (i.e., integrable, by Frobenius’ Theo-

rem 1.44), then the constraint is said to be holonomic,and(11.13) turns into

Newton’s equation (11.2) on the integral manifolds of the distribution. Thus,

a system with a holonomic constraint reduces to one with no constraint on a

manifold of lower dimension.

When the distribution β fails to be involutory (i.e., it is not integrable), the

constraint is called non-holonomic. In this case, some extra eﬀort is needed

to study the mechanical system.

Deﬁnition 11.23. A constraint β is totally non-holonomic if the Lie brackets

of the admissible vector ﬁelds generate the entire Lie algebra of vector ﬁelds

on M.

268 11 Newtonian Mechanics

Remark 11.24. One may introduce the notion of the degree of non-

holonomity [225], which we do not consider here. Note also that linear con-

straints are admissible and ideal in the sense of [224].

11.6.2 Reduced connections

Consider the orthonormal frame bundle ¯π : O

(M) → M of β (i.e., b ∈ O

is an orthonormal frame in β

). It is clear that O

(M) is a principal bundle

with structure group O(k), k =dimβ

Theorem 11.25 The reduced covariant derivative

∇ has all four properties

of the regular covariant derivative described in Theorem 2.24.

Proof. Since the operator P is linear on the ﬁbers of TM, only the fourth

property deserves a proof. For admissible X, Y and a smooth function f,we

have

∇

(fY)=P ∇

(fY)=P



f∇

Y +(Xf)Y



= f

∇

Y +(Xf)Y,

because PY = Y . 

Thus,

∇ is the covariant derivative on admissible vectors and, in particular,

it gives rise to the parallel translation of admissible vectors along admissible

curves. The deﬁnition of such a parallel translation is quite similar to the

standard one. Since P is orthogonal, it is clear that the parallel translation

preserves the inner product on ﬁbers of β and (2.29) holds for

∇. Therefore,

on the ﬁbers of β, the parallel translation of orthonormal frames is deﬁned

along admissible curves. Consider now the sub-bundle

H of T O

(M), which

is deﬁned as follows. The ﬁber of

H over a point (m, b) ∈ O

(M)isformed

by “inﬁnitesimal” parallel translations of the frame b.Itiseasytocheckthat

the sub-bundle is invariant with respect to the right action of O(k)andthe

ﬁbers of

H have zero intersection with the vertical subspaces

(m,b)

.Thus,

H can be thought of as an analog of a connection.

Deﬁnition 11.26. The sub-bundle

H is called the reduced connection .

Remark 11.27. If the constraint is holonomic, the reduced connection is the

Levi-Civit´a connection on the integral manifolds with respect to the induced

Riemannian metric. (Compare the construction of the reduced connection

with that of the connection on the adapted frame bundle [161].)

Theorem 11.28 Let X

,...,X

be orthonormal admissible vector ﬁelds

in a chart U (i.e., at every m ∈ U the vector ﬁelds X

,...,X

form an

orthonormal basis in β

). Then

∇

◦

,where

◦

are the tetrad

Christoﬀel symbols (see Remarks 2.37 and 2.56) taken for an orthonormal

frame in U which contains X

,...,X

as a subframe.

11.6 Geometric Mechanics with Linear Constraints 269

The result follows from Remark 2.56.

Corollary 11.29 The reduced connection of a non-holonomic constraint de-

pends on the Riemannian metric on the entire manifold M rather than only

on the restriction of the metric to β.

Remark 11.30. A variety of open problems concerning reduced connections,

as well as their additional properties, is discussed in, e.g., [224]. Here we only

point out that the torsion tensor of a reduced connection cannot be deﬁned

in the standard way. The reason is that even though the Christoﬀel symbols

of the reduced connection are symmetric by deﬁnition, the diﬀerence

∇

Y −

∇

X − [X, Y ]=P



[X, Y ]



− [X, Y ]

is zero only if the distribution is involutory.

11.6.3 Length minimizing and least constrained

non-holonomic geodesics

Let β be a non-holonomic constraint on M.

