Indeterminism, Acausality, and Time Irreversibility


August 2, 2006

Abstract added October 17, 2006

Indeterminism, Asymptotic Reasoning, and Time Irreversibility

in Classical Physics

Alexandre Korolev1

Abstract

A recent proposal of Norton (2003) to show that a simple Newtonian system can exhibit stochastic

acausal behavior by giving rise to spontaneous movements of a mass on the dome of a certain shape

is examined. We discuss physical significance of an often overlooked and yet important Lipschitz

condition  the violation of which leads to the existence of anomalous non-trivial  solutions in this

and similar cases. We show that the Lipschitz condition is closely linked with the time reversibility

of  certain  solutions  in Newtonian  mechanics and the  failure  to incorporate  this condition  within

Newtonian  mechanics  may  unsurprisingly  lead  to  physically  impossible  solutions  that  have  no

serious  metaphysical  implications.  To  further  support  this  view  we  also  discuss  how  certain

solutions in hydrodynamics associated with first order differential equations (ODEs) with spatially

non-Lipschitz right-hand side lead to lack of important properties such as stability with respect to

perturbations and Markovianity in time.

1 Introduction

2 The Mass on the Dome

2.1 Setting the Stage: The Original Formulation

2.2 The Mass on the Dome: Cartesian Perspective

2.3 The Mass on the Dome vs. the Mass on the Pinnacle

1 Department of Philosophy, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada, email:
korolev@interchange.ubc.ca.

1



2.4 The Mass on the Dome or the Mass in the Air?

2.5 The Mass on the Dome, Modified

3 The Lipschitz Condition

4 Asymptotic Reasoning and Time Irreversibility

4.1 Models and Idealizations

4.2 Stability and Parameter Sensitivity

4.3 The Mass on the Dome and Time Irreversibility

5 Lipschitz Indeterministic Solutions and the Markov Condition

5.1 Generalized Flows in Hydrodynamics

5.2 Non-Lipschitz Velocity Fields, Regularizations, and the Markov Condition

1. Introduction

Abstruse theories like quantum mechanics and general relativity routinely violate common

intuitions about causality and determinism. In contrast, classical physics is often assumed to

be  a paradigm example  of  a fully deterministic  physical theory that  never violates these

intuitions, or that violates them only in the most extreme circumstances which render such

situations as plainly unphysical. A number of authors have argued that this is not so, and

that classical physics is a poor choice of hunting ground for such beliefs. A definitive guide

to  the  discussion  is  John  Earman's  A  Primer  on  Determinism (1986)  that  collects  and

discusses various situations that threaten uniqueness of solutions for common differential

equations governing dynamics of ordinary classical systems. A more recent attempt by John

Norton (2003) presents another simple Newtonian system that seems to exhibit stochastic

acausal behavior in that it allows generation of spontaneous motion of a mass without any

2



external  intervention  or any change in  the physical  environment. The latter  system is  of

particular interest since, unlike most of Earman's examples, it does not seem to involve, at

least directly, any singularities, wild divergences, or any other bullying with infinities  of

physically  meaningful  parameters  in  any  way  that  often  leave  the  true  believer  of

determinism unsatisfied. Norton uses this example to support his vision of causality as a

notion belonging more in folk science rather than being a fundamental principle underlying

all natural processes and unifying all the domains of science at some deeper level.

While by and large sympathetic with the general thrust of the "anti-fundamentalist"

program of Norton, we intend to demonstrate that his mass on the dome example fails to

provide  support  for  such  a  view.  We  show  that  the  existence  of  anomalous  non-trivial

solutions  in  this  case  is  due  to  the  violation  of  an  often  overlooked  and  yet  important

Lipschitz  condition  and  discuss  its  physical  significance.  We  show  that  the  Lipschitz

condition is closely linked with several important temporal properties of certain solutions

such  as  time  reversibility and  Markovianity in  time  so  that  the  failure  to  recognize  this

condition  as  an  important  implicit  assumption  within  Newtonian  mechanics  may

unsurprisingly lead  to  physically impossible  solutions  that  have  no serious  metaphysical

import, as, for instance, in Norton's causal skeptical anti-fundamentalist program.

The  rest  of  the  paper  is  organized  as  follows.  Subsection  2.1  introduces  Norton's

mass-on-the-dome  example  and  the  time  reversal  argument.  In  Subsections  2.2-2.5  we

expose  the loopholes  of this example, some of  them  potentially fatal  to  the project,  and

introduce  a  cleaner  version  free  of  loopholes  and  unnecessary complications.  Section  3

discusses  the  Lipschitz  condition  as  it  appears  in  the  theory  of  ordinary  differential

equations  (ODEs)  and  as  it  enters  the  mass-on-the-dome  example.  Section  4  applies

(infinite) asymptotic reasoning to display the asymmetry of time-reversal arguments for the

3



non-Lipschitz velocity fields and an unphysical singular nature of the spontaneous motion

generation is revealed. In Section 5 we draw several results from fluid dynamics to further

illustrate  how  certain  solutions  associated  with  first  order  differential  equations  with

spatially non-Lipschitz  right-hand  side  may lead  to  lack  of  important  properties  such  as

stability with respect to perturbations and Markovianity in time.

2. The Mass on the Dome

2.1 Setting the Stage: The Original Formulation

A  unit  point  mass  slides  frictionlessly  on  the  surface  under  the  action  of  gravity.  The

surface is shaped as a symmetric dome described by the equation:

3 22
3( ) gy r r= - , (1)

where  r is the radial coordinate in the surface, i.e., the distance traveled by the mass from

the highest point of the dome along the surface, | |y  specifies how far the dome surface lies

below the apex as a function of r, and g is the acceleration due to gravity (Fig. 1).

4



Fig. 1 Mass sliding on a dome.

At  any point,  the  magnitude  of  the  gravitational  force  tangential  to  the  surface  is

1 2( )d gyF r
drt

= - =  and  is  directed  outward.  Newton's  second  law  of  motion,  F ma= ,

applied to the mass on the surface gives

2
1 2

2

d r
r

dt
= . (2)

If the mass is initially located at rest at the apex 0r = , then one obvious solution to

(1) for all times t is a trivial one:

( ) 0r t = . (3)

The mass  simply remains at rest for all times.  However, there exists another large

class of unexpected solutions. For any radial direction,

41
144 ( ) ,  for all ( )
0,  for all 

t T t T
r t

t T
м - іп

= н
Јпо

, (4)

5



where  0T і  is  an  arbitrarily  chosen  constant.  By  direct  computation  one  can  readily

confirm that (4) satisfies Newton's second law (2).

Note that equation (4) describes a point mass sitting at rest at the apex of the dome,

whereupon at an arbitrary time 0T і  it spontaneously moves off in some arbitrarily chosen

radial direction.

The solutions (4) appear to be fully in accord with Newton's second and first laws, if

one takes the first law in its instantaneous form as follows:

In the absence of a net external force, a body is unaccelerated.

Indeed, for all times t T< , there is no net force applied, since the body is at position 0r = ,

the force free apex; and the mass is unaccelerated.

For  all  times  t T> ,  there  is  a  non-zero  net  force  applied,  since  the  mass  is  at

positions  0r >  not the  apex,  the  only  force  free  point  on  the  dome;  and  the  mass

accelerates in accord with F ma= .

