Post on 07-Feb-2016
description
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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