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

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xx Introduction

velocities with linear or quadratic growth). Besides, it is a well-known fact

that for forces with quadratic growth, all trajectories starting at a certain

point may be conﬁned to some bounded domain for all time, unable to reach

an exterior point. Examples of all these cases of non-solvability are described

in the ﬁrst section of the chapter.

A certain geometric condition is found for the geometry of a manifold, a

pair of points and a force ﬁeld, having quadratic growth, such that if the

points are not conjugate along at least one geodesic and the condition is sat-

isﬁed, the points can be connected by a trajectory at least in a small enough

time interval. It is shown that if the force has less than quadratic growth,

this condition is always satisﬁed and so the problem is solvable for all pairs

of points, non-conjugate along at least one geodesic. If the manifold is ﬂat

(in particular, this means that conjugate points are absent) and the force

is uniformly bounded, the construction yields the classical result that the

problem is solvable for any pair of points in any time interval. For forces with

quadratic growth an additional condition is found which, in combination with

the ﬁrst condition, ensures that the problem is solvable in any time interval.

All results are proved for the general case of set-valued force ﬁelds with vari-

ous continuity conditions (ordinary single-valued results follow as corollaries)

and so they are connected with the problem of controllability for mechanical

systems. A generalization to systems with non-holonomic constraints is also

presented. In this case it is natural to investigate the problem of connecting

a point with a certain submanifold.

A modiﬁcation of the constructions of this chapter is later applied to a

certain analogous problem on Lorentz manifolds.

In Chapter 13 we deal with the general theory of relativity. The material of

the ﬁrst section can be considered as an introduction to the subject, presented

axiomatically, as is habitual for mathematicians. This part of the chapter is

the basis for all following relativistic problems. We then investigate a certain

two-point boundary value problem arising in A. Poltorak’s concept of refer-

ence frame. In this concept the reference frame is a certain manifold equipped

with a connection. On the basis of the machinery developed in Chapter 16,for

two particular cases of reference frame some geometric conditions are found

under which we can conclude from the fact that two events are connected by

a time-like geodesic in the reference frame, that the same can be done in the

space-time (i.e., if the second event belongs to proper future of the ﬁrst one

in the reference frame, the same is true in the space-time).

In the last section we describe the motion of a classical particle in a classical

gauge ﬁeld in terms of a special version of Newton’s law on a ﬁber bundle

with a connection (recall that mathematically the notion of a gauge ﬁeld

coincides with the notion of a connection on a ﬁber bundle). This section

also contains a short introduction to the geometric theory of gauge ﬁelds.

In Chapter 14 we consider mechanical systems with random perturbation

of either the force ﬁelds or of the velocities. Newton’s laws for such systems

Introduction xxi

are expressed in terms of forward mean derivatives and corresponding integral

operators with stochastic parallel translation.

First we investigate the so-called Langevin equation on a Riemannian man-

ifold. This is Newton’s law for a mechanical system on a nonlinear conﬁgura-

tion space whose force ﬁeld takes the form a(t, m(t), ˙m(t)) +

A(t, m(t), ˙m(t)) ˙w(t), where a(t, m, X) is a deterministic vector force ﬁeld,

A(t, m, X)isa(1, 1)-tensor ﬁeld (i.e., a ﬁeld of linear operators in tangent

spaces), depending on velocity X,and ˙w(t)isanItˆo white noise in tangent

space. In particular such an equation describes the motion of a physical Brow-

nian particle in a non-linear conﬁguration space. We present a well-posed

mathematical description of this equation in terms of mean derivatives and

of integral operators with parallel translation that avoids using distribution

theory. Existence theorems for weak and strong solutions are proved. Nat-

ural analogs of Ornstein-Uhlenbeck processes on Riemannian manifolds are

described. Generalizations to the case of the so-called Langevin diﬀerential

inclusion (where both a and A are set-valued) are also considered.

