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Owen Lynch (Nov 11 2021 at 20:29):It just hit me while I was working on the stat mech of the Carnot engine that force is the derivative of momentum in a different way than I normally think about it. I.e., I normally think about it in terms of force producing a change in momentum for a specific object. But another way of thinking about it is that it measures the flow of momentum. This became clear to me when I derived the pressure of an ideal gas by calculating the momentum flow into its boundary from collisions by particles.
Two gases that are at equilibrium because they have matched pressures have a balanced equilibrium of momentum flowing in to each other.
I feel like there is an essentially "open" quality to analysis of flow like this that is absent from the "closed" perspective of Hamiltonian mechanics, and I would like to understand how to bring this kind of thing into the more general setting of Hamiltonian mechanics.
Owen Lynch (Nov 12 2021 at 18:31):Similarly, two gases that are in equilibrium of temperature have a balanced equilibrium of kinetic energy flowing in to each other.
Owen Lynch (Nov 12 2021 at 18:32):This implies that temperature is the rate at which heat energy flows out of the boundary
John Baez (Nov 14 2021 at 00:24):I feel like there is an essentially "open" quality to analysis of flow like this that is absent from the "closed" perspective of Hamiltonian mechanics, and I would like to understand how to bring this kind of thing into the more general setting of Hamiltonian mechanics.
You might like my work on open Hamitonian systems. But so far it's just getting started - there's a lot of stuff I know should be done, that we didn't get around to! For example we worked out the open Euler-Lagrange equations but didn't write about them, and there should also be an open version of Hamilton's equations.