Deﬁnition 11.31. An admissible curve m(t)inM is called a least con-

strained non-holonomic geodesic if it satisﬁes the equation

˙m(t)=0.

It is clear that the least constrained geodesics are, in fact, trajectories of

constrained mechanical systems with a zero force ﬁeld. Deﬁnition 11.31 is

analogous to the standard deﬁnition of geodesics of a connection.

Deﬁnition 11.32. An admissible curve m(t)isalength minimizing non-

holonomic geodesic if it is an extremum of the action functional





˙m(t), ˙m(t)



dt,

an analog of the functional A

, see (11.8), on the space of admissible curves

with ﬁxed end-points (for simplicity of presentation we consider the curves

parametrized by the interval [0,T] as in (11.8)).

For a non-holonomic constraint, the notions of least constrained and length

minimizing geodesics are not equivalent. Moreover, if the constraint is non-

holonomic, the equation of the least constrained geodesics is not equivalent to

any variational principle, even if the force is conservative (cf. Theorem 11.12).

A more detailed discussion of this matter can be found, for example, in [224].

270 11 Newtonian Mechanics

Remark 11.33. The two notions of geodesics we discuss here are due to

Heinrich Hertz, who was apparently the ﬁrst to notice that Newton’s equation

and the variational principle become non-equivalent to each other for a system

with constraint [224, 225].

Theorem 11.34 (Chow-Rashevsky, see [225, 226]). Let a constraint β be

totally non-holonomic. Then for any two points m

∈ M, there exists an

admissible curve which joins m

and m

Corollary 11.35 (see [226]) Let a constraint be totally non-holonomic. Then

any two points in M can be joined by a length minimizing non-holonomic

geodesic.

Remark 11.36. The diﬀerential equation of length minimizing geodesics

(see, e.g, [225]) involves admissible vectors as well as their annihilators (i.e.,

vectors in TM orthogonal to β). Therefore, once the beginning m

∈ M

of a length minimizing geodesics is speciﬁed, the space of initial conditions

has dimension n. Thus, as mentioned above, if the constraint is totally non-

holonomic, length minimizing geodesics (beginning at m

) ﬁll the entire man-

ifold M. This question is discussed in more detail in [225, 226].

Theorem 11.37 On a complete Riemannian manifold, the reduced con-

nection is complete in the sense that all non-holonomic least constrained

geodesics extend to (−∞, ∞).

Indeed, since the Riemannian norm of all “velocity vectors” of a least con-

strained geodesic is constant, the Riemannian length of the curve is bounded

on every ﬁnite interval. Thus, the arc of the geodesic taken over a ﬁnite in-

terval is relatively compact because the manifold is complete. This, in turn,

means that the geodesic extends to (−∞, ∞).

Corollary 11.38 (see [224]) Let the constraint β on M be totally non-

holonomic. Then any two points of M can be joined by a piecewise least

constrained geodesic.

Remark 11.39. The corollary is sharp: two generic points in M cannot be

joined by a least constrained geodesic even if the constraint is totally non-

holonomic. The equation of least constrained geodesics is a second order

diﬀerential equation on the total space of the bundle β. (The equation is

given by the vector ﬁeld TP(Z)from(11.14).) Thus, the space of initial

conditions of least constrained geodesics starting at a given point m

∈ M

has dimension k =dimβ

and the geodesics cannot ﬁll the entire manifold

In the contemporary mathematical literature the study of least-constraint

geodesics (or, more generally, of solutions of equations of type (11.13))

is called non-holonomic dynamics while the study of length minimizing

geodesics (or more generally, of variational problems with non-holonomic con-

straints) is called vakonomic dynamics.

11.7 Discontinuous forces. Diﬀerential inclusions 271

Remark 11.40. We should draw the reader’s attention to the paper [37]

where some geometric fundamentals of non-holonomic and vakonomic dy-

namics are investigated. For non-holonomic dynamics a certain invariant,

the torsion of a special extension of a reduced connection, is found such

that if it equals zero, the distribution β of the constraint is integrable. If

the orthogonal complements to β

form an integrable distribution, a (local)

Ehresmann connection is constructed whose torsion characterizes the vako-

nomic dynamics. If this characteristic equal zero, the distribution β is also

integrable.