We then consider the case where the velocity of a mechanical system tra-

jectory is subjected to random perturbation. This situation is motivated by

the motion of a particle, subjected to a deterministic force, that in addition

moves in a medium under random inﬂuence. Such systems are described by

Newton’s law in terms of mean derivatives, whose form is diﬀerent from that

in the Langevin case. The stochastic integrals with stochastic parallel trans-

lation are applied to the investigation of such systems on manifolds. We also

consider such systems in linear spaces (in particular, with set-valued forces)

since some more general results can be obtained for them.

Another stochastic version of Newton’s second law, namely the so-called

Newton-Nelson equation, is considered in Chapter 15. It is given in terms of

mixed second order mean derivatives and describes the motion of a quantum

particle in the framework of Nelson’s stochastic mechanics. The main result

here is the existence of solution theorem where the force ﬁeld is the sum of

a vector ﬁeld, independent of velocities, and a (1,1)-tensor ﬁeld (i.e., a ﬁeld

of linear operators in tangent spaces), applied to the current velocity of the

process. We investigate the non-relativistic case (in R

and on a manifold)

as well as the relativistic case (in Minkowski space and on a space-time of

general relativity). In fact we obtain a revised version of stochastic mechanics

that is free of the defects found by Nelson within his initial approach to this

theory.

In Chapter 16 we describe hydrodynamics via the modern Lagrangian

formalism suggested in the works of V.I. Arnold, D. Ebin and J. Marsden.

This formalism arises from Newton’s law on the group of Sobolev diﬀeomor-

phisms of a ﬁnite-dimensional manifold, formulated in terms of the covariant

derivative of the Levi-Civit´a connection of a weak Riemannian metric (de-

termining the topology of the functional space L

). The basic system here

is the one of so-called diﬀuse matter. By considering a special force ﬁeld we

obtain the description of a perfect barotropic ﬂuid and, by deﬁning a special

xxii Introduction

constraint, the description of a perfect incompressible ﬂuid. If we collect the

velocity vectors of a solution in the tangent space at the unit to the group

by right translations, the curve that we obtain in that tangent space satisﬁes

the Euler equation. The diﬀerential equation of motion (Newton’s law) on

the group usually has a smooth right-hand side. Passing to the Euler equa-

tion yields loss of derivatives. We prove, among other results, local existence

of solutions theorems, regularity of solutions theorems (including the case

of ﬁnite-dimensional manifolds with boundary) and a version of Noether’s

theorem.

For the description of viscous incompressible ﬂuids we use stochastic anal-

ysis, particularly the machinery of mean derivatives. We mainly deal with the

model problem of a ﬂuid moving on an n-dimensional ﬂat torus. A special

second order equation is found with backward mean derivatives on the group

of diﬀeomorphisms, subjected to a certain constraint (a special stochastic

analog of Newton’s law), such that the expectations of its solutions are ﬂows

of a viscous incompressible ﬂuid on the torus. (It should be noted that the

stochastic Newton law here is expressed in terms of backward mean deriva-

tives and so it has a form diﬀerent to that appearing in Chapters 14 and

15.) Passing to the Euler description yields a Navier-Stokes equation in the

tangent space at the unit in complete analogy to the appearance of the Euler

equation in the case of a perfect incompressible ﬂuid.

In complete form this construction is realized under the assumption that

the backward mean derivative of the process satisfying the stochastic Newton

law, mentioned above, is generated by a right-invariant vector ﬁeld. If this

is not the case, in the “algebra”, after passing to the Euler approach, some

other types of hydrodynamical equations may arise.

We ﬁnish the chapter by introducing a special stochastic perturbation

of a ﬂow of diﬀuse matter on the group of diﬀeomorphisms such that the

perturbed ﬂow satisﬁes the stochastic Newton law and the corresponding

curve in the tangent space at the unit satisﬁes the Burgers equation. The

same perturbation of a perfect incompressible ﬂow without external force

satisﬁes the stochastic Newton law with zero force, but yields a curve in the

tangent space at the unit that is a solution of a Reynolds type equation.

Nevertheless, under the action of a certain special external force on the ﬂow,

this curve becomes a solution of a Navier-Stokes equation without external

force. As above, we consider a ﬂuid motion on the ﬂat n-dimensional torus T

Everywhere in the book we use Einstein’s summation convention:

Einstein’s Convention.

