The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. The real period is of course the time it takes the pendulum to go through one full cycle. Paul Appell pointed out a physical interpretation of the imaginary period:  if θ 0 is the maximum angle of one pendulum and 180° − θ 0 is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other. This interpretation, involving dual forces in opposite directions, might be further clarified and generalized to other classical problems in mechanics with dual solutions. 
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Prior to the early few centuries of the first millennia, planetary measurement was always referenced to the fixed stars (sidereal) as was the astrology of the ancients (Babylonia, etc). Astronomy and astrology were always one and the same until the Greco-Roman church-state astronomers began to record planetary ephemerides referenced to the moving vernal point rather than to the fixed stars, but ignoring that the vernal point moved with precession, and while maintaining that all other heavenly bodies rotated around the Earth—a statement certainly affirming true egoic consciousness. (This was due in particular to Claudius Ptolemaeus (c. 90 - c. 168 CE) who was employed by the Greco-Roman controlling faction. Ptolemaeus was a mathematician, not a true astronomer. Some assert he never even looked into the heavens as an astronomer would look (Allen).) To say anything other than Earth was the center was heresy, so only those willing to sell out to the manipulative ploys of the church-state, and perpetuate the asserted geocentric dogma of western Christendom, stayed on the pay-role and kept their heads (actually, heretics were burned at the stake!). Here was planted the bastardization of true soul level astronomical astrology, and here began the Hellenistic, Ptolemaic, tropical astrology during an age of suppression, and which is still in vogue today. This is not to say either way that Ptolemy was ill-intended or was deliberately trying to mislead—a topic of controversy amongst historians. However, it was merely the nature of the times and the natural course of evolution, a time of trying to re-emerge from the lost wisdom of the ancients amalgamated with manipulative powers attempting to control empires, land and people. People from the future may say the same for us now.