Finally,  when  t T= ,  the  direct  computation  of  the  mass  acceleration  from  the

equation (4) gives us

21
12 ( ) ,  for all ( )
0,  for all 

t T t T
a t

t T
м - іп

= н
Јпо

, (5)

so that at  t T= , the  mass is  still at the force-free apex  0r =  and the mass  acceleration

(0)a  is equal to zero. Again, no force, no acceleration, exactly as the first law requires.

What about the initiating cause that sets the mass in motion in the first place? Surely

the instant  t T=  is  not the first instant at which the mass moves; it is the  last instant at

which the mass does  not move. In fact, one can name no first instant at which the mass

6



moves. So, if there is no first instant of motion, then there is no first instant at which to seek

the initiating cause.

Yet  another  powerful  argument  can be given  in  support  of  acausality of the  mass

motion.  This  argument  involves  the  time  reversal  trick.  Since  the  Newtonian  dynamical

laws of gravitational systems are invariant under time reversal we can invert the sliding-

down-the-dome scenario to produce another legitimate solution which insults the principle

of causality. Instead of having the mass starting at the apex of the dome, we will imagine it

starting at the rim and that we give it some initial velocity directed exactly at the apex. If we

give it too much initial velocity, it will pass right over the apex  to the other side of the

dome.  If we  give  it  too  small  initial  velocity, it  will  rise  toward  the  apex,  but  before  it

reaches the apex it halts and slides back to the rim. Now if we give the mass just the right

amount of initial velocity, it will rise up and momentarily halts exactly at the apex.

The proper mathematical analysis of the latter situation reveals that the time required

for the mass to reach the apex moving along the surface of this particular shape is  finite.

That this time is finite is essential for the time reversal trick to succeed. Infinite time would

mean that the mass never actually arrives at the apex, and the time reversal scenario would

display a mass that has been in motion at all past times, without any spontaneous launches.

It  should  be  emphasized  that  by  no  means  this  feature  is  common  to  all  domes.  For

hemispherical or parabolic domes, for instance, the time  taken for the mass  to reach the

apex to its momentary halt is unbounded. In the case of the dome of Fig. 1 the time reversal

trick does work.

2.2 The Mass on the Dome: Cartesian Perspective

7



Defining the surface by  
3 22

3( ) gy r r= - , in terms  of the distance  r  traveled by the mass

along the surface, conceals important  details about the actual geometry of the dome and

directionality of motion as viewed from the "external" Cartesian perspective. Special care

should be taken not to overlook these details since they may prove crucial in the general

case.  Indeed, as  the  present  section  shows,  the  original  formulation  of  the  mass-on-the-

dome example  harbors  several  loopholes, some  of  which potentially fatal  to  the  project.

Fortunately,  most  of  these  difficulties  are  reparable  and  can  be  overcome  by  slightly

modifying the original formulation of the problem. As the next section shows, the modified

version  inherits  all  the  strangeness  of  being  a  source  of  spontaneous  motion  generation

without our having to deal with the loopholes and unnecessary qualifications.

Consider an axially symmetric dome, and  x is  a Cartesian radial variable. In what

follows, we consider only non-negative  x's, and then extend the results to incorporate the

negative values  of the  x-coordinate. For  the  curvilinear  coordinate  r measured  along the

surface slice cut vertically through the apex at the origin we have

21 [ ( )]
dr

y x
dx

ў= + ,

so that

1 2
2

1( ) ( )
1 [ ( )]

g
dy r dy dx y x

r
dr dx dr y x

ў
= - = Ч =

ў+
, ( ) 0y xў Ј .

Expressing the coordinate r we can write down

2
2

2

[ ( )]
1 [ ( )]

y x
r g

y x
ў

=
ў+

. (6)

Putting v yў=  and differentiating both sides of (6) gives us

8



2 5 2

2

1 (1 )
2

dv v
dx g v

+
= , so that 2 5 2 2

1
(1 ) 2

vdv
x Const

v g
= +

+т .

Integrating the left-hand side of the last expression we get:

2 3 2 1(1 [ ( )] )y x
C kx

ў+ =
-

, (7)

for 232gk =  and some constant C.

The first observation to make is that the dome surface appears to be defined not for

all x's, but only for x's out of some (final) interval [0, L). Indeed, the left-hand side of (7) is

always greater or equal to 1, so it must be the case that 0 1C kxЈ - Ј  for all x out of some

interval [0, L). That provides the constraint on the possible values the constants C and L can

take:

0 1kL C< Ј Ј , where 1L k< .

Finally, integrating (7) and incorporating negative values of x, we can express y(x) as

a function of the Cartesian coordinate x:

2 3 3 2
1

1( ) [1 ( | |) ]ky x C k x C= - - - + , (8)

where 0 1kL C< Ј Ј , 1L k< , and the constant 2 3 3 21 1 (1 )kC C= - - .

The tangential gravitational force acting on the mass as a function of x is

2

( )
1 [ ( )]

y x
F g

y x
t

ў
=

ў+
. (9)

The normal force exerted upon the mass by the surface at the point x is

2

1
1 [ ( )]

N g
y x

=
ў+

. (10)

2.3 The Mass on the Dome vs. the Mass on the Pinnacle

9



Having expressed the shape of the dome in the linear Cartesian coordinates, it is easy to see

that  not all the domes described by formula (8) would fit well for generating spontaneous

motion within Newtonian mechanics.

Indeed, having fixed some "rim", 1L k< , and depending on the value of the constant

C, two distinct cases are possible.

Case 1: C ≠ 1. Substituting 0x =  into (7) we obtain 2 3 2 1(1 [ (0)] ) 1Cyў+ = > , so that

the first derivative of the function tends to some non-zero constant, d, as x approaches zero.

Geometrically this means that the tangent line to the dome surface at zero hits the y-axis at

some non-zero angle – the mass arrives at the apex not exactly horizontally but at some

non-zero angle. As we pass through zero into negative  x's, the tangent line to the surface

experiences a sudden step-like jump:

 

0 0
lim ( ) lim ( ) 2
x x

y x y x d
® + ® -

ў ў- =  for some 0d № .

So ( )y xў  is simply not defined at 0x = . We shall refer to this case as the mass-on-

the-pinnacle scenario (Fig. 2).

10



Fig. 2 The mass on the pinnacle.

Since  ( )y xў  enters the expressions (9) and (10) for the tangential gravitational and

normal forces acting on the mass, these forces appear not to be defined at zero either. As

Newton's second law of motion " F ma= " cannot be written for the mass at zero, the mass-

on-the-pinnacle scenario simply does not belong in Newton's mechanics jurisdiction, and

should be isolated from the discussion by an appropriate stipulation. Yet, we shall return to

this case again in section 4, where it appears in regard with the Lipschitz condition.

Case  2:  1C = .  Since  the  constant  2 3 3 21 1 (1 )kC C= - - ,  we  have  1 0C = ,  and  the

expression (8) for the shape of the dome takes a relatively simple form:

2 3 3 21( ) [1 (1 | |) ]ky x k x= - - - , (11)

for all x out of the interval (–L, L), 1L k< , for which the surface is defined.