Remark 11.41. We refer the reader to [107, Section 16] where stochastic

diﬀerential equations in Belopolskaya-Daletskii form with constraints are in-

troduced. The main idea is to use the exponential mapping of the reduced

connection instead of the general exponent to get a constrained analog of

the equations from Section 7.3. The constrained analogs of stochastic inte-

gral operators and the equations of Section 7.7 are also presented in [107,

Section 16]. In their construction the parallel translation with respect to the

reduced connection along the corresponding stochastic processes is applied.

Note that the constrained Belopolskaya-Daletskii and integral approaches to

SDEs with constraint have to be applied even for constraint equations in R

assuming of course that the constraint is not trivial (i.e., does not consists of

subspaces parallel to each other).

11.7 Mechanical Systems with Discontinuous Forces and

Systems with Control. Diﬀerential Inclusions

Consider a mechanical system with a discontinuous force ﬁeld. Such ﬁelds

appear, for example, in systems with dry friction, switching, or with motion

in several media having diﬀerent resistance forces. When the conﬁguration

space is linear, the following method is often used to study systems with

a discontinuous force. First, one extends the discontinuous force ﬁeld to a

set-valued vector ﬁeld with convex images. Then one passes from (11.2)toa

diﬀerential inclusion whose solutions are trajectories of the system (see [74]).

This approach in linear space is knows as Filippov’s method. In this section,

we develop a similar method for non-linear conﬁguration spaces [165].

The equation of motion of a mechanical system with feedback control may

also be reduced to a diﬀerential inclusion. In this case, the set-valued force,

a subset in every tangent space, is formed by all values of the force for all

possible values of the controlling parameter at a given point.

The requisite notions and results on set-valued mappings we use here can

be found in Chapter 4.

Consider a locally bounded vector ﬁeld f on a ﬁnite-dimensional manifold

M. The vector ﬁeld f is not assumed to be continuous, nor even measurable.

272 11 Newtonian Mechanics

For each point m

, let us deﬁne a subset R(m

)ofT

M as follows. The

set R(m

) is formed by the limits of all sequences f(m

)asm

→ m

with

= m

.Itiseasytoseethat

R(m



>0







m∈U



f(m)



\ f(m

)



where U

is the ε-neighborhood of the point m

and cl denotes the closure.

Deﬁnition 11.42. The set F (m

)=coR(m

) ⊂ T

M,whereco denotes

the convex hull, is called the essential extension of the ﬁeld f at m

Deﬁnition 11.43. A set-valued map f : R × TM  TM such that for any

point (m, X) ∈ TM (meaning that X ∈ T

M, i.e., X is a tangent vector

to M at the point m ∈ M) the relation πf(t, m, X)=π(m, X)=m holds

is called a set-valued vector force ﬁeld (cf. the Deﬁnition 11.2 of vector force

ﬁelds).

The essential extension F is a set-valued mapping which assigns a subset

of T

M to m

∈ M . One can easily see that F is a set-valued vector ﬁeld.

Note that F = f if f is continuous.

Theorem 11.44 The set-valued vector ﬁeld F is upper semicontinuous.

Proof. Let δ>0 be a real number. Fix a metric ρ on TM which gives

rise to a topology equivalent to that on the tangent bundle. Denote the δ-

neighborhoods of R(m

)andF (m

)byR

)andF

), respectively. We

prove that for any δ and any m ∈ M , there exists a neighborhood U(m) ⊂ M

of m such that R(m



) ⊂ R

(m) for every m



∈ U(m) and, therefore, F(m



) ⊂

(m).

By the deﬁnition of the set R(m), there exists a neighborhood U(m)of

m such that for all m



∈ U (m)wehaveρ(f(m



),R(m)) <δ. Then there

exists an open neighborhood V (m



) ⊂ U(m) of the point m



such that the

inequality ρ



f(m



),R(m



)



<δis satisﬁed for every m



∈ V (m



). Pick a

sequence m



→ m



in V (m



). We have

lim ρ



f(m



),R(m)



= ρ lim



f(m



),R(m)



<δ .

