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What relation does the phase of a wave have with the phase space? Namely, how are they related historically and/or physically?

P.S. if it helps, I came across this question while thinking about the phase-space formulation of QM and the pilot-wave theory.

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    $\begingroup$ There's no relationship. Two different uses of the same word. $\endgroup$
    – garyp
    Feb 20, 2018 at 14:41
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    $\begingroup$ Phase space in mechanics (hence QM) was probably introduced by Boltzmann in 1872: for periodic motions, the classical state is specified by energy and phase--think of the classical oscillator elliptic trajectories. Waves are periodic phenomena, so the wave phase quantifies position in the cycle, like the phases of the moon.... In phase-space QM, the two usage are aggressively disjoint, though. $\endgroup$
    – Cosmas Zachos
    Feb 20, 2018 at 15:19

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There is no relation between the phase of a wave and the so-called phase space of a mechanical system which consists of the space of all possible generalized coordinate and conjugate generalized momentum variables.

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