Looking at (7), we see that the first derivative,  ( )y xў , turns to zero as  x  approaches

zero. Geometrically this means that the mass arrives at the apex exactly horizontally. Even

though the second and higher derivatives of  y(x) do diverge at zero, this fact by itself, at

11



least directly, seems to generate no singularity or divergence in any physically meaningful

parameter. We shall refer to this situation as the (proper) case of the mass on the dome (Fig.

3). All discussion that follows below will have this situation in mind.

Fig. 3 The mass on the dome.

At this point we also note that in the vicinity of the apex, i.e., at the limit | | 0x ® , the

graph behaves as a fractional power function:

3 2 3 22
3( ) 8 27 | | | |gy x k x x» - = - , (12)

which, of course, should have been expected, since, for small  x's, the  x-coordinate almost

coincides with the radial coordinate r measured along the practically horizontal surface. We

shall return to this point later in this section when trying to modify the original formulation

of the problem to avoid some of its loopholes.

On  the  other  side  of  the  interval,  as  x  approaches  the  "rim",  | |x L® ,  the  graph

descends steeper and steeper until it hits the vertical wall at   | |  x L= . (The graph of this

function appears in Fig. 1.)

12



2.4 The Mass on the Dome or the Mass in the Air?

Further observations about the behavior of the mass on the so defined dome are in order.

Consider again the three classes  of  trajectories  produced if  we give  the  mass  at  the  rim

some  initial  velocity directed  at  the  apex  along the  surface:  those  where  the  mass  halts

before it reaches the apex and falls back to the rim; those where the mass halts exactly at

the apex; and those where the mass passes the apex with some non-zero velocity and rushes

over to the other side of the dome.

There is an easy and instructive way to see how things may go astray by considering

the  trajectories  from  the  third  class  when  the  mass  passes  over  the  apex.  As  the  mass

proceeds through the apex with a non-zero velocity into negative x's, it continues its motion

along an artillery shell like ballistic  trajectory. However,  in the  vicinity of  the apex,  the

dome surface descends faster than any such parabolic trajectory so that the mass necessarily

detaches itself  from  the  surface  once  it  enters  the  negative  x's,  refusing to  follow  the

prescribed track. (The results of numerical simulation for different velocities appear below

in Fig. 4.)

13



Fig. 4 Passing over the dome apex.

Similar  detachment  occurs in the  second, more important, class of trajectories, for

which the mass should halt exactly at the apex. To see this, recall that at the "rim", x L= ,

the  tangential  to  the  surface  plane  is  exactly  vertical.  It  means  that  the  mass,  initially

positioned at the rim and given any initial velocity directed at the apex along the surface,

will go up (and then fall back) precisely vertically,  detaching itself from the surface and

thus, again,  refusing to follow its curvature. A careful analysis reveals that this is true not

only for the rim x L=  taken as an initial position of the mass, but also for some vicinity of

the rim. Indeed, for  the so  defined  domes,  there always exists  a (finite)  interval  (L1,  L),

10 L L< < ,  such  that,  for  all  initial  positions  of  the  mass  (the  "rims")  taken  within  this

interval,  the  ballistic  trajectory  of  the  mass  descends  slower than  the  dome  surface

immediately under  it,  thus  causing  the  mass  to  detach  from  the  surface.  (The  results  of

numerical simulation for different initial positions are shown below in Fig. 5.)

14



Fig. 5 Mass detachment in the vicinity of the rim.

Clearly, such initial positions would be a poor choice for being the "rim" since, as

mentioned  in  Section  2.1,  letting  the  mass  fly  along  its  ballistic  parabolic  path  would

irreparably ruin the time reversal argument.

Any attempt  to  bypass  this  difficulty by pushing  the  mass  back  on  the  prescribed

track (e.g., switching to a bead-on-the-wire example) would necessarily involve additional

forces  (viz.,  the  elasticity forces  of  the  wire  along which  the  bead  would  slide)  without

which  the  mass  will  simply refuse  to  follow  the  track.  Adding  new  forces  (external  or

internal),  as  we  shall  soon  see,  brings  new  and  undesirable  complications  into  Norton's

original problem.2

These complications become especially important (and more subtle) when we move

away from the rim to the vicinity of the apex. Unlike the previous situation, for any initial

position of the mass taken within the interval (0, L1), the ballistic trajectory of the mass now

2 This phenomenon of mass detachment is closely connected with a mechanical system's being ideal
holonomic since an ideal holonomic constraint can be taken as the limiting case of a system with a large
potential enegry, or, equivalently, the limiting case of an infinite force field in a neighborhood of the curve,
directed toward the curve to ensure the moving point remains exactly on the curve (see Arnold 1978). I intend
to expand on this point elsewhere.

15



descends  faster than the  dome surface, and  no detachment  of  the mass  from  the surface

occurs. (The results of numerical simulation are shown in Fig. 6.)

Fig. 6 No mass detachment in the vicinity of the apex.

Though we need not resort to elastic wires going through the mass to keep it from

detaching itself from the surface of the dome, this is the elasticity of the dome that acquires

special importance here.

At this point we can discern a general pattern that begins to emerge. We don't want to

let the mass  to move along its  free-flight parabolic trajectory; parabolic  trajectories  give

infinite past times for the time reversal scenarios, thus stripping the whole argument of its

force. In the vicinity of the rim this can be done the invoking the elasticity forces of the wire

through which the bead-mass  now  slides  (or  applying some  other external force). In the

vicinity of the apex these are the elasticity forces of the dome that would "straighten up" the

mass path sufficiently to yield the required curvature; were it not for the elasticity forces of

the dome, the mass would choose to follow its free-flight ballistic path. In any case, there is

a (very strong)  force  field  in  the  neighborhood  of,  and directed  toward  the surface,  that

16



ensures that the mass moves along the required path. Yet, as the following sections  will

show,  no finite, however large, elasticity coefficient of  the dome can allow for  the time

reversibility of  the mass  motion;  only  absolutely  rigid dome  can make the time  reversal

trick possible; thus the singularity in a physically meaningful parameter of the situation.

2.5 The Mass on the Dome, Modified

Many of the difficulties mentioned above can be overcome and will disappear if we make

the following minor change in the original formulation of the problem. Namely, instead of

defining the surface by  
3 22

3( ) gy r r= - , in curvilinear terms of the distance  r  traveled by

the mass along the surface, we define it, just as (12) suggests, in the usual linear Cartesian

coordinates by

3 22
3( ) | |gy x x= - .

This  way,  first,  the  situation  no  longer  harbors  the  mass-on-the-pinnacle  case  in

which the slope of the surface experiences a sudden step-like jump at the apex. The mass

moving  along  the  surface  will  now  always  arrive  at  zero  exactly  horizontally  and  no

qualifications to the contrary are necessary.

Second, the surface is now defined for  all x's, not just for all  x out of some interval

( ,  )L L- , 1L k< ; no more "preferred" "rims" or vertical walls.

Third, with any arbitrary (x-positive) point on the dome taken as the motion starting

point (the "rim"), no detachment of the mass from the surface when it starts at its "rim" ever

occurs; no need to resort to elastic wires to keep the mass on the prescribed track.

17



This  does  not,  however,  prevent  the  mass  to detach from the surface for  the third

class  of  trajectories  once  the  mass  passes  the  apex  with  non-zero  velocity as  in  Fig. 4.

Fortunately, these trajectories play no role in Norton's time reversal argument, so we shall

simply let the mass disappear from our attention once it vanishes behind the apex into the

other side.