Hence, R(m



) ⊂ R

(m)andF (m



) ⊂ F

(m). 

Now consider a mechanical system with conﬁguration space M and kinetic

energy K(X)=X, X/2, where ·, · is the Riemannian metric on M.Let

¯α(t, m, X) be a force ﬁeld that we require only to be locally bounded in

all variables. (As above, we do not assume that ¯α is continuous or even

measurable.) Consider the vector ﬁeld Z(m, X)+¯α(t, m, X)

(i.e., a second

order diﬀerential equation on M ; see Section 3.3), where Z is the geodesic

spray of the Levi-Civit´a connection of ·, · and ¯α(t, m, X)

is the vertical lift

11.7 Discontinuous forces. Diﬀerential inclusions 273

of ¯α(t, m, X) to the point (m, X) ∈ TM (see (4.3)). It is easy to see that

the essential extension (with respect to all variables) of Z(m, X)+¯α(t, m, X)

may be written in the form

Z(m, X)+a(t, m, X)

, (11.15)

where a(t, m, X)

is the vertical lift of the essential extension a(t, m, X)of

¯α(t, m, X)tothepoint(m, X). Note that a(t, m, X)=

coQ(t, m, X), where

Q(t, m, X) is the set of limit points of all sequences ¯α(t

) such that

) → (t, m, X), X

∈ T

M,and(t

) =(t, m, X).

From now on, we focus on the diﬀerential inclusion in TM given by the

formula

˙γ(t) ∈ Z (γ(t)) + a (t, γ(t))

. (11.16)

Deﬁnition 11.45. A solution of (11.16) is an absolutely continuous curve

γ(t)inTM which almost everywhere satisﬁes (11.16).

Alternatively, making use of covariant derivatives, we consider the follow-

ing diﬀerential inclusion on M:

˙m(t) ∈ a (t, m(t), ˙m(t)) . (11.17)

Deﬁnition 11.46. A solution of (11.17)isaC

-curve m(t)inM such that

˙m(t) is absolutely continuous and (11.17) is almost everywhere satisﬁed.

Taking into account (11.15) and the deﬁnition of

,itiseasytocheck

that (11.16)and(11.17) are equivalent. More precisely, this means that m(t)

is a solution of (11.17) if and only if ˙m(t), regarded as a curve in TM,isa

solution of (11.16).

Deﬁnition 11.47. A solution of (11.17) is called a trajectory of the mechan-

ical system with discontinuous force ﬁeld ¯a.

It is easy to see that Deﬁnition 11.47 is justiﬁed from the physical point

of view. As mentioned above, for a ﬂat conﬁguration space the reasons for

supporting the deﬁnition are discussed, for example, in [74].

The right-hand side of (11.16) is an upper semicontinuous set-valued vector

ﬁeld with convex images. This implies that locally there exists a solution

of the Cauchy problem for (11.16) (see Chapter 4). Thus, for any initial

conditions m ∈ M and X ∈ T

M, inclusion (11.17) has a solution on a

suﬃciently small interval.

Note that an interesting question for applications in physics is whether or

not the local solution of (11.17) is unique. Certain uniqueness conditions are

found in [74].

Another class of mechanical systems involving inclusions like (11.17)is

the class of mechanical systems with feedback control. Let the force ﬁeld

274 11 Newtonian Mechanics

¯α(t, m, X, u) depend on the parameter u ∈ U. We deﬁne the set-valued vector

ﬁeld a(t, m, X)onTM as

a(t, m, X)=



u∈U

¯α(t, m, X, u).

Now we have to assume that this ﬁeld is upper semicontinuous and has closed

convex images. The solution of (11.17) is a trajectory of the control system for

a time-dependent control u(t). Let us point out that, since the conﬁguration

space is non-linear, we cannot assume the control force to be independent of

time, coordinates or velocity, as is usually the case for linear spaces. Some

very particular examples where such an assumption does make sense will be

considered in Section 11.9 andinChapter12.

