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

Подождите немного. Документ загружается.

256 11 Newtonian Mechanics

Let ¯α(t, m, X) be the vector ﬁeld on M (depending on X ∈ T

M)phys-

ically equivalent to the 1-form ˜α(t, m, X) with respect to the Riemannian

metric ·, ·, which gives rise to the kinetic energy of the system. In other

words, ¯α(t, m, X),Y =˜α(t, m, X)(Y ) for any Y ∈ T

Deﬁnition 11.2. A vector ﬁeld a(t, m, X)wheret ∈ R, m ∈ M and X ∈

M such that πa(t, m, X)=π(m, X)=m is called a vector force ﬁeld.

One can easily see that ¯α(t, m, X), as deﬁned above, is an example of a

vector force ﬁeld.

Remark 11.3. In the remainder of the book, with the exception of Chap-

ter 16, we shall consider only mechanical systems with a ﬁnite-dimensional

conﬁguration space. The force ﬁelds are usually introduced as vector ﬁelds

and the passage to 1-forms is left to the reader as a simple exercise.

The motion of a mechanical system is governed by Newton’s second law,

i.e., the equation:

˙m(t)=¯α (t, m(t), ˙m(t)) , (11.2)

where

is the covariant derivative of the Levi-Civit´a connection of the metric

·, · (see Section 2.6). Recall also that a curve m(t) is a solution of (11.2)if

andonlyifthecurve(m(t), ˙m(t)) in TM is an integral curve of the vector ﬁeld

Z +¯α(t, m, X)

, (11.3)

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

the vertical lift of ¯α(t, m, X)tothespaceV

(m,X)

⊂ T

(m,X)

TM.

Acurvem(t)onM is called a trajectory of the mechanical system if it is

a solution of (11.2). For any initial conditions m(0) = m

and ˙m(0) = X

∈

M, there exists a trajectory m(t) on a suﬃciently small interval of time

provided that, for example, ˜α(t, m, X) satisﬁes the Carath´eodory condition

[74]. To see this, observe that, since Z is smooth, the ﬁeld (11.3)onTM sat-

isﬁes the Carath´eodory condition and the local existence of a solution follows

from the classical existence theorem for ordinary diﬀerential equations. Below

we investigate the existence of trajectories for more general force ﬁelds (for

instance, in the case of a discontinuous force ﬁeld, this question is related to

the passage from (11.2) to diﬀerential inclusions). Assume, for example, that

¯α(t, m, X) is locally Lipschitz. Then the trajectory is unique for any given

initial conditions.

The existence of trajectories on (−∞, ∞) may be analyzed using the meth-

ods developed in Section 3.1. Let us point out that if the Riemannian metric

which gives rise to the kinetic energy is complete, then, by the Hopf-Rinow

theorem, the trajectories of the system with zero force ﬁeld (i.e., geodesics)

are deﬁned on (−∞, ∞). From the physical point of view, this means that

a free particle in the system does not escape to inﬁnity in ﬁnite time. Note

that the assumptions that the Riemannian metric is complete and that a

11.2 Mechanical Systems on Lie Groups 257

solution of the Cauchy problem for trajectories is unique are easy to justify

for systems arising in physics. Here, however, we usually do not require in

advance that the solution should be unique.

The cotangent vector p physically equivalent to the velocity ˙m with respect

to the metric ·, · (i.e., where p =  ˙m, ·) is called the momentum of the

system. Consider the operator L: TM → T

∗

M sending a vector X ∈ T

to the covector X, · ∈ T

∗

M. The operator L is called the inertia operator

(or the inertia tensor). In particular, p = L(˙m). In terms of the inertia

operator, the kinetic energy of the system is given by the relation K(X)=



L(X)



(X)/2.

The manifold M is often equipped with a canonical Riemannian metric

(·, ·) (e.g., the standard inner product in R

), which is not related to the

metric ·, · giving rise to K. In this case, we may identify vectors and cov-

ectors by means of (·, ·). Then X, Y  =(LX, Y )whereL is a (1, 1)-tensor

ﬁeld of self-adjoint linear operators in tangent spaces. L is also called the

inertia tensor; in fact this tensor is physically equivalent to L. Sometimes it

is convenient to deﬁne the kinetic energy not by the metric ·, · but by the

self-adjoint operator L: TM → TM once the canonical metric (·, ·) is given.

