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Unformatted text preview: ail for each spring, but in fact the springs are completely
symmetric, and the choice can be reversed without any eﬀect. (Hopefully you will discover why it
is convenient to use the incidence matrix A to specify the topology of the system.)
The total energy is the sum of the gravitational energies, over all the masses, plus the sum of the
elastic energies, over all springs. The equilibrium positions of the masses is the point that minimizes
the total energy, subject to the constraints that the ﬁrst p positions are ﬁxed. (In the equilibrium
positions, the total force on each mass is zero.) We let Emin denote the total energy of the system,
in its equilibrium position. (We assume the energy is bounded below; this occurs if and only if each
mass is connected, through some set of springs with positive stiﬀness, to a ﬁxed mass.)
The total energy Emin is a measure of the stiﬀness of the structure, with larger Emin corresponding
to stiﬀer. (We can think of Emin = −∞ as an inﬁnitely...
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 Fall '13
 F.Borrelli
 The Aeneid

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