Indeterminism
in systems with infinitely and finitely many
degrees of freedom
John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
1
Indeterminism is generic among
systems with
infinitely many
degrees of freedom.
Source: Appendix to Norton, “Approximation and Idealization…”
2
The mechanism that generates pathologies
system of infinitely many
coupled components
0
1
2
3
Solution that manifests
pathological behavior for a
few components, e.g.
spontaneous excitation
Enforced by
embedding in
larger solution.
4
5
Enforced by
embedding in still
larger solution.
…and so on
indefinitely.
3
Masses and Springs
Motions
governed by
Expected
solution
d2xn/dt2 = (xn+1 – xn) - (xn – xn-1)
xn(t) = 0
for all n, all t
with initial conditions
dxn(0)/dt = xn(0) = 0
for all n
4
Masses and Springs
Motions
governed by
Unexpected
solution with
same initial
conditions
d2xn/dt2 = (xn+1 – xn) - (xn – xn-1)
x1(t) = x2(t) = (1/t) exp (-1/t)
Solve for remaining variables iteratively
x3 = d2x2/dt2 + 2x2 - x1
dx3/dt = d3x2/dt3 + 2dx2/dt - dx1/dt
Non-analytic
functions
needed to
ensure initial
conditions
preserved.
x4 = d2x3/dt2 + 2x3 – x2
dx4/dt = d3x3/dt3 + 2dx3/dt – dx2/dt
etc.
5
Indeterminism is exceptional among
systems with
finitely many
degrees of freedom.
6
The Arrangement
A unit mass sits at the apex of a
dome over which it can slide
frictionless. The dome is
symmetrical about the origin
r=0 of radial coordinates
inscribed on its surface. Its
shape is given by the
(negative) height function h(r)
= (2/3g)r3/2.
The mass experiences an outward directed force field
F = (d/dr) potential energy = (d/dr) gh = r1/2.
The motion of the mass is governed by Newton’s “F=ma”:
d2r/dt2 = r1/2.
7
Possible motions: None
r(t) = 0
solves Newton’s equation of motion
since
d2r/dt2 = d2(0)/dt2 = 0 = r1/2.
8
Possible motions: Spontaneous Acceleration
The mass remains at
rest until some
arbitrary time T,
whereupon it
accelerates in some
arbitrary direction.
r(t) = 0, for t≤T and
r(t) = (1/144)(t–T)4, for t≥T
solves Newton’s equation of motion
d2r/dt2 = r1/2.
For t≤T, d2r/dt2 = d2(0)/dt2 = 0 = r1/2.
For t≥T
d2r/dt2 = (d2 /dt2) (1/144)(t–T)4
= 4 x 3 x (1/144) (t–T)2
= (1/12) (t–T)2
= [(1/144)(t–T)4]1/2 = r 1/2
9
The computation again
For t≤T, d2r/dt2 = d2(0)/dt2 = 0 = r1/2.
For t≥T
d2r/dt2 = (d2 /dt2) (1/144)(t–T)4
= 4 x 3 x (1/144) (t–T)2
= (1/12) (t–T)2
= [(1/144)(t–T)4]1/2 = r 1/2
10
Without Calculus
Imagine the
mass
projected
from the
edge.
Close…
11
Without Calculus
Imagine the
mass
projected
from the
edge.
Closer…
12
Without Calculus
Now consider the
time reversal of
this process.
Spontaneous
motion!
Imagine the
mass
projected
from the
edge.
BINGO!
BUT there is a loophole.
Spontaneous motion fails for a
hemispherical dome. How can the
thought experiment fail in that case?
13
What should we think of this?
14
15
16