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VII. Exercise. Coupled Pendulums. For two coupled pendulums of equal mass m and equal length l the equations of motion read $mddot{x} = -alpha x - k(x - y)$, $mddot{y} = -alpha y - k(y - x)$, with $alpha = mg/l$ (g is the gravitational constant, k is the spring constant). Here the coordinate systems are chosen in such a way that x = y = 0 corresponds to the equilibrium point, and it is assumed that the pendulums hang vertically at rest. These are linearized equations, which are valid for small oscillations. Give a fundamental system of solutions (either by transforming to a system of first order and cal- culating the characteristic polynomial or by making two physically suggestive ans

          VII. Exercise. Coupled Pendulums. For two coupled pendulums of equal
mass m and equal length l the equations of motion read
$mddot{x} = -alpha x - k(x - y)$,
$mddot{y} = -alpha y - k(y - x)$,
with $alpha = mg/l$
(g is the gravitational constant, k is the spring constant). Here the coordinate
systems are chosen in such a way that x = y = 0 corresponds to the equilibrium
point, and it is assumed that the pendulums hang vertically at rest. These are
linearized equations, which are valid for small oscillations. Give a fundamental
system of solutions (either by transforming to a system of first order and cal-
culating the characteristic polynomial or by making two physically suggestive
ans

VII. Exercise. Coupled Pendulums. For two coupled pendulums of equal
mass m and equal length l the equations of motion read
mddotx = -alpha x - k(x - y),
mddoty = -alpha y - k(y - x),
with alpha = mg/l
(g is the gravitational constant, k is the spring constant). Here the coordinate
systems are chosen in such a way that x = y = 0 corresponds to the equilibrium
point, and it is assumed that the pendulums hang vertically at rest. These are
linearized equations, which are valid for small oscillations. Give a fundamental
system of solutions (either by transforming to a system of first order and cal-
culating the characteristic polynomial or by making two physically suggestive
ans

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