As  the  following  sections  will  show,  the  so  modified  mass  on  the  dome  example

inherits  all  the  strangeness  of  being a source  of  spontaneous  motion  generation  with  no

need to deal with the above discussed loopholes and unnecessary qualifications.

3. The Lipschitz Condition

We recall that the function  ( )x t  satisfying the initial condition  0 0( , )t x  is a solution of the

differential equation determined by a vector field v

( )x v x=& (13)

if the following identity holds for all t in the interval I on which ( )x t  is defined:

( ( ))
dx

v x t
dt

= , and 0 0( )x t x= . (14)

Every differential equation (13) defines a direction field of this equation in the plane:

the line attached at the point  ( , )t x  has slope  ( )v x . If  0x  is a singular point of the vector

field, i.e.,  0( ) 0v x = , then  0( )x t x=  is a solution of the equation (13) satisfying the initial

condition  0 0( )x t x= .  Such  a  solution  is  called  an  equilibrium  position or  stationary

solution.

18



Let  3 4( ) sgn( ) | |v x x x= . For such a field, (13) has more than one solution, e.g., the

solutions  1( ) 0x t =  and 2
4( ) 4 ||x t t=  satisfy the same initial condition  (0, 0) . In fact, (13)

has a whole 1-parameter family of solutions obtained by gluing together the corresponding

halves of the two solutions, 1( ) 0x t =  and 
4

2 ( ) [( ) 4]x t t T= - , at some arbitrary time 0T > .

(This  situation  is  typical  in  that  if  (13)  has  more  than  one  solution,  then  it  has  a

"continuum" (i.e., a closed connected set) of solutions.) In the general theory of ordinary

differential equations it is hardly a surprising fact; if the direction field v is continuous but

nondifferentiable  (only  Hölder  continuous),  the  solution  with  initial  condition  in  the

equilibrium  position  may  fail  to  be  unique.  Indeed,  for  any  ( ) sgn( ) | |v x x x a= ,  where

0 1a< < , there always exists a family of branching solutions for (13) that satisfy the same

initial condition (0, 0)  (Fig. 7):

1 (1 )]( ) [(1 )( )x t t T aa -+= ± - - , (15)

where max( , 0)f f+ = , for an arbitrary T.

19



Fig. 7 The direction field for 3 4a = .

Geometrically,  the  reason  for  non-uniqueness  in  these  cases  is  that  the  velocity

decreases too slowly when approaching the equilibrium position. As a result the solution

manages to reach the singular point in a finite time. It turns out that the smoothness of  v

guarantees the uniqueness in these cases. This observation needs more elaboration.

Let us assume that  ( )x t  is a solution of the equation  ( )x v x=&  with a smooth right-

hand side v. We shall suppose that 0 0( )x t x=  is an equilibrium position and 1 1( )x t x=  is not

(Fig. 8).

20



Fig. 8

On  the  interval  between  0t  and  1t  consider  the  instant  2t  closest  to  1t  such  that

2( ( )) 0v x t = . By Barrow's formula for any point 3t  between 2t  and 1t  we have

3

1

3 1 ( )

x

x

d
t t

v
x
x

- = т , 3 3( )x x t= .

If  the  function  v is  smooth,  then  the  integral  tends  to  infinity as  3x  tends  to  2x .

Indeed, the slope of the chord of the graph of a smooth function on an interval is bounded

(Fig. 9), so that    2| ( ) | | |v k xx xЈ - , where the constant k is independent of the point x  of

the interval 1 2[ , ]x x .

21



Fig. 9

Thus

  

3

1

3 1
2

| | | |
( )

x

x

d
t t

k x
x

x
- і

-т .

The latter integral is easily calculated; it tends to infinity as 3x  tends to 2x . It is easy

to verify this without even calculating the integral: it must be equal to the time of transit

between the two points in the linear field, and this time tends to infinity when one of the

points tends to the equilibrium position.

Thus the number 2 1| |t t-  is larger than any preassigned number. So the solution with

initial condition in an equilibrium position cannot assume values that are not equilibrium

positions.  Therefore  if  0( )x t  is  an  equilibrium  position,  we  have  ( ( )) 0v x t є  for  all  t.

Consequently ( ) 0x t є& , i.e., ( )x t  is a constant. The uniqueness is now proved. 3

Note that the main point of the proof was the comparison of a motion in a smooth

field with a more rapid motion in a suitable linear field (i.e., parabolic trajectories). For the

3 The proof is due to Arnold (1992).

22



latter motion the time to enter an equilibrium position is infinite, and consequently it is  a

fortiori infinite for the slower motion in the original field. Indeed, it can be shown that a

sufficient condition for uniqueness of the solution with initial value  0x  is that the integral

0
( )

x

x

d
v

x
xт  diverge at 0x .

The  condition  that  2| ( ) | | |v k xx xЈ - ,  where  the  constant  k is  independent  of  the

point x  of the interval 1 2[ , ]x x  (i.e., the condition that the slope of the chord of the graph be

bounded), is called a Lipschitz condition and the constant k a Lipschitz constant.4 It can be

shown that a sufficient condition for uniqueness is that the right-hand side function v satisfy

a Lipschitz condition    | ( ) ( ) | | |v x v y k x y- Ј -  for all x and y.

It is by no accident that we chose the function 3 4| | | |v x=  to exemplify the Lipschitz

discontinuity. Writing down the energy conservation relation for a unit mass sliding on the

surface of the axially symmetric dome defined by the equation 3 2( ) | |y x x= -  gives us

2

| ( ) |
2
x

g y x=
&

,

so that

1 2 3 4| | 2 | ( ) | 2 | |x g y x g x= =& ,

4 More generally, if | ( ) ( ) | | |f f x k x bx x- Ј -  for given x and all | |xx d- < , where k, b  are independent
of x , and 0b > , and a  is the upper bound of all the b  such that a finite k exists, ( )f x  is said to satisfy a
Lipschitz condition of order a  at xx = . If a Lipschitz condition of order a  is satisfied at x, it can be shown
that ( ) 0f xў = . If at every point of the interval it is satisfied for some 1a > , then ( ) 0f xў =  throughout the
interval. Hence ( )f x  is constant; consequently only the case 0 1a< Ј  is of much interest.

23



which, to the factor of 2g , coincides with our | |v . The right-hand side of the expression

is  non-Lipschitz:  its  derivative,  1 43 2 | |
4

g
x - ,  is  unbounded  in  the  neighborhood  of  the

origin. As  a result, the uniqueness theorem does not apply. Gluing together in a smooth

manner the corresponding halves of the two solutions, 1( ) 0x t =  and 
4

2 ( ) [( ) 4]x t t T= - , at

some  arbitrary  (positive)  time  T reproduces  Norton's  anomalous  solutions.  Thus  the

existence of anomalous non-trivial solutions in Norton's case can be wholly attributed to

(spatial) Lipschitz-discontinuity of the potential well wherein the mass moves. We shall call

such solutions (spatial) Lipschitz indeterministic.

