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  • Hamiltonian mechanics for discrete systems?

    From Julio Di Egidio@julio@diegidio.name to sci.physics on Sat Jul 18 15:34:29 2026
    From Newsgroup: sci.physics

    Can we use Hamiltonian mechanics for discrete systems?

    By discrete system I mean a system with a finite number
    of states and where time itself is discrete (stroboscopic).

    E.g. what is the Hamiltonian of a coin's flipping where
    the law of motion is H->H, T->T (i.e. the coin doesn't ever flip)?

    E.g. what is the Hamiltonian of a coin's flipping where
    the law of motion is H->T, T->H (i.e. the coin flips at every step)?

    I am confused since, on a side, generator functions in the Hamiltonian/Poisson-bracket formulation are generators of
    infinitesimal variation, which seems to imply/necessitate
    continuity (a local/global principle).

    OTOH, isn't a coin's flipping a classical system that can be
    described in terms of a (discrete) phase space and flows in
    phase space?

    TIA for any clarification/explanation.

    Julio

    P.S. I do have tried a web search first, but couldn't find
    anything to the point.

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  • From Julio Di Egidio@julio@diegidio.name to sci.physics on Sun Jul 19 12:50:58 2026
    From Newsgroup: sci.physics

    On 18/07/2026 15:34, Julio Di Egidio wrote:
    Can we use Hamiltonian mechanics for discrete systems?

    By discrete system I mean a system with a finite number
    of states and where time itself is discrete (stroboscopic).

    E.g. what is the Hamiltonian of a coin's flipping where
    the law of motion is H->H, T->T (i.e. the coin doesn't ever flip)?

    E.g. what is the Hamiltonian of a coin's flipping where
    the law of motion is H->T, T->H (i.e. the coin flips at every step)?

    I am confused since, on a side, generator functions in the Hamiltonian/Poisson-bracket formulation are generators of
    infinitesimal variation, which seems to imply/necessitate
    continuity (a local/global principle).

    OTOH, isn't a coin's flipping a classical system that can be
    described in terms of a (discrete) phase space and flows in
    phase space?

    TIA for any clarification/explanation.

    I have asked the question to ChatGTP, here is the answer: <https://chatgpt.com/s/t_6a5bbc8d91b88191bfa37010bde2c7cf>

    Besides the more or less obvious things, it is interesting
    that ChatGTP finds a way to do it with Quantum Mechanics,
    which I think is the answer I was looking for, though I am
    not there yet.

    Have fun,

    Julio
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