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Galileo’s Simple Pendulum

T he simple pendulum is first reviewed from an historical point of
view, exposing the most elegant mathematics adopted in the last
few centuries to rationalize all known cases of pendulum motion,
which, during that time, could only be resolved by the so called elliptic
functions. After this brief history, the author retracts from any further
recourse to the elliptic functions intends to solve the equations of
motion of the simple pendulum in another way. In particular, it is shown
that the simple pendulum can be solved exactly using only high-school
algebra and known transcendental functions, the consequence of which
has led to demonstrating integrability of elliptic integrals – analytically.
Finally, a discussion of the actual back and forth motion in the
amplitude of the pendulum’s swing is viewed as representing two
different directions that appear to make a difference to the equations of
motion of the pendulum, and as such will be postponed for analysis
toward the end of the chapter.

3.1 Introduction

§. A Brief History of the Simple Pendulum. Galileo Galilei’s
(1564-1642) fascination with motion and dynamics of bodies, as
recollected by Seeger in Galilei and Seeger (1966) [1], which is based on
dialogues taken from Galileo’s scientific testament, the Two New
Sciences, stands out as one of his most outstanding achievements;
mainly, because Galileo had challenged the Aristotolean beliefs upon
which the foundations of physics, mathematics, astronomy and
philosophy had been built on at that time. Galileo’s perceptions in
science not only exceeded many of the Aristotolean ideas of that time
but his inventions are known also to have definitely dawned a new
technological era in Rome in the 17th century – such that this most