A  further  observation  clarifying the  role  of  the  Lipschitz  condition  is  in  place.  A

Lipschitz  condition  is  weaker  than  that  for  the  function  v to  be  smooth.  Indeed,  the

uniqueness  theorem  holds  in  the  case  when  the  first  derivative  of  v exists  but  is

discontinuous.  What  that  means  with  respect  to  our  situation  is  that  the  mass-on-the-

pinnacle  case  cannot  be  amended by  defining  the  otherwise  undefined  values  of  the

derivative  ( )y xў . Try, for instance, to define it as the right-hand limit of the derivative at

zero (i.e., regard the mass at apex positioned as if it is still on the right-hand side of the

surface only):

0
(0) : lim ( ) 0

x
y y x

® +
ў ў= < .

Now that the derivative exists  but is  discontinuous, the uniqueness theorem comes

into play, and only one solution survives. Which one and why? Substituting this value of

(0)yў  into (9) we see that, at the apex, there is always a non-zero tangential gravitational

force  Ft  pushing the mass downward. It is to say that, after having reached the apex and

24



having momentarily halted, the mass will necessarily turn back and fall down to where it

came from. The situation becomes no more paradoxical than throwing a stone vertically in

the air, seeing it  halt after  some finite time,  and then catching it  back again. The  trivial

solution 1( ) 0x t =  is impossible on so defined pinnacle surface exactly for the same reasons

that make it impossible to have the stone hang in the midair forever once it has reached its

maximal altitude. (A similar argument applies if one tries to define  (0)yў  as the left-hand

limit of the derivative at zero.)

Defining the derivative  by zero,  instead,  leaves  only the  trivial  solution. Once  the

mass is perched on the "flat" top of the dome, it will never move away. Any slightest shift

in  its  position  will  lead  to  a  sudden  jump  in  both  magnitude  and  directionality  of  the

tangential and normal forces acting on the mass. No non-trivial solutions of the equation

can accommodate these jumps. Causality reigns.

4. Asymptotic Reasoning and Time Irreversibility

4.1 Models and Idealizations

Scientific theories using mathematical models "approximate" or "idealize" in one way or

another. Whereas much attention in philosophy of science has been drawn to the role of

idealization  in  the  development  of  scientific  theories  (e.g.,  Cushing  (1990)),  it  is  the

applicability of the theories or models that shall be of main interest in this section. More

specifically, we shall focus on the role that certain idealizations play in Norton’s stochastic

motion generation.

In  his  recent  book  (2002)  Robert  Batterman  discusses  what  he  calls  asymptotic

reasoning in  physics, i.e.,  the qualitative  analysis of  behavior  of physical theories  in the

25



neighborhood of singular limits,  and its relevance to philosophical issues of explanation,

reduction  and  emergence.  Batterman  argues  that  many  physically  and  philosophically

important theories and models involve a new and powerful category of explanation based

on asymptotic reasoning that has been totally overlooked by philosophers of science.

Whereas much of Batterman's efforts have been directed on issues of intertheoretical

reduction and the development of the new type of explanation with the idea of emergence,

we shall be applying the (infinite) asymptotic reasoning to see whether a particular model,

namely,  Norton’s  mass-on-the-dome  example,  involves  any  physically  inadmissible

idealizations.

One of the key ideas involved in infinite asymptotic reasoning is a commonplace fact

of mathematics that finite and infinite compositions may differ in their essential properties.

(Thus, a finite intersection of open sets is always an open set, but an infinite intersection of

open sets can yield a closed set). If some property is preserved at any finite stage of a finite

sequence  of  operations,  there  is  no guarantee that  this  property will  be  preserved  in  the

transition to the infinite stage. One may argue that many paradoxes in philosophy can be

traced down to exactly this illegitimate projection – incorrectly projecting properties from

finite  to  infinite  compositions.5 This  can  be  seen  as  a  particular  application  of  a  more

general  pattern  encountered  routinely  in  such  mathematical  disciplines  as  the  system

stability  and  control  theory,  the  theory  of  parameter  sensitivity  in  dynamical  systems,

catastrophe theory, and robotics.

Unlike  the  more  traditional  approaches  in  which  to  determine  the  properties  of  a

system has been to exhibit a complete set of exact solutions of the equations describing this

system,  and  then  to  study  the  properties  of  these  solutions,  in  catastrophe  theory  it  is
5 See, e.g., Earman and Norton (1999) for applying this reasoning to argue against known supertask
paradoxes.

26



realized that in many instances it is only information of a qualitative nature, or only limited

quantitative  information,  which  is  the  ultimate  goal  of  the  study  of  some  systems  of

equations. In such cases a full spectrum of solutions to an equation, obtained by much hard

work  (if  at  all),  may be  a  hindrance  rather  than  a  help  in  understanding  the  qualitative

properties of the equation or system of equations (Gilmore 1981).

As a part of mathematics,  catastrophe theory is  a theory about singularities. Many

interesting  phenomena  in  nature  (or  their  mathematical  models)  involve  some

discontinuities – breaking of a wave, the division of a cell or the collapse of a bridge. When

applied to scientific theories, it deals with the properties of discontinuities directly, without

reference to any specific underlying mechanism. This makes it especially appropriate for

the  study of  systems  whose  inner  workings  are not  known,  or  too  complicated,  and  for

situations in which the only reliable observations are of discontinuities.

4.2 Stability and Parameter Sensitivity

In robotics, to maintain system stability has been the prime concern when designing any

practical  machine.  There  always  exists  a  certain  discrepancy  between  an  actual  (real-

operating)  and  the  nominal  (theoretical)  trajectories  of  any  system.  This  discrepancy  is

partly  due  to  various  inherently  approximational  schemes  in  system  identification,  and

partly due to possible further parameter  variations  stimulated  by environmental  changes.

Thus,  special  attention  should  be  paid  to  the  evaluation  of  possible  system  parameter

variations, and their effects on system’s functional performance or "output".6

To  illustrate  the  concepts  of  stability  and  parameter  sensitivity  let  us  consider  a

familiar classical system – a pendulum. A simple gravity pendulum – a weight on the end of

6 See, e.g., Eshlami (1991).

27



a rigid rod, which, when given an initial push, swings back and forth under the influence of

gravity over its central (lowest) point. As is known, the oscillations (not necessarily small)

of the "ideal" pendulum are described by the following system of differential equations:

1 2x x=& , 
2

2 1sinx xw= -& , l gw = , (16)

where  1x  is  the angle of  deviation  from  the  vertical,  2x  is  the angular  velocity,  l is  the

length of the pendulum, and g is the acceleration due to gravity.

The corresponding vector field in the phase plane with coordinates 1x , 2x  is just

1 2v x= , 
2

2 1sinv xw= - ,

with singular points 1x mp= , 2 0x =  (Fig. 10).

Fig. 10 Phase-space of a simple gravity pendulum

If  we  restrict  the  motion  of  the  pendulum  to  a  relatively  small  amplitude,  i.e.,

 1| | 1x << , the solution is a well-known harmonic oscillatory function:

1 0( ) cos( )x t x t w= , 2 0 ( ) (1 ) sin( )x t x tw w= - ,

where 0x  is the largest angle attained by the pendulum.

Period of the small oscillations is

28



0 2T p w= .

For  amplitudes  beyond the  small  angle  approximation,  the  exact  period  cannot  be

evaluated  in  terms  of  elementary functions  and  can  only be  written  in  the  form  of  the

elliptic function of the first kind:

0
 

4 sin
,

2 2
x

T E
p

w
ж ц= з ч
и ш

,

where ( , )E y j  is Legendre's elliptic function of the first kind:

2 2
0

1
( , )

1 sin
E d

j

y j x
y x

=
-

т .