11.2 Mechanical Systems on Lie Groups

Consider a mechanical system which has a Lie group as the conﬁguration

space with the kinetic energy given by a right- or left-invariant metric. We

shall call such a system a mechanical system on the group.

The best known example of a mechanical system on a group is the system

describing the motion of a rigid body which rotates about a stationary point

in R

. It is easy to see that the conﬁguration space of this system, i.e., the

set of all possible positions of the rigid body with a stationary point, is

the special orthogonal group SO(3). Choosing an orthonormal basis in R

we may identify SO(3) with the group of all orthogonal matrices with unit

determinant. Recall that the Lie algebra so(3) of SO(3) at e =idisidentiﬁed

with R

(see Section 1.2) so that after this identiﬁcation, the Killing form

(A, B)=tr(A◦B) and the commutator [A, B]=A◦B−B◦A on so(3) become

the standard inner and vector products on R

, respectively. The Riemannian

metric obtained from the Killing form by left translations turns out to be bi-

invariant. This metric is used as the canonical metric to express the kinetic

energy via the inertia tensor.

Let g(t) be a trajectory on SO(3) corresponding to the motion of the rigid

body. The velocity vector ˙g(t) ∈ T

g(t)

SO(3) can be translated to the algebra

so(3) = T

SO(3) (and so to R

) in two diﬀerent (and non-commuting) ways:

by left translation and by right translation (see Section 1.2). The vectors

(t)=TL

−1

g(t)

˙g(t)andω

(t)=T R

−1

g(t)

˙g(t)

belong to so(3), which we have identiﬁed with the space R

where the body

moves (see above). The vector ω

is the angular velocity with respect to the

258 11 Newtonian Mechanics

body coordinates (i.e., the coordinate system “attached” to the body and

moving along with it) and the vector ω

is the angular velocity in the space

coordinates (i.e., with respect to a coordinate system ﬁxed in space).

Let L: R

→ R

be the inertia operator (tensor) of the rigid body. Recall

that L depends on the shape of the body and on the distribution of its mass.

The operator L is self-adjoint with respect to the standard inner product

on R

. Let us deﬁne a new inner product A, B

=(LA, B)onso(3). This

inner product gives rise to the left-invariant Riemannian metric ·, ·,which

determines the kinetic energy by the general formula K(X)=X, X/2.

Note that the choice of the left-invariant metric is motivated by physics: the

kinetic energy depends on the angular velocity with respect to the body’s

coordinates, but not on the position of the body in space.

Remark 11.4. In Chapter 16 we consider a mechanical system on the

inﬁnite-dimensional group of diﬀeomorphisms. This system describes the hy-

drodynamics of an ideal incompressible ﬂuid and the energy is given by a

right-invariant (weak) Riemannian metric.

The classical Euler equation of motion of a rigid body (with a stationary

point) describes the time variation of the angular velocities ω

and ω

,as

well as the angular momenta M

= L(ω

)andM

= L(ω

). Thus, the Eu-

ler equation is an equation in the algebra so(3) or in the dual space so(3)

∗

Throughout this book, we adopt the following terminology of [47]. The equa-

tions in SO(3) are said to describe the motion in the material coordinates or

in the Lagrangian representation. The equations for ω

in so(3) are said to

be with respect to the space coordinates or in the Eulerian representation,

whereas the equations for ω

are in the body coordinates or in the convective

representation. Similar terminology is used for the corresponding angular

momenta equations.

Analogous terms are used to describe a mechanical system on an arbitrary

Lie group G. Thus, Newton’s equation on G is called the equation of motion in

the material coordinates (the Lagrangian representation). The corresponding

equation on the Lie algebra is called the Euler equation. If the equation

is obtained by right translations, then it is said to be with respect to the

space coordinates (the Eulerian representation), while for left translations it

is with respect to the body coordinates (the convective representation), etc.

This also applies to the so-called Eulerian and Lagrangian speciﬁcations in

hydrodynamics.

Mechanical systems on groups, especially the Euler equation on Lie alge-

bras, have been studied very intensively in the last few decades and there is a

substantial literature devoted to the subject. A detailed introduction to this

ﬁeld may be found in Appendix 2 of [3].

11.3 Conservative Mechanical Systems

Deﬁnition 11.5. A mechanical system is called conservative if its force ﬁeld

is independent of velocities (and is usually autonomous) and is equal to −dU,

11.3 Conservative Mechanical Systems 259

the negative diﬀerential of a certain real function U on M which we call the

potential energy.