Note that in systems with control the inclusions of type (11.17) with lower

semicontinuous a can also arise. Let us consider an example of such an inclu-

sion.

Consider a set-valued bounded and Hausdorﬀ continuous vector force ﬁeld

a(t, m, X) with convex closed values.

Deﬁnition 11.48. A point a of a convex set a is called extreme if there does

not exist an open interval of a straight line in a that contains a. Denote

by Ext a(t, m, X) the set-valued vector force ﬁeld whose values at all points

(t, m, X) consist of extreme points of a(t, m, X).

Lemma 11.49 For a set-valued bounded Hausdorﬀ continuous vector force

ﬁeld a(t, m, X) with convex closed values the set-valued vector force ﬁeld

Ext a(t, m, X) is lower semicontinuous.

Lemma 11.49 is a well-known statement from set-valued analysis. A proof

can be found in [222, Lemma 2.1.1] and in [48, Proposition 6.2]. Note that

Ext a(t, m, X) is bounded and may not have convex values.

Deﬁnition 11.50. We say that a trajectory m(t) of a mechanical system

with Hausdorﬀ continuous vector force ﬁeld a(t, m, X) occurs under ex-

tremal values of controlling force if almost everywhere

˙m(t) belongs to

Ext a(t, m(t), ˙m(t)).

Thus for a mechanical system with control given by (11.17), with set valued

vector force ﬁeld as above, the problem of the existence of solutions that

occurs under extremal values of controlling force is reduced to the diﬀerential

inclusion

˙m(t) ∈ Ext a(t, m(t), ˙m(t)) (11.18)

with a lower semicontinuous right-hand side having non-convex values.

11.8 Integral Equations of Geometric Mechanics. The Velocity Hodograph 275

11.8 Integral Equations of Geometric Mechanics. The

Velocity Hodograph

In this chapter we use the integral operators with parallel translation in-

troduced in Section 3.2 to ﬁnd integral equations equivalent to Newton’s

equation of geometric mechanics (see Section 11.1). One of these equations

describes the velocity hodograph in the sense of [217]. This is an ordinary

integral equation in a speciﬁed tangent space. We also introduce analogous

integral equations for a system with constraint. Our approach is based on the

results of [88, 90, 94, 98].

Integral versions of Newton’s equation and, in particular, the equation of

the velocity hodograph turn out to be useful in the study of certain qualita-

tive problems concerning the behavior of mechanical systems, for example,

the existence of special trajectories. It is important to emphasize that the

equation of the velocity hodograph is an integral equation in a linear space,

and therefore standard methods may be applied to study it. Integral equa-

tions are used in Chapter 12 and also in Chapters 14 and 15 wherewework

with versions of integral equations for random force ﬁelds.

11.8.1 General constructions

Consider a mechanical system as in Section 11.1. We assume here that the

Riemannian metric ·, · is complete (and so a trajectory of a free particle can-

not escape to inﬁnity in ﬁnite time) and that the vector force ﬁeld ¯α(t, m, X)

is jointly continuous in all variables. The case of discontinuous force ﬁelds

will be studied in Chapter 12.

Since the metric is complete, we can use the operator S introduced in

Section 3.2

Let Γ ¯α



t, m(t), ˙m(t)



denote the curve in T

M such that the vector

Γ ¯α(t, m(t), ˙m(t)) is parallel to ¯α(t, m(t), ˙m(t)) along m(·) for every t. Specify

a point m

∈ M and a vector C in T

m(0)

M and consider the integral equation

m(t)=S





Γ ¯α (τ,m(τ ), ˙m(τ)) dτ + C



(11.19)

on I =[0,l].

Theorem 11.51 The solutions of (11.2) with the initial conditions m(0) =

and ˙m(0) = C coincide with the solutions of (11.19).

Proof. It is easy to show that for a given v ∈ C

(I,T

M), the C

-curve

m(t)=S





v(τ)dτ + C