The  value  of  the  elliptic  function  can  also  be  computed  numerically by using the

following series:

( ) ( ) ( ) ( ) ( ) ( )2 22 2 4 60  0 0 013 13 512 2 2 4 2 2 4 6 21 sin sin sin
x x x

T T Ч Ч ЧЧ Ч Ч
й щ= + + + +
к ъл ы

K .

A number of assumptions are built in this model: the bob of the pendulum is a point-

mass; the rod on which the bob is swinging is massless and absolutely rigid; motion occurs

in a vertical plane; there is no air resistance and friction at the nail, the material of which

the bob is made is irrelevant to the study of the question, acceleration due to gravity does

not  depend  on  the  position  of  the  bob;  there  is  no  gravitational  influence  of  the  nearby

objects at the mass, etc. Taken seriously, many of these idealizations are plainly unphysical

(or  physically inadmissible)  in  that  they can  never  be  achieved  in  practice  for  principle

reasons, but, of course, no one is tempted to think that this "unphysicality" is indispensable

to the relevant theory or that the theory would be absolutely unworkable without them.

A typical way how such conceptual "frauds" are dealt away in the teaching of science

can be illustrated by the following excerpt:

29



One of the fundamental concepts of mechanics is that of a [material point]. By

this  we  mean  a  body  whose  dimensions  may  be  neglected  in  describing  its

motion. The possibility of so doing depends, of course, on the conditions of the

problem  concerned.  For  example,  the  planets  may  be  regarded  as  [material

points]  in considering their motion  about the Sun, but not in considering their

motion about their axes. (Landau and Lifschitz 1976, p. 1)

Implicit in such stipulations are our intuitions about the  stability (or  robustness) of

the  behavior  of  the  system  with  respect  to  disturbances  or  changes  made  to  various

parameters  of  the  system.  That  is  this  feature  of  a  system  to  change  its  behavior

insignificantly when the various parameters of the system are changed insignificantly that

legitimizes some physical features to be idealized, or "neglected", as in the example above.

This point needs more elaboration.

We recall that an equilibrium point 0x  of a system of differential equations,

( ( ))t=&x v x , (16)

is locally Lyapunov stable at 0t t=  if all solutions of this equation which start near 0x  (i.e.,

with their initial conditions in a neighborhood of 0x ) remain near 0x  for all time, i.e., if for

any 0e >  there exists a 0( , ) 0td e >  such that

if 0 0|| ( ) ||t d- <x x  then 0|| ( ) ||t e- <x x , for all 0t tі ,

for some appropriate choice of the norm || ... || .

The equilibrium point  0x  is said to be locally asymptotically stable if  0x  is locally

stable  (in  the  sense  of  Lyapunov)  and,  furthermore,  all  solutions  starting  near  0x  tend

30



towards  0x  as  t ® Ґ . Thus, the pendulum has a locally stable equilibrium point (but not

asymptotically  stable)  when  the  pendulum  is  hanging  straight  down  and  an  unstable

equilibrium point when it is pointing straight up. (If the pendulum is damped, the stable

equilibrium point is locally asymptotically stable.)

Let us frame this situation differently in the following terms. Instead of disturbing the

initial conditions of the system, we can talk of changing (smoothly) the total system energy

taken as a parameter, and observe the qualitative changes in the behavior of the system. As

we start with small system energies (i.e., the initial conditions are around the point (0,0) in

the  phase-space  diagram),  the  trajectories  of  the  pendulum  bob  are  closed  curves;  the

pendulum performs  back-and-forth oscillations. Increasing, in a smooth manner, the total

energy of the system, we will eventually reach a point when, suddenly, the pendulum bob

ceases to track a closed curve in the phase-space; in fact, it ceases to move at all once it

takes an (unstable) vertical position with the zero velocity, and remains in this unmovable

state ever since. This position is unstable – any, however light, disturbance of the pendulum

returns it to the back-and-forth oscillatory motion.

If we  increase  the  total  energy even  more,  the  pendulum  starts  a rotating motion,

corresponding to non-closed curves in the phase-space diagram, resembling more and more

straight lines as we push the energy still up. Thus, depending on the (non-zero) value of the

chosen parameter (system's total energy), there exist three distinct behaviors of the system –

that  of  a  back-and-forth  oscillations  (closed  phase  trajectories),  that  of  a  halted  (though

unstable) position (a phase trajectory is just a single point), and that of a rotational motion

about the axe of the pendulum (non-closed phase trajectories).

Depending on a particular  problem in question,  any other parameter  of the system

may be chosen to be disturbed or manipulated. An interesting (and important to us) case is

31



when  we  choose  to  manipulate  the  rod's  elasticity  coefficient  k,  and  see  whether  the

behavior of the system will be, in some appropriate sense, robust under these perturbations.

Suppose, for example, that we are given a system of differential equations describing

the behavior of a gravitational pendulum, in which the elasticity of the rod is not assumed

infinite  but  appears  explicitly.  Suppose  further  that,  in  this  system  of  equations,  the

elasticity coefficient  k of  the  rod  is  replaced  by a  new  coefficient  k k kdў = + ,  x(t)  is  a

solution of the original equation, and ( )tўx  is a solution of the equation with the changed kў

. For x(t) to be a robust solution, we could require that for any appropriate disturbances kd

there exists a 0( , ) 0td e >  such that

 || ( ) ( ) ||t t dў - <x x , for all 0t tі .

This  is  the  latter  sense  of  robustness  that  is  essential  for  the  possibility  of  the

"unphysical" models' being used in simulating the behavior of physical systems. Thus, by a

pendulum  with  infinitely  rigid  rod we  could  now  understand  a  series of  (physically

legitimate) approximations to the original problem, with finite but arbitrarily large and ever

increasing  elasticity  coefficients,  k's,  as  long  as  (1)  the  series  converges  (in  some

appropriate sense) to some limiting solution (e.g., if the modulus of any two consecutive

solutions  can be made arbitrarily small by further increasing the corresponding elasticity

coefficients), and (2) this limiting solution has the same essential properties that any finitely

approximate solution has.

4.3 The Mass on the Dome and Time Irreversibility

What does it all have to do with the mass-on-the-dome example? As we have seen earlier,

the generation of  non-uniqueness  in  this  situation  is  due to  the violation of  the (spatial)

32



Lipschitz condition. The violation occurs at one point of the dome only – the apex. If we

grant  infinite  divisibility  of  matter  and  neglect  the  issues  with  the  vanishing  Lebesgue

measures of the initial conditions, a more philosophically interesting issue arises, namely,

that of time irreversibility of the solutions.

Note that, even though the Lipschitz condition is violated at a single point, the apex,

this violation depends on the behavior of the dome at the infinitesimal vicinity of the apex;

it  is  crucial  that  the  dome  shape's  first  derivative  experiences  unbounded  growth  in  the

infinitesimal  proximity  of  the  apex.  An  arbitrarily  small  vicinity  of  the  apex  has  been

chosen, the particular shape of the path that the mass moves along outside of this vicinity is

absolutely immaterial to the spontaneous motion at the apex to begin; no points of the dome

outside of this area in any way affect the spontaneous generation of motion at the apex.