A monomial with a shared upper and lower index represents the summation

where the common index ranges from 1 to n, n being the dimension of the

manifold under consideration.

Introduction xxiii

For example, a



i=1

. Further to this convention, we treat upper

indices appearing in a denominator as lower indices and lower indices ap-

pearing in a denominator as upper indices. Examples of such expressions are:

X = X

∂

∂q

and P = P

∂

∂p

Part I

Global Analysis

Chapter 1

Manifolds and Related Objects

1.1 Manifolds, Vectors and Covectors. A Glossary

A manifold will generally be denoted by the symbol M . Recall that a ﬁnite-

dimensional manifold, as a topological space, is assumed to be Hausdorﬀ and

satisfy the second countability axiom. Unless otherwise stated, dim M = n

for a ﬁnite-dimensional manifold M.

We denote the charts of a maximal atlas by the symbols (V

,ϕ

), where

is an open ball in R

, and the corresponding neighborhoods in M by

= ϕ

. Usually we do not distinguish between V

and U

and so the

latter are also called charts. Recall that in this case R

is called the model

space.

The change of coordinates between U

and U

(i.e., between V

and V

)is

denoted by ϕ

βα

= ϕ

−1

. If the changes of coordinates are homeomorphisms,

the manifold is called topological,andifC

-smooth, a C

-manifold.Ifk = ∞,

the manifold is said to be smooth.

Convention 1.1 Everywhere below, unless stated to the contrary, all mani-

folds are assumed to be smooth, i.e., C

∞

-smooth.

Theorem 1.2 (Whitney) Every smooth manifold with dimension n can be

embedded as a smooth surface into a linear space with dimension 2n +1.

Finite dimensional manifolds are clearly locally compact. It is a well-known

fact that all locally compact spaces satisfying the second countability axiom

are paracompact (see, e.g., [30]). Thus every ﬁnite-dimensional manifold is

paracompact.

We introduce coordinates q

,...,q

in a chart V

as in a domain in R

and analogously coordinates q



,...,q



in another chart V

. As there is no

chance of confusion, the corresponding coordinates in U

and U

, obtained

by the homeomorphisms ϕ

and ϕ

, are denoted by the same symbols. They

are called local or curvilinear coordinates. The change of coordinates ϕ

βα

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

to Mathematical Physics, Theoretical and Mathematical Physics,

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

1, © Springer-Verlag London Limited 2011

4 1 Manifolds and Related Objects

given by expressing the coordinates q



of every point m ∈ U

αβ

in terms of

its coordinates q

: q



= q



,...,q

). Of course the inverse mapping ϕ

αβ

is determined by expressing the coordinates q

in terms of q



. The change

of coordinates is smooth if all q



,...,q

)andq



,...,q



) are jointly

smooth functions of all their variables.

On smooth manifolds the notion of a smooth mapping is well-deﬁned. A

mapping f : M → N is determined not only by a chart U

on M but also by

achartU

on N and is denoted by f

γα

. For other charts U

on M and U

on N we obtain f

δβ

= ϕ

δγ

◦f

γα

◦ϕ

αβ

. Recall that the changes of coordinates

δγ

and ϕ

αβ

are smooth. Thus, if f

γα

is smooth at the point ϕ

−1

(m), f

δβ

smooth at the point ϕ

−1

(m).

A particular case of the above deﬁnition is a mapping from an interval of

R to a manifold, called a curve (or path) in the manifold. Thus, the notion

of a smooth curve is well-deﬁned.

A submanifold M



of a manifold M is a subset of M that is a manifold

with the property that each point of M



belongs to a chart U of M such that

for the corresponding ball V of U (where V is a ball in a space E,say)the

intersection U





corresponds to an open ball of a linear subspace of E.

For certain manifolds M the charts for some points are not open balls but

“half-balls”, i.e., intersections of open balls with a closed half-space. Such

points form the boundary of M, which is usually denoted by the symbol ∂M.