Thus the vector force ﬁeld of a conservative mechanical system is −grad U

where the gradient is calculated with respect to a Riemannian metric deﬁning

the kinetic energy, i.e., X, grad U=dU(X) for any vector ﬁeld X on M.

For a conservative system Newton’s law (11.2) is transformed into

˙m(t)=−grad U. (11.4)

Recall that X, grad U= XU for any vector ﬁeld X on M,whereXU denotes

the derivative of the function U along the vector ﬁeld X. In particular, for a

trajectory m(t) (i.e. a solution of (11.4)) this means that

U(m(t)) =  ˙m, grad U. (11.5)

Recall also that the gradient of U depends both on U and on the Rieman-

nian metric ·, ·, i.e., on the form of the kinetic energy.

Now we can derive the principal laws of dynamics as theorems.

There is a notable diﬀerence in the deﬁnitions of kinetic and potential

energy: the kinetic energy is a function on TM (indeed, it assigns the value

X, X to any vector X ∈ TM) while the potential energy is a function

on M. This is not convenient because, for example, we cannot sum these

functions. To overcome this obstacle we extend U to TM by the formula

U◦π : TM → R (11.6)

where π : TM → M is the natural projection.

Deﬁnition 11.6. The function E = K + U◦π : TM → R is called the total

energy.

Theorem 11.7 (Conservation of energy law). The total energy is con-

stant along any trajectory m(t) of a conservative mechanical system.

Proof. Since the Levi-Civit´a connection is Riemannian, from (2.30) it follows

that

 ˙m(t), ˙m(t) = 

˙m(t), ˙m(t) +  ˙m(t),

˙m(t) =2

˙m(t), ˙m(t).

Taking into account formula (11.5)weget

E(m(t)) =

(K(˙m(t)) + U(m(t))) =

 ˙m(t), ˙m(t)+

U(m(t))



˙m(t), ˙m(t)



+ grad U, ˙m(t).

Since by (11.4)

˙m(t)=−grad U,weget

E(m(t)) = 0. 

260 11 Newtonian Mechanics

11.4 Hamilton’s Principle of Least Action

The Lagrangian of a natural mechanical system is the function L = K−U ◦π.

Let us recall some notions and formulae from the calculus of variations. A

function whose argument is a curve (for the sake of deﬁniteness we assume

all curves to be parametrized by t in the interval t ∈ [0,T]), not a point, is

(in the calculus of variations) called a functional.

The functional of action of the above-mentioned Lagrangian L is

A(m(t)) |



L(m(t))dt =





 ˙m(t), ˙m(t)−U(m(t))



dt. (11.7)

We shall also deal with two more functionals: the functional of length

s(m(t)) =



 ˙m(t)dt =





 ˙m(t), ˙m(t)dt (cf. formula (1.19)) and the so-

called functional of action A

(which is a particular case of (11.7) with U =0)

(m(t)) =



 ˙m(t)

dt =



 ˙m(t), ˙m(t)dt. (11.8)

Foracurvem(t)itsvariation m(t, s) is a smooth mapping from [0,T] ×

(−ε, ε)intoM such that m(t, 0) = m(t). The term “variation” refers to the

fact that by varying s, we vary the curve m(t), i.e. we obtain curves near to

it. As in Section 2.6 (see Lemma 2.57), we consider the vector ﬁelds

∂

∂t

and

∂

∂s

on the image of a variation. In particular, the ﬁeld of vectors

∂

∂s

along the

curve m(t)=m(t, 0) will be denoted by W (t) and called the ﬁeld of variation.

Deﬁnition 11.8. A variation m(t, s) is called a variation with ﬁxed end-

points if m(0,s) = const and m(T,s) = const for all s.

In particular, for a variation of a curve with ﬁxed ends we have

∂

∂s

(0,s)=0

∂

∂s

(T,s)=0. (11.9)

The variation of a functional with respect to the variation of a curve is an

analog of the derivative of a function in ordinary analysis. It is denoted by

. To ﬁnd the variation of a functional one should substitute the variation of

the curve into the functional and diﬀerentiate the obtained expression with

respect to s at s =0.

The extremal (an analog of the extremum) is a curve at which the variation

of the functional with respect to each variation of the curve equals zero. An

extremal with ﬁxed end-points of a functional is a curve at which the variation

of the functional with respect to each variation of the curve with ﬁxed ends

equals zero.