Now, we  know  that  in a (real)  body that  is  not  deformed,  the  arrangement  of  the

molecules  correspond  to  a  state  of  thermal  equilibrium;  all  parts  of  the  body  are  in

mechanical  equilibrium.  This  means that,  if some  portion  of  the body is  considered, the

resultant of the forces on that portion is zero. When a deformation occurs, the arrangement

of the molecules is changed, and the body ceases to be in its original state of equilibrium.

Forces therefore arise which tend to return the body to equilibrium. These internal forces

which occur when a body is deformed are called internal stresses. If no deformation occurs,

there are no internal stresses.

The  internal  stresses  in  a  real  physical  body are  due  to  molecular  forces, i.e.,  the

forces  of  (electro-magnetic)  interaction between  the  molecules.  An  important  fact  in the

theory of  elasticity is  that  the  molecular forces  have  a very short  range  of  action.  Their

effect extends only to the neighborhood of the molecule exerting them, over a distance of

the same order as that between the molecules, whereas in the theory of elasticity, which is a

33



macroscopic  theory,  the  only  distances  considered  are  those  large  compared  with  the

distances  between  the  molecules.  The  range  of  action  of  the  molecular  forces  should

therefore be taken as zero in the theory of elasticity. We can say that the forces which cause

the internal stresses are, as regards the theory of elasticity, "near-action" forces, which act

from any point only to neighboring points. Hence it follows that the forces exerted on any

part  of  the  body by surrounding  parts  act  only on  the  surface  of  that  part  (Landau  and

Lifshitz, 1959). Another consequence of such treatment of infinitely divisible matter is that

no  internal  stresses  in  the  theory of  elasticity can  be  non-smooth  or  have  divergent  or

singular derivatives.

Coming  back  to  the  mass-on-the-dome  example,  the  trajectories  of  the  first  class

(where a mass is given an initial push up and it stops exactly at the apex) can now be shown

robust under the ever increasing rigidity of the dome in the following sense.

Take any arbitrarily large elasticity coefficient  1k  of the dome. Place a mass on the

"rim" (any fixed point on the dome other than the apex), give it a push up so that the mass

halts exactly at the apex. Write down the formal exact solution of such a motion (non-trivial

at all, given that we are not to neglect any dome deformation effects). Call this solution, as

a function of time, 1( )tx . Double (triple, or whatever) the elasticity coefficient 1k  to get a

new 2k . Get a similar solution 2 ( )tx . Proceed in the similar manner to get an infinite series

of solutions: 1( )tx , 2 ( )tx , ..., ( )i tx , .... It can be shown that, in some appropriate sense, this

sequence converges to some limiting solution, ( )tx , which we would call a solution for an

absolutely rigid dome (the elasticity coefficient is infinite).

34



Notice now what happens if we try to time-reverse the situation. Fix some finite 1k ,

place a (point) mass on the top of the dome, and … Will the mass move down? Given that

the  mass  is  a point  mass  (to  hit  exactly the  apex) and  the  internal stresses  in  the  (non-

absolutely rigid) dome are smooth, there exists no way the surface of the dome hits the apex

in a non-smooth manner. No Lipschitz-discontinuity – no non-uniqueness – the mass will

stay on top forever! Try doubling, tripling, etc., the elasticity of the dome – you get the

same picture. As long as the dome elasticity stays finite (and no matter how big), there is no

spontaneous motion generation when the mass is on the top!

The  situation  changes  essentially  if  we  take  an  absolutely  rigid dome  with  no

deformations at all; only then, given the nature of the internal forces that we have, could the

Lipschitz condition be violated and the mass slide down. Thus, a  limiting solution of this

family of solutions exhibits a property (of being non-unique), whereas any solution in an

approaching series (corresponding to a physically plausible situation) fails to exhibit this

property. The only way to generate non-Lipschitz spontaneous motion of the mass down the

dome is to have an absolutely rigid dome; any diversion from actual infinity in the elasticity

coefficient would result in the mass staying on top forever.

There is an important lesson to be learned from this example to be very careful in

using time-reversibility tricks. As Norton's mass on the dome shows, these are the cases the

real  metaphysical power  of  which may essentially depend  on  the  nature of  idealizations

done  in  those  models,  and  the  techniques  of  asymptotic  reasoning may prove  crucial  in

elucidating these issues.

5. Non-Lipschitz Indeterministic Solutions and the Markov Condition

35



5.1 Generalized Flows in Fluid Dynamics

In this section we draw the results from fluid dynamics to further support the view that the

Lipschitz condition should be taken is an important implicit assumption within Newtonian

mechanics and that the failure to recognize it as such may unsurprisingly lead to physically

impossible situations.

Consider  the  following  transport  equation  for  the  scalar  field  ( , )x tq in

( , ) [0, )dx t О ґ ҐR :

( ( , ) ) 0v x t
t
q

q
¶

+ ЧС =
¶

,   0|t oq q= = . (17)

In the classical theory of partial differential equations it is known that if the velocity

field  dv ОR  is bounded and continuous in  ( , )x t  and Lipschitz continuous in  x, then (17)

can be solved uniquely by the method of characteristics. Denote by , ( )s t xj  the solution of

( , )
dx

v x t
dt

= , (18)

starting at x at time s, i.e., with the initial condition  ( )x s x= . The solution of (17) is then

given by the following expression:

1
0 0, 0 ,0( , ) ( ( )) ( ( ))t tx t x xq q j q j

-= = . (19)

The map , :
d d

s tj aR R  satisfying the following four properties:

(a) , ( )s s x xj =  for all s;

(b) , ( )s t xj  is continuous in s, t, x;

(c) , , ,( ( )) ( )t s s tx xt tj j j=  for all s, t, τ and all x;

(d) , ( ) :
d d

s t xj aR R  is a homeomorphism for all s, t

36



is called a flow of homeomorphism or, in short, a flow.7

These classical results are inapplicable when  v, though bounded and continuous in

( , )x t , fails to be Lipschitz continuous in x. In such cases no standard flow satisfying (a)–(d)

can be associated with the ODE in (18) since the solution of this equation may fail to be

unique. Since the solutions of (18) typically branch (i.e., (18) have more than one solution

for  the  same  initial  condition),  no  forward-in-time  map  can  be  associated  with  such

solutions. Similarly, no backward-in-time map can be associated with the solutions of (18)

because they may coalesce on each other in finite time. This situation is unfortunate since

transport in non-Lipschitz velocity fields may be physically motivated, e.g., for the problem

of turbulence.8

Formally, the standard way to deal with this situation in general case is to randomize

the  set  of  maps  which  can  be  associated  with  the  solutions  in  (18)  by selection  at  the

branching points thus defining a random field. Then a generalized flow associated with (18)

can  be  defined  as  a  random  field  with  parameter  (s,  t,  x)  constructed  by  assigning  a

probability  measure  on  the  set  of  all  maps  associated  with  the  solutions  of  (18).9 The

generalized flows obtained in this way are typically non-degenerate random fields, i.e., the

measure is not concentrated on a single point, due to branching.

As far as the modeling of the underlying physical processes is concerned, to pick the

probability  measure  and  single  out  physically  relevant generalized  flows  the  following

regularization  procedure is used. Instead of the original problems described by (17) and

(18), the  regularized  problems with unique  solutions  are considered and the generalized

7 Exposition is due to W. E and Vanden-Eijnden (2003).
8 The classical theory of Kolmogorov (1941) predicts that the solution of the Navier-Stokes equation in three
dimensions in only Hölder with exponent 1 3  in the limit of zero viscosity.
9 For a more rigorous definition of generalized flows and their properties see Appendix A, W. E and Vanden-
Eijnden (2003).

