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# This example simulates a `swinging Atwood's machine'. An Atwood's
# machine consists of two masses joined by a taut length of cord. The cord
# is suspended from a pulley. The heavier mass (M) would normally win
# against the lighter mass (m), and pull it upward. A `swinging' Atwood's
# machine is an Atwood's machine with an additional degree of freedom: it
# allows the lighter mass to swing back and forth in a plane, at the same
# time as it is being drawn upward.
# Let `a' denote the angle by which the cord extending to the lighter mass
# deviates from the vertical. Let `l' denote the distance along the cord
# between the pulley and the lighter mass. Then the system of differential
# equations below will describe the evolution of the system.
# You may run this example, with output to an X window in real time, by doing
#
# ode < atwoods.ode | graph -T X -x 9 11 -y -1 1 -m 0 -S 1
#
# The plot will trace out `l' and `ldot' (its time derivative). The `-m 0
# -S 1' option requests that successive datapoints not be joined by line
# segments, but rather that marker symbol #1 (a point) be plotted at the
# location of each datapoint.
# You may have some difficulty believing the results of this simulation.
# Allowing the lighter mass to swing, it turns out, may prevent the heavier
# mass from winning against it. The system may oscillate,
# non-periodically.
m = 1 # lighter mass
M = 1.0625 # heavier mass
a = 0.5 # initial angle of cord from vertical, in radians
adot = 0
l = 10 # initial distance along cord from pulley to mass m
ldot = 0
g = 9.8 # acceleration due to gravity
ldot' = ( m * l * adot * adot - M * g + m * g * cos(a) ) / (m + M)
l' = ldot
adot' = (-1/l) * (g * sin(a) + 2 * adot * ldot)
a' = adot
print l, ldot
step 0, 400
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