Lemma 11.9 The variation of the functional of action A

at a smooth curve

m(t) with respect to the variation m(t, s) with ﬁxed ends is equal to

11.4 Hamilton’s Principle of Least Action 261

= −2





˙m(t),W(t)



dt, (11.10)

where W (t) is the ﬁeld of variation m(t, s).

We refer the reader to [181] for a proof of Lemma 11.9.Notethatin[181]

the variation is found in the class of piecewise smooth curves, from which

(11.10) follows as a particular case. This equation suﬃces for our purposes,

since the trajectories of mechanical systems are smooth.

Theorem 11.10 The geodesics of the Levi-Civit´a connection, and only these

geodesics, are extremals with ﬁxed ends of the functional of action A

Proof. Let m(t) be a geodesic, i.e.,

˙m(t) = 0. Then expression (11.10)

equals zero for any W (t). Thus the variation of A

at m(t) equals zero with

respect to each variation of this curve with ﬁxed ends. By deﬁnition this

means that m(t)isanextremalofA

with ﬁxed ends.

Let m(t) be an extremal with ﬁxed ends, i.e., expression (11.10) equals zero

for each W (t). This can happen only if the second factor under the integral

is zero. Hence

˙m(t) = 0 for all t, i.e., m(t) is a geodesic. 

Theorem 11.11 The geodesics of the Levi-Civit´a connection, and only these

geodesics, are extremals with ﬁxed ends of the functional of length.

Proof. First we derive an analog of formula (11.10):

∂

∂s





∂

∂t



|s=0

∂

∂s







∂

∂t

∂

∂t



|s=0



∂

∂s





∂

∂t

∂

∂t



|s=0



∂

∂s



∂

∂t

∂

∂t







∂

∂t

∂

∂t



|s=0

Recall that the length (unlike the action A

) does not depend on the

parametrization. Hence we can change the parametrization t in the varia-

tion m(t, s)sothat

∂

∂t

(t, s) =





∂

∂t

∂

∂t

 does not depend on t, i.e., is equal

to some constant C(s). Then

∂

∂s





∂

∂t

dt

|s=0

2C(s)



∂

∂s



∂

∂t

∂

∂t

dt

|s=0

which diﬀers from (11.10) only by the presence of the factor

2C(s)

before the

integral. Thus the remaining argument coincides word-for-word with that of

the proof of Theorem 11.10. 

So, unlike the straight lines of Euclidean spaces, the geodesics of the

Levi-Civit´a connection realize only local minimums of length. In addition

we should mention that sometimes several geodesics can connect two points

262 11 Newtonian Mechanics

and yet it can happen that there exists no geodesic with length equal to

the Riemannian distance between those points. Conditions under which such

geodesics exist are given by the Hopf-Rinow Theorem (Theorem 3.68).

The next statement is often regarded as the central law of mechanics (see,

e.g., [171]).

Theorem 11.12 (Hamilton’s principle of least action). The trajecto-

ries of a natural mechanical system, and only these trajectories, are the ex-

tremals of the functional of action (11.7) with ﬁxed ends.

Proof. The variation of A with respect to the variation m(t, s) of the curve

m(t), t ∈ [0,T], is calculated as follows:

∂

∂s





m

(t, s),m

(t, s)−U(m(t, s))



|s=0

∂

∂s





m

(t, s),m

(t, s)dt

|w=0

−

∂

∂w



U(m(t, s))



|s=0

The ﬁrst integral on the right-hand side here is one half of the variation of

the functional A

and so we can substitute formula (11.9) divided by 2 for

its value. For the second integral we have:

∂

∂s



U(m(t, s)))dt

|s=0



∂

∂s

U(m(t, s)))dt

|s=0

which is equal to



grad U,

∂

∂s

m(t, s)dt

|s=0

by the deﬁnition of gradient.

Recall (see above) that

∂

∂w

m(t, s)

|s=0

is the ﬁeld of variation denoted by

W (t). Thus, the formula of variation with ﬁxed ends of A takes the form

−





˙m(t),W(t)



dt −



grad U,W(t)dt. (11.11)

Now let m(t) be a trajectory, i.e.,

˙m(t)=−grad U by (11.4). In this

case, by (11.11) the variation of A is equal to zero for any W , i.e., m(t)is

an extremal. If (11.11) equals zero for every W (t),

˙m(t)=−grad U and so

the extremal is a trajectory. 