If two copies of a manifold M with boundary ∂M are pasted together in

such a way that identical points of ∂M are pairwise identiﬁed and so that the

resulting manifold has no boundary, the latter manifold is called the double

of the manifold M. In the double of M the boundary ∂M is a submanifold

of codimension 1.

The case where the model space of a manifold is inﬁnite-dimensional has

some special features. First of all the model space is assumed to be a Hilbert

or a Banach space since in more general inﬁnite-dimensional spaces the notion

of diﬀerentiability is not completely well-deﬁned. Manifolds with Hilbert or

Banach model spaces are respectively called Hilbert or Banach manifolds.

In addition, an inﬁnite-dimensional manifold is not assumed to satisfy the

second countability axiom: it is a well-known fact that a topological space

satisfying the second countability axiom is separable, while many manifolds

arising in applications have non-separable model spaces. Usually (but not

necessarily) the manifolds are assumed to be paracompact.

A scalar ﬁeld f on a manifold M is a mapping from M to a vector space

, f : M → R

. If this map is continuous (smooth), it is called a continuous

(smooth, respectively) scalar ﬁeld.Ascalar is a value of a certain scalar ﬁeld

at some point m ∈ M, but usually (when it does not lead to confusion) a

scalar ﬁeld also is called a scalar. Note that according to this convention a

scalar may have dimension greater than 1 (this corresponds to the use of the

term scalar by physicists).

1.1 Manifolds, Vectors and Covectors. A Glossary 5

The characteristic feature of any scalar f(m) is that its presentation in

a chart does not transform under coordinate changes, unlike all other ﬁelds

(see below).

For the sake of future applications we should also give another presentation

of scalar ﬁelds. Consider the direct product M × R

and denote by π the

projection onto M. Then a scalar ﬁeld can be considered as a map f : M →

M × R

such that for each m ∈ M the relation π(f(m)) = m holds. This

relation is often expressed in operator form: π◦f = id where id is the identity

map.

The tangent space to M at m ∈ M is denoted by T

M and the total

tangent bundle by TM.Byπ : TM → M we denote the natural projection.

In every T

M there is a basis generated by coordinates (q

,...,q

)ina

chart U

 m. This basis is denoted by

∂

∂q

,...,

∂

∂q

, where the vector

∂

∂q

is the derivative of the i-th coordinate axis passing through m, with respect

to the corresponding parameter q

. Thus every tangent vector X ∈ T

M is

represented in U

 m with coordinates (q

,...,q

)asX

= X

∂

∂q

and in

 m with coordinates (q



,...,q



)asX

= X



∂

∂q



. Note that the indices

of vectors are subscripts while those of vector coordinates are superscripts.

The relation between X

and X

is described by the formula

= ϕ



βα

, i.e., in coordinates X



(1.1)

where ϕ



βα

is the Jacobi matrix of ϕ

βα

. Formula (1.1) is the transformation

rule for vectors under a change of coordinates.

Given the basis

∂

∂q

in T

M, we can create a coordinate system in T

with respect to this basis. Denote by ˙q

the coordinate corresponding to

∂

∂q

and consider its coordinate axis passing through a certain point X ∈ T

Consider the tangent space T

M to T

M at X. The coordinate system

(˙q

,..., ˙q

)inT

M generates in T

M the basis

∂

∂ ˙q

,...,

∂

∂ ˙q

by complete

analogy with the procedure that creates the basis

∂

∂q

in T

M from the

coordinate system (q

,...,q

)inU

Obviously the linear space T

M is naturally isomorphic to the linear

space T

M (visually they diﬀer only by the location of the origin: the origin of

M is located at the point X ∈ T

M). Denote by p : T

M → T

the linear isomorphism between them which is deﬁned by the rule



∂

∂ ˙q



∂

∂q

. (1.2)

Note that the construction of p is valid for any linear vector space and its

tangent. For example, consider the space R

and (using the previous notation)

denote a basis in it by

∂

∂q

,...,

∂

∂q

. Then at a given point X ∈ R

create