37



flows  are  obtained  as  limits  of  the  standard  (stochastic)  flows  associated  with  these

regularized problems.

Consider first the  regularization  by  smoothing of  the velocity around the  points  of

Lipschitz discontinuity (the  ε-limit process). Here the original equation (17) is understood

as the limiting equation for the following motivating (and physically legitimate) problem:

( ( , ) )v x t k
t

eq q q
¶

+ ЧС = D
¶

,   0|t oq q= = , (20)

where k is the molecular diffusivity and ve  is a mollified version of v on the scales | |x e=

(e.g., if v solves Navier-Stokes equation, ε is the characteristic length scale associated with

the kinematic viscosity). Unlike the original transport equation, (20) has a unique solution if

either  k or  ε  are positive. The generalized flow is then taken as the limit as  0e ®  of the

stochastic flow associated with (20), provided this limit can be defined in a suitable way.

Secondly, some Brownian motion (the k-limit process) can added to the dynamics in

(18) to obtain a unique (stochastic) flow associated with the solutions of

( , ) 2 ( )dx v x t dt k d te b= + , (21)

where  ( )b Ч  is a  d-dimensional  Wiener process (Stroock and Varadhan, 1969, 1979). The

fact  that  the  term  2 ( )k d tb  regularizes  (21)  can  be  understood  more  intuitively  as

thermodynamical fluctuations (always present in any real physical system unless we freeze

it  to  the  absolute  zero),  with  probability  1,  kicking  instantaneously  out  any  path  that

happens to go to the points x for which the solution of (18) is non-unique, thereby resolving

the ambiguity associated with these positions. The generalized flow can now be defined,

38



similarly, as the limit as  0k ®  of the stochastic flow associated with (20), provided this

limit is defined in a suitable way. 10

Mixed limits where both smoothing of the velocity field and Brownian motion are

used can be considered as well.

The limiting generalized flows obtained in this manner, however, appear to depend

sensitively  on  the  regularization  procedure,  and  they  are  non-Markov  for  generic

regularizations. The latter fact raises an interesting issue regarding the connection of the

Lipschitz indeterministic solutions with the Markov condition.

5.2 Non-Lipschitz Velocity Fields, Regularizations, and the Markov Condition

Consider  first  the  following  ODE  we  met  in  Section  4  (with  3 4a =  corresponding to

Norton's original formulation):

sgn( ) | |
dx
dt

x x a= , ,x ОR  (0,1)a О . (22)

As mentioned before, the set of solutions of this equation is given by

1 (1 )]( ) [(1 )( )x t t T aa -+= ± - - , (23)

where max( , 0)f f+ = , for an arbitrary T.

By resolving the ambiguity of where to map the point x = 0, the following family of

forward maps , :
d d

s tj aR R  can be constructed:

1 (1 )

1 (1 )

1

,

sgn( )(| | (1 )( )) ,  if 0
( )

( )((1 )( ) ) ,             if 0
s t

x x t s x
x

f t x

a

a

a
w

w

a
j

t a t

-

-

-

+

м + - - №п
= н

- - =по
, (24)

10 Typically, the ε-limit is a weaker limit than the k-limit in the sense that the regularization by smoothing is
more subtle due to the lack of stability of solutions to perturbations and issues with the choice of appropriate
convergence. See W. E and Vanden-Eijnden (2003) for more details.

39



where  inf( : ( ) 0)t s f swt = і № . Each of the maps  ,s t
wj  is a weak form of a flow, a  quasi-

flow, satisfying only the following three properties:

(a') , ( )s s x x
wj =  for all [0, ]s TО ;

(b') , ( )s t x
wj  is continuous in s, t, x;

(c') , , ,( ( )) ( )t s s tx x
w w w
t tj j j=  for all x and for all s, τ, [0, ]t TО  with s ttЈ Ј ;

By superimposing these quasi-flows and assigning a suitable probability measure a

generalized flow can be defined. 

In this particular case, the generalized flows, obtained as limits of the standard flows

by regularization either via the k-limit process or via the ε-limit process, can be shown to be

Markov in time. However, as in the next example, Markovianity is not a generic property of

generalized flows.

Consider a further generalization of the previous example:

sgn( ) | | ( )
dx
dt

x x g ta= , ,x ОR  (0,1)a О , (25)

where g is a bounded function. Some solutions of (25) branch at the origin x = 0 on the time

intervals where ( ) 0g t > , and other collapse at x = 0 where ( ) 0g t <  (see Fig. 11).

40



Fig. 11 The set of all solutions of (25) for 3 4a =  and ( ) cos( )g t t= .

Quasi-flows and the corresponding generalized flow have been properly defined, it

can be shown that the generalized flows associated with (25) are not necessary Markovian

in time. Though the generalized flows obtained by regularization by the k-limit process are

Markov, the generalized flows obtained via the ε-limit process are not, so that this feature

appears to depend sensitively on the regularization procedure used.

Finally, consider  a further  relaxation of  the  condition  on the  velocity field.  In the

following ODE the velocity field is continuous in (x, t) but non-Lipschitz at (x, t) = (0, 0)

and not even Hölder continuous (x, t) = (0, 0):

( , )
dx

v x t
dt

= , ( , )x v О ґR R , (26)

where ( , )v x t  is given by

2

2

2
,              if | | ,

( , )
2 sgn( ) ,   if | | .

x
x t

v x t t
x t x t

м >п
= н

п Ч Јо
(27)

The set of solutions of (27) can be parameterized as

41



2( ) sgn( )x t a t a= + , 2( )x t bt= (28)

with a > 0, [ 1,1]b О -  (see Fig. 12):

Fig. 11 The set of all solutions of (26).

In this  case,  the  generalized  flows  associated  with  (26)  can  be  shown  to  be  non-

Markov in time for both the  k-limit process flows and most of the  ε-limit process flows.

The only regularization procedures that do produce a Markov generalized flows in the  ε-

limit process are those for which, in the limit 0e ® , all the paths that collapse on a single

node (0,0) exit on a single trajectory.

To summarize the main points of this section, the purpose of drawing these results

from fluid dynamics is to illustrate how properties specific to generalized flows associated

with first order differential equations with spatially non-Lipschitz right-hand side may lead

to  interesting  and  non-trivial  features  in  terms  of  transport  by  such  fields.  Namely,

generalized flows, constructed as limits of regularized standard flows, typically appear to

lack desirable properties such as stability with respect to perturbations or Markovianity in

42



time  so  that  the  failure  to  recognize  the  Lipschitz  condition  as  an  important  implicit

assumption within Newtonian mechanics may unsurprisingly lead to physically impossible

solutions  that  have  no  serious  metaphysical  import,  as,  for  instance,  in  Norton's  causal

skeptical anti-fundamentalist program.

Acknowledgments

I thank Steven Savitt of the Philosophy Department at the University of British Columbia

for  drawing  my  attention  to  the  Lipschitz  condition,  and  Alexei  Cheviakov  of  the

Mathematics Department at the University of British Columbia for useful discussions.

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