Indeterminismin systems with infinitely and finitely many

degrees of freedom

John D. NortonDepartment of History and Philosophy of Science

University of Pittsburgh

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Indeterminism is generic amongsystems with

infinitely many degrees of freedom.

2Source: Appendix to Norton, “Approximation and Idealization…”

Enforced by embedding in still larger solution.

Enforced by embedding in

larger solution.

Solution that manifests pathological behavior for a few components, e.g. spontaneous excitation

The mechanism that generates pathologies

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system of infinitely many coupled components

…and so on indefinitely.

Expected solution

dxn(0)/dt = xn(0) = 0 for all n

xn(t) = 0 for all n, all t

with initial conditions

Masses and Springs

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Motions governed by

d2xn/dt2 = (xn+1 – xn) - (xn – xn-1)

Unexpected solution with

same initial conditions

x1(t) = x2(t) = (1/t) exp (-1/t) Non-analytic functions needed to ensure initial conditions preserved.

Masses and Springs

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Motions governed by

d2xn/dt2 = (xn+1 – xn) - (xn – xn-1)

Solve for remaining variables iteratively

x3 = d2x2/dt2 + 2x2 - x1

dx3/dt = d3x2/dt3 + 2dx2/dt - dx1/dt

x4 = d2x3/dt2 + 2x3 – x2

dx4/dt = d3x3/dt3 + 2dx3/dt – dx2/dt

etc.

Indeterminism is exceptional amongsystems with

finitely many degrees of freedom.

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The Arrangement

The mass experiences an outward directed force fieldF = (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.

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.

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Possible motions: None

r(t) = 0solves Newton’s equation of motion

since

d2r/dt2 = d2(0)/dt2 = 0 = r1/2.

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Possible motions: Spontaneous Acceleration

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

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≥Tsolves Newton’s equation of motion

d2r/dt2 = r1/2.

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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

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Without CalculusImagine the mass

projected from the edge.

Close…

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Without CalculusImagine the mass

projected from the edge.

Closer…

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Without Calculus

BUT there is a loophole. Spontaneous motion fails for a hemispherical dome. How can the thought experiment fail in that case?

Now consider the time reversal of this process.

Spontaneous motion!

Imagine the mass projected from the edge.

BINGO!

What should we think of this?

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