For conservative systems we also have the principle of least action in Mau-

pertuis form. According to this principle, trajectories with total energy h are

simply the geodesics of a new metric (the so-called Jacobi metric):

(h −U)·, ·. (11.12)

Note that the Jacobi metric (11.12) is Riemannian only when h>U

everywhere on M . Hence, in this case, the analysis of trajectories reduces

to the description of geodesics on M with the metric given by (11.12). If

11.5 Noether’s Theorem 263

there are points where the value of U is greater than or equal to h, then the

trajectories lie in the domain Ω



m ∈ M



U(m) <h



called the domain

of possible motion (with boundary). The geometry of this domain has been

studied by many authors in connection with certain problems of mechanics.

Particularly active research in this area was initiated by the works of Kozlov.

In [169] the reader may ﬁnd a detailed review of this subject.

Remark 11.13. A more complex case is where one has the so-called gyro-

scopic force in addition to the potential one. A gyroscopic force ﬁeld has the

form ˜α

(m, X)=ω(X, ·), where ω is a 2-form on M. (Recall that the 1-form

˜α

(m, X) takes the value ω

(X, Y )onY ∈ T

M.) Usually one assumes that

the form ω is closed or even exact. An example of a gyroscopic force is the ac-

tion of a magnetic ﬁeld on a charge. Substantial progress in the study of such

systems was achieved by Novikov [193]. Another approach was developed in

[169].

11.5 Noether’s Theorem

Recall the following:

Deﬁnition 11.14. A non-constant smooth function on a manifold, on which

a diﬀerential equation is given, is called the ﬁrst integral or an integral of

motion of the equation if it is constant along every solution of the equation.

Sometimes one says that the equation has a conservation law .

Examples of ﬁrst integrals (conservation laws) are the Hamiltonian for a

Hamiltonian system (see Remark 3.69) and the total energy for a conservative

mechanical system (see Theorem 11.7).

In this section, for the case of conservative mechanical systems, we

prove one of the most important theorems in contemporary natural science,

Noether’s theorem, which gives a standard method of ﬁnding conservation

laws.

Deﬁnition 11.15. A one-parameter diﬀeomorphism group g

is a mapping

g : R × M → M, jointly smooth in all variables, for which the following

conditions are satisﬁed:

(i) for each s ∈ R the mapping g

: M → M is a diﬀeomorphism (i.e., g

is one-to-one and smooth and g

−1

is also smooth);

(ii) for each m ∈ M and all s, u ∈ R the relation g

s+u

(m)=g

(m))

holds (this relation is usually written as g

s+u

= g

◦ g

From (ii) above it is clear that g

(m)=m for each m ∈ M and that

−1

= g

−s

Denote by g(m) the set of points obtained from m by the action of g

for all s ∈ R; such a set is called a trajectory or orbit of the one-parameter

diﬀeomorphism group g

264 11 Newtonian Mechanics

For m ∈ M consider the vector Y

(m)|

s=0

. Assigning, for each

m ∈ M, the vector Y

to the point m we obtain a smooth vector ﬁeld Y on

M that is called the generator of the group g

. Note that the orbits of the

group, and only the orbits, are integral curves of its generator Y .

Recall that on a compact manifold M every smooth autonomous vector

ﬁeld is a generator of a one-parameter diﬀeomorphism group that is the ﬂow

of the vector ﬁeld. In the general case of non-compact manifolds, a smooth

vector ﬁeld is a generator of a one-parameter diﬀeomorphism group if the

ﬂow is complete.

Deﬁnition 11.16. Agroupg

is said to preserve a non-constant function

f : M → R if f is constant along the orbits of g

It is clear that g

preserves a smooth function f if and only if for every

m ∈ M the relation

f(g

(m))|

s=0

= Yf|

g=0

= 0 holds where Y is the

generator of g

Let g

be a one-parameter diﬀeomorphism group on M .Thenitiseasyto

see that Tg

is a one-parameter diﬀeomorphism group on the tangent bundle

TM where Tg

is the tangent mapping to g

.LetL = K−U◦π : TM → R

be the Lagrangian of a conservative mechanical system.

Deﬁnition 11.17. A one-parameter diﬀeomorphism group g

is said to pre-

serve the Lagrangian L if Tg

preserves L according to Deﬁnition 11.16.

Theorem 11.18 (E. Noether) Let on the conﬁguration space M of a con-

servative mechanical system with Lagrangian L = K−U◦π there be a one-

parameter diﬀeomorphism group g

that preserves the Lagrangian L.Then

this system has a conservation law.

Proof. Consider a trajectory m(t) of the mechanical system for t ∈ [0,l].

Recall that m(t) satisﬁes Newton’s second law (11.4). Applying the diﬀeo-

morphisms in g

,fors ∈ (−ε, ε), to m(t), we obtain the set N = g

m(t)in

M as the image of the mapping m :[0,l] × (−ε, ε) → M, m(t, s)=g

m(t).

Consider the vector ﬁelds

∂

∂t

∂

∂t

m(t, s)and

∂

∂s

∂

∂s

m(t, s)onN. It is clear

that

∂

∂s

coincides with the generator Y of the group g

Lemma 11.19 For any given s

∗

∈ (−ε, ε) the curve m(t, s

∗

) is a trajectory

of the mechanical system.

Proof. [of Lemma 11.19] By Hamilton’s principle of least action (Theo-

rem 11.12) the trajectory m(t) is an extremal with ﬁxed ends of the ac-

tion functional with Lagrangian L.Sinceg

preserves L, the values of the

action functional on the curves under the action of g

are preserved. Thus

m(t, s

∗

)=g

∗

γ(t) is also an extremal, i.e., it is a trajectory of the mechanical

system. 

11.5 Noether’s Theorem 265

By the hypothesis, g

preserves the Lagrangian L, i.e., for all t the value

L(m(t, s)) = K(

∂

∂t

) −U(m(t, s)) is constant in s, i.e.,

∂

∂s

L(m(t, s)) =

∂

∂s



∂

∂t



−

∂

∂s

U(m(t, s)) = 0.

Taking into account the deﬁnition of K, formula (2.29) deﬁning the notion of a

Riemannian connection and Lemma 2.57 on the second covariant derivative

(recall that they are both valid for the Levi-Civit´a connection), we obtain

that

∂

∂s

∂

∂t

∂

∂s



∂

∂t

∂

∂s

 = ∇

∂

∂s

∂

∂t

∂

∂t

 = ∇

∂

∂t

∂

∂s

∂

∂t

.

Applying consequently formula (11.5), Newton’s second law (11.4)and

Remark 2.27 we obtain

∂

∂s

U(m(t, s)) = dU



∂

∂s





∂

∂s

, grad U



= −



∂

∂s

∂

∂t



= −



∂

∂s

, ∇

∂

∂t

∂

∂t



Thus

∂

∂s

L(m(t, s)) =



∇

∂

∂t

∂

∂s

∂

∂t





∂

∂s

, ∇

∂

∂t

∂

∂t



By formula (2.29) this expression equals

∂

∂t



∂

∂s

∂

∂t

. Hence,

∂

∂t



∂

∂s

∂

∂t

 =0,

i.e., 

∂

∂s

∂

∂t

 is constant in t.

We have shown that along every trajectory m(t) of the mechanical system

the product Y, ˙m is constant where ˙m is the velocity vector of the trajectory

and Y is the generator of the one-parameter diﬀeomorphism group g

. 

As the ﬁrst application of Noether’s theorem we prove the momentum

conservation law for some special systems.This law is not as universal as the

conservation of energy law since it is not valid for all conservative systems.

Consider the motion of two material particles in R

.Letx be the vector

in R

corresponding to the ﬁrst particle and y the vector corresponding to

the second particle. It is clear that the system’s state at a given time is

described by the pair x, y of vectors in R

, i.e., the conﬁguration space is

= R

× R

. We allow the particles to be located at the same point R

(otherwise the conﬁguration space would have the form (R

× R

)\ where

 is the diagonal set {(x, y) ∈ R

× R

|x = y}).

The kinetic energy here is the sum of the kinetic energies of both particles.

It depends only on the velocities and does not depend on the positions of the

particles. This situation is natural for Euclidean spaces, where it is possible

to translate the vectors along the space.

Suppose that the force depends only on the distance between the par-

ticles and does not depend on the positions of the particles (this situation

occurs, for example, in Newton’s law of gravitation and in Coulomb’s law in

electrostatics). Denote the force by ¯α(t, x

